Full-field derivations: 45 complete records

the axiom (V01-PB-0000)

LAYER: 0 FORCED BY: NOTHING — this is the input

The axiom

04 · AXIOM — the only input

(unity+unity)(e₀(∂₀Ψ) − e₁(∂₁Ψ) − e₂(∂₂Ψ) − e₃(∂₃Ψ) − e₄(∂₄Ψ) − e₅(∂₅Ψ) − e₆(∂₆Ψ))B
    = Ψ e₀ (Ψ̃ − Ψ*)

over seven generators, stepwise arithmetic from finitely many symbols.

The equation is asserted. Everything below fixes what the symbols are; what may be done with them is not fixed here. Nothing beyond what is written here is assumed.

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Two scalar primitives.

void    nothing
unity   one

Coefficients are scalar expressions, built step by step from unity and declared symbols by addition, negation, multiplication, and inversion (a·a⁻¹ = unity) under a declared nonvoid guard, together with finitely many declared identities, nonvoid conditions, and derivative rules.

Eight rules, asserted of each written instance and of no other:

1.  x + (y + z) = (x + y) + z
2.  x + void = x
3.  for each written x there is a written −x, with x + (−x) = void
4.  x(yz) = (xy)z
5.  x(y + z) = xy + xz
6.  (x + y)z = xz + yz
7.  unity·x = x
8.  cx = xc                     for a scalar expression c

x − y abbreviates x + (−y).

One declared condition. unity + unity is nonvoid.

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Indices. μ, ν range over 0,…,6. i ranges over 1,…,6.

Seven generators.

e₀e₀ = unity
eᵢeᵢ = −unity
e_μe_ν + e_νe_μ = void              for μ ≠ ν

Blades. For I ⊆ {0,…,6}, e_I is the product of the generators indexed by I, written in increasing order; e_∅ = unity, the empty product.

The order gives each index set exactly one blade. Without it, e₁e₂ and −e₂e₁ would be two writings of one blade, and no statement about distinct blades would name anything.

Written elements. A written element is a finite sum Σ_I a_I e_I over the blades, with scalar-expression coefficients. Ψ is a written element. The sum, the product, the negation, the reversion, the star, and the ∂_μ of written elements are written elements again. Inversion is written only for scalar expressions under a declared nonvoid guard.

B = e₁e₂. Every rule in this statement is written uniformly in 1,…,6 — no spatial index occurs in them except as a free letter — so a uniform relabelling of 1,…,6 applied to the whole statement, this declaration included, carries any written derivation to a written derivation of the corresponding statement. It is not an internal relabelling of this statement unless the declared ORDERED pair is unchanged — e₂e₁ is not e₁e₂, so exchanging 1 and 2 does not fix the declaration. The equation distinguishes 1 and 2 from the other four spatial indices, and a conclusion turning on that distinction is content.

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Reversion and star. Both fix every written scalar expression.

reversion   (x+y)~ = x̃ + ỹ      (xy)~ = ỹ x̃      ẽ_μ = e_μ
star        (x+y)* = x* + y*     (xy)* = x* y*     e₀* = −e₀ ,  eᵢ* = eᵢ

Seven derivatives ∂₀,…,∂₆. The ∂_μ of a scalar expression is again a scalar expression.

∂_μ(X + Y) = ∂_μX + ∂_μY
∂_μ(XY)    = (∂_μX)Y + X(∂_μY)
∂_μ(e_ν)   = void
∂_μ(∂_νX)  = ∂_ν(∂_μX)     for each written X and each pair (μ,ν)

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End of the axiom. The text above is the whole of it.

What it means

One equation relates the written derivatives of Ψ to a product built from Ψ, the distinguished generator e₀, and the difference between reversion and star. The surrounding declarations fix the generators, their signs, the ordered blade B, the written arithmetic, and the operations appearing in that equation. They are part of the asserted input, not consequences already proved by this node.

The ordered choice matters: a uniform relabelling of 1,…,6, applied to the whole statement with the declaration of B included, carries any written derivation to a written derivation of the corresponding statement. It is an internal relabelling of this statement only when the declared ordered pair is unchanged.

This node supplies the starting statement. The programme asks what predictions this input forces, that is all.

The arithmetic of written elements: cancellation, the left forms, and the sign scalar (V01-PB-0001)

LAYER: 1 FORCED BY: THE AXIOM

Declares

NONE. Every symbol used here — void, unity, −x — is written by 04_AXIOM.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, for written elements x, y, z, and with −x as the axiom declares it:

1.  (−x) + x = void                          the inverse is two-sided
2.  void + x = x                             the identity is two-sided
3.  x + y = x + z  implies  y = z            cancellation on the left
    y + x = z + x  implies  y = z            cancellation on the right
4.  x + y = void   implies  y = −x           the inverse is unique
5.  x + y = y + x                            addition commutes
6.  void·x = void   and   x·void = void
7.  (−unity)x = −x  and  x(−unity) = −x
8.  (−x)y = −(xy)   and   x(−y) = −(xy)
9.  (−unity)(−unity) = unity

10. x·unity = x
11. −(−x) = x and −void = void
12. −(x+y) = (−x)+(−y)
13. x−y = void iff x=y x+y = x iff y=void

The forcing

Rules cited by their numbers in 04_AXIOM: 1 is additive associativity, 2 is x + void = x, 3 is the written inverse with x + (−x) = void, 5 and 6 are the two distributivities, 7 is unity·x = x, 8 is cx = xc for a scalar expression c.

1 — the inverse is two-sided. Write y for −x, so x + y = void by rule 3. Write z for −y, so y + z = void by rule 3. Then:

y + x = (y + x) + void          rule 2
      = (y + x) + (y + z)       rule 3, for y
      = ((y + x) + y) + z       rule 1
      = (y + (x + y)) + z       rule 1
      = (y + void) + z          rule 3, for x
      = y + z                   rule 2
      = void                    rule 3, for y

2 — the identity is two-sided.

void + x = (x + (−x)) + x       rule 3
         = x + ((−x) + x)       rule 1
         = x + void             result 1
         = x                    rule 2

3 — cancellation. Suppose x + y = x + z. Then:

y = void + y                    result 2
  = ((−x) + x) + y              result 1
  = (−x) + (x + y)              rule 1
  = (−x) + (x + z)              the supposition
  = ((−x) + x) + z              rule 1
  = void + z                    result 1
  = z                           result 2

Suppose instead y + x = z + x. Then:

y = y + void                    rule 2
  = y + (x + (−x))              rule 3
  = (y + x) + (−x)              rule 1
  = (z + x) + (−x)              the supposition
  = z + (x + (−x))              rule 1
  = z + void                    rule 3
  = z                           rule 2

4 — the inverse is unique. Suppose x + y = void. By rule 3, x + (−x) = void also, so x + y = x + (−x), and result 3 gives y = −x.

5 — addition commutes. Expand (unity + unity)(x + y) two ways.

(unity + unity)(x + y) = (unity + unity)x + (unity + unity)y       rule 5
                       = (unity·x + unity·x) + (unity·y + unity·y) rule 6
                       = (x + x) + (y + y)                         rule 7

(unity + unity)(x + y) = unity(x + y) + unity(x + y)               rule 6
                       = (x + y) + (x + y)                         rule 7

So (x + y) + (x + y) = (x + x) + (y + y). Regrouping both sides by rule 1:

x + (y + (x + y)) = x + (x + (y + y))

Cancel x on the left by result 3, then regroup both sides by rule 1 again:

(y + x) + y = (x + y) + y

Cancel y on the right by result 3: y + x = x + y.

6 — void annihilates. By rule 2 with x written as void, void + void = void. Then:

void·x = (void + void)x         the line above
       = void·x + void·x        rule 6

And void·x = void·x + void by rule 2. So void·x + void·x = void·x + void, and result 3 cancels void·x on the left, giving void·x = void. The same argument with rule 5 in place of rule 6 gives x·void = void.

7 — the sign scalar. By rule 3, unity + (−unity) = void. Then:

void = void·x                   result 6
     = (unity + (−unity))x      the line above
     = unity·x + (−unity)x      rule 6
     = x + (−unity)x            rule 7

So x + (−unity)x = void, and result 4 gives (−unity)x = −x.

For the other factor, first x·unity = x: rule 7 gives unity·x = x, and rule 8 with the scalar expression unity gives unity·x = x·unity. Then the mirrored argument runs:

void = x·void                   result 6
     = x(unity + (−unity))      rule 3
     = x·unity + x(−unity)      rule 5
     = x + x(−unity)            x·unity = x, above

So x + x(−unity) = void, and result 4 gives x(−unity) = −x.

8 — a negation in either factor. By rule 3, x + (−x) = void. Then:

xy + (−x)y = (x + (−x))y        rule 6
           = void·y             the line above
           = void               result 6

So xy + (−x)y = void, and result 4 gives (−x)y = −(xy). The same argument with rule 5 gives x(−y) = −(xy).

9 — the square of the sign scalar. By result 7 with x written as −unity:

(−unity)(−unity) = −(−unity)

By result 1, (−unity) + unity = void, and result 4 gives unity = −(−unity). Therefore (−unity)(−unity) = unity.

10 — right unity. Rule 8 with the scalar unity gives x·unity=unity·x; rule 7 gives unity·x=x. Thus x·unity=x. This is the identity used in step 7.

11 — double negation and the negative of void. Result 1 gives (−x)+x=void; rule 3 applied to −x gives (−x)+(−(−x))=void. Result 4 identifies the two additive inverses of −x, so −(−x)=x. With x written as void, rule 2 gives void+void=void; result 4 therefore gives −void=void.

12 — the negative of a sum. Additive associativity, result 5, rule 3 at x and y, and rule 2 give

(x+y)+((−x)+(−y))
  = (x+(−x))+(y+(−y))
  = void+void
  = void.

Result 4 applied to x+y yields −(x+y)=(−x)+(−y).

13 — the two elementary equivalences. If x=y, substitution and rule 3 give x−y=y+(−y)=void. Conversely, if x−y=void, append y to that equality:

(x+(−y))+y = void+y,
x+((−y)+y) = y,
x+void = y,
x = y.

These steps use additive associativity, result 1, result 2 and rule 2. If x+y=x, compare it with x+void=x and use result 3 to obtain y=void. Conversely y=void gives x+y=x+void=x. Each implication retains its displayed hypothesis.

What is not forced

The claims are arithmetic identities of written elements under rules 1, 2, 3, 5, 6, 7 and 8, with the equality reading stated above. Their proofs require no generator relation, use of the equation, or nonvoid guard. Unity is already the empty blade. Multiplicative associativity is not needed for this arithmetic interface.

Scalar commutation is exactly rule 8's scope. The results about addition and signs neither extend multiplication to general commutation nor supply an inversion operation. Multiplicative cancellation requires an additional argument about the specified factor.

The item numbers locate the statements and proofs; they are not scalar premises.

What it means

The written rules yield two-sided additive identity and inverse, additive cancellation, commutative addition, void annihilation, and the arithmetic of a sign factor. These permissions are earned before any generator relation or the equation is used. Unity is present as the empty blade; no nonempty blade is needed.

This is the arithmetic that later records may use without silently adding the familiar laws they need. Its scope stays exact: multiplication has not been shown to commute in general, and no general inversion operation has been licensed.

Every blade is undone by the same generators written backwards, up to a sign (V01-PB-0002)

LAYER: 2 FORCED BY: V01-PB-0001

Declares

• σ_I := e_I ẽ_I — the blade–reversion product. It is initially a written element; the claim establishes that it is one of the two scalar signs.
• b_I := σ_I ẽ_I — the written blade-undoing expression.
• ε_m := unity for m=0 and ε_m := −unity for m in 1,…,6; these abbreviate the supplied generator-square values.
• I denotes a supplied subset of the seven generator labels. Local W,X,Y are the actual word prefixes and suffixes in the displayed reductions; an absent word is represented by unity. Local c denotes the scalar generator square reached at the step where it is used.

Reversion and the increasing-order blade e_I are the axiom's operations and objects. The reversed word is identified with ẽ_I in the proof; it receives no second permanent name.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, for each blade e_I:

0.  ẽ_I is its generators written in decreasing order; ẽ_∅=unity.

1.  e_I ẽ_I  is unity or −unity.            This element is σ_I.
2.  ẽ_I e_I  = σ_I.                          The same scalar on either side.
3.  σ_I σ_I  = unity, and therefore
    e_I b_I = b_I e_I = unity.           Undoing from either side.

σ_∅ = unity.

The forcing

Rules cited by their numbers in 04_AXIOM: 4 is multiplicative associativity, 7 is unity·x = x, 8 is cx = xc for a scalar expression c. The generator squares are the axiom's Seven generators: e₀e₀ = unity, and eᵢeᵢ = −unity for i in 1,…,6.

Reversion gives the reversed word. It fixes the empty word unity because unity is scalar. It fixes each individual generator. If W is the actual prefix of the increasing listing and e_μ its next generator, the supplied rule gives

(W e_μ)̃ = ẽ_μ W̃ = e_μ W̃.

At the empty prefix this starts with unity; at each actual appended label it places that generator before the already reversed prefix. Following the specified finite listing of I reaches exactly the decreasing word. This proves claim 0, including its empty case, without using an involution theorem or anticommutation.

0 — three small facts, each used more than once.

(a) x·unity = x. This is PB-0001(10).

(b) a scalar factor may be carried to the front of a product. For a scalar expression c and written elements X, Y:

X(cY) = X(Yc)      rule 8
      = (XY)c      rule 4
      = c(XY)      rule 8

With X absent the factor is already at the front; with Y absent, Xc = cX by rule 8.

(c) a product of two written elements each unity or −unity is unity or −unity. Four cases, and they are all the cases, since each factor is one of two written elements:

unity·unity      = unity        rule 7
unity·(−unity)   = −unity       rule 7
(−unity)·unity   = −unity       V01-PB-0001(7), with x written as unity
(−unity)(−unity) = unity        V01-PB-0001(9)

1 — what is being reduced. e_I ẽ_I is the increasing listing of the generators indexed by I followed by the decreasing listing of the same generators. ẽ_I e_I is the decreasing listing followed by the increasing listing. Each of the two reads the same in either direction, and in each the two innermost factors are one generator written twice — the largest index of I in the first, the smallest in the second. This is a description of the written product, fixed the moment the product is written; nothing is being iterated.

2 — the index this is checked over. The blades are a derived finite index: the axiom's Blades paragraph supplies them by taking each generator in turn and writing every index set already listed both with it and without it, seven times over. So the claim above is a finite conjunction, one conjunct per written blade, and what discharges it is finite exhaustive verification over a derived index. The reduction below is written once with I free; for each written blade it is a written finite sequence of steps, one for each member of I, and no step differs in kind between blades.

3 — the reduction of e_I ẽ_I. Take a blade e_I with I nonempty, and let μ be the largest index in I. Then e_μ e_μ are the two innermost factors of e_I ẽ_I, and by rule 4 they may be grouped alone:

e_I ẽ_I = X (e_μ e_μ) Y

where X is the increasing listing of I without μ, Y is the decreasing listing of I without μ, and either or both may be absent (both are, when I has one member). By the axiom's generator squares, e_μ e_μ is unity if μ is 0 and −unity otherwise; write c for it, a scalar expression. By 0(b):

e_I ẽ_I = c (X Y)

and X Y is the increasing listing of I without μ followed by the decreasing listing of I without μ — a product of the same shape over the index set I without μ.

The step just written is available at each of the index sets so reached, one step for each member of I, and after the last of them no generator remains: what is left is the empty product, unity. The steps are as many as the members of I, they are written down in advance of performing them, and each is an equality to what preceded it — so the last expression equals the first. Collecting the scalar produced at each step to the front by 0(b) and regrouping by rule 4, the accumulated scalar and the next factor c are each unity or −unity, so by 0(c) the accumulated scalar is unity or −unity after every step, including the last. The last expression is that scalar times unity, which is that scalar by 0(a).

Therefore e_I ẽ_I is unity or −unity. It is σ_I. For I = ∅, e_∅ ẽ_∅ = unity·unity = unity by rule 7.

4 — the reduction of ẽ_I e_I. ẽ_I e_I is the decreasing listing followed by the increasing listing; its two innermost factors are e_ν e_ν with ν the smallest index in I, and removing them leaves the decreasing listing of I without ν followed by the increasing listing of I without ν — again a product of the same shape. Every step of 3 applies with "largest" read as "smallest" and nothing else changed: the grouping is rule 4, the square is the axiom's, the carry is 0(b), the accumulation is 0(c). So ẽ_I e_I is unity or −unity.

5 — the two products have the same sign. By rule 4,

ẽ_I (e_I ẽ_I) = (ẽ_I e_I) ẽ_I.

The left side is ẽ_I σ_I, equal to σ_I ẽ_I by rule 8. Multiply this established equality on the right by e_I and regroup:

σ_I (ẽ_I e_I) = (ẽ_I e_I)(ẽ_I e_I) = unity.

The last equality uses the two possibilities for ẽ_I e_I established in step 4 and the four sign products in 0(c). Therefore

σ_I = σ_I·unity
    = σ_I((ẽ_I e_I)(ẽ_I e_I))
    = (σ_I(ẽ_I e_I))(ẽ_I e_I)
    = unity·(ẽ_I e_I)
    = ẽ_I e_I.

The steps use right unity, the just-established sign square, rule 4, the preceding equality, and rule 7. This proves claim 2 without cancelling an unproved factor.

6 — the element that multiplies e_I to unity. By 0(c), σ_I σ_I = unity. Then

e_I (σ_I ẽ_I) = e_I (ẽ_I σ_I)   rule 8
              = (e_I ẽ_I) σ_I   rule 4
              = σ_I σ_I         result 1
              = unity

(σ_I ẽ_I) e_I = σ_I (ẽ_I e_I)   rule 4
              = σ_I σ_I         result 2
              = unity

What is not forced

The scope is each supplied increasing-order blade, its reversion, the sign of their product, and the displayed two-sided undoing b_I. The input's scalar inversion notation is not extended to nonscalars: b_I is the specified product σ_I ẽ_I, with its two undoing equations proved.

The proof uses arithmetic, generator squares and the supplied reversion rules. No distinct-generator exchange, coefficient recovery, or nonvoid guard is needed. It gives σ_∅=unity and a finite written reduction for each σ_I; it does not give a separate numerical index formula. General products of two blades are handled in PB-0005, where the required exchanges and repeated-generator removals are written.

A consumer can use the undoing equations on their specified blade factor. A sum of blades is a different operand and needs its own argument. The result here neither classifies such operands nor grants their cancellation.

What it means

Reversion provides exactly the backward word needed to undo a blade. The blade–reversion product supplies a sign; attaching that sign to the reversion gives b_I, which multiplies the blade to unity on either side.

The operation and the word are now explicitly connected at their introduction. Later cancellation uses these written products, rather than treating a blade as though the scalar-inversion permission automatically applied to it.

The normal form: any product of distinct generators is the increasing-order blade, up to a sign the reordering produces (V01-PB-0003)

LAYER: 2 FORCED BY: V01-PB-0001

Declares

• For a supplied finite ordered sublist U of the seven generator labels, ρ(U) starts at unity and is multiplied by −unity once at each listed member, in that order. In particular ρ(∅)=unity. This names an actual finite sign accumulation, not a scalar power of a count.
• s — a scalar of a reordering: a factor, equal to unity or to −unity, relating a product of distinct generators to the same generators written in increasing order. That it is the same factor whichever reordering is taken is not forced here.
• n_word — the last-position label of the supplied nonempty generator listing, used to write e_{μ₁}···e_{μ_{n_word}}. It is not a scalar coefficient. The empty listing is handled separately. No arithmetic conclusion is inferred from this label.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, let μ₁, …, μ_{n_word} be indices from 0, …, 6, no two the same, and let I be the set of them.

1.  e_μe_ν = −(e_νe_μ)                    for μ ≠ ν

2.  X(e_μe_ν)Y = −(X(e_νe_μ)Y)            for μ ≠ ν, and X, Y written elements or absent:
                                          an exchange of two adjacent distinct generators
                                          anywhere in a product reverses the sign of the
                                          whole product and changes nothing else

3.  e_{μ₁}e_{μ₂}···e_{μ_{n_word}} = s · e_I       where e_I is the product written in increasing
                                          order and s is unity or −unity

For the empty listing, the input word is unity, I=∅, and s=unity; result 3 then reads unity=unity·e_∅=unity·unity.

The forcing

1 — the exchange. The axiom declares, under Seven generators, that e_μe_ν + e_νe_μ = void for μ ≠ ν. By V01-PB-0001(4), the inverse is unique, so e_νe_μ = −(e_μe_ν). Applying V01-PB-0001(11), the double-negation identity, gives the exchange in either direction:

e_μe_ν = −(e_νe_μ)        for μ ≠ ν

2 — an exchange inside a longer product. Let X and Y be written elements, possibly absent. By rule 4 the product X e_μ e_ν Y may be grouped so that e_μe_ν stands alone, and by result 1 it may be replaced:

X(e_μe_ν)Y = X(−(e_νe_μ))Y = −(X(e_νe_μ)Y)

the last step by V01-PB-0001(8), a negation in either factor, used once on each side. So an exchange of two adjacent distinct generators anywhere in a product reverses the sign of the whole product and changes nothing else.

3 — reaching the increasing order. For the empty listing the input word and its increasing-order blade are both unity. Take s=unity: rule 7 gives unity=unity·unity, so result 3 holds without an exchange. For a nonempty listing, the statement is about a written product, and every rule of the axiom is asserted of each written instance. Take that product of n_word distinct generators, the indices written in whatever order they stand.

Its generators can be listed in increasing order — they are n_word written symbols with distinct indices — and that listing fixes, for this product, which generator must end up in each position. The exchanges that carry the written order to the listed one can therefore be written out for this product: for the generator that belongs first, one exchange with each generator standing to its left; then the same for the generator that belongs second; and so on through the listing. Each exchange is licensed by result 2 and reverses the sign of the whole.

Nothing here is repeated until a condition holds, so nothing needs to be proved about when it stops. The exchanges are as many as the listing dictates, written down in advance of performing them, and each step is an equality to what preceded it — so the last expression equals the first.

Each exchange contributes (−unity) as a factor by result 2; by V01-PB-0001(7) and (9), a product of such factors is unity or −unity. Write s for it.

What is not forced

The normal-form claim covers the supplied word of distinct generators and its empty case. Results 1 and 2 separately expose the contextual exchanges needed by later records. A consumer uses these statements at their stated scopes; the forcing supplies their complete justification.

The prescribed exchanges produce a sign relating the original word to the increasing-order blade. Agreement of signs from two routes is supplied by PB-0005. That record also handles general products of two blades, where shared indices produce repetitions requiring the generator-square rules. Neither result is a premise of the present construction.

A position label identifies the next written exchange; no scalar equality is inferred from a number of positions. The original order remains part of the operand even when different words reach the same blade index.

What it means

The seven generators supply an alphabet. For a written product in which no generator occurs twice, increasing index order supplies its canonical spelling. The derived exchange rule carries the original word to that ordered blade, with a scalar sign produced by the exchanges.

The construction reaches that form by a route written for the original word. This record does not give a formula for the sign from the indices, and it does not prove that other exchange routes agree. Agreement for the same original word is established separately in PB-0005. Products with repeated generators and general products of blades remain outside the result here.

The complete change of the axiom's source and residual determines exactly which replacements preserve them (V01-PB-0006)

LAYER: 2 FORCED BY: V01-PB-0001

Declares

All declarations below abbreviate written expressions. They add no equation satisfied by the distinguished Ψ and select no physical field or preferred decomposition.

• u := unity, v := void, and h := unity+unity.
• X, Y, H, and A are free written elements; U, V, Z_add, and U' are local free written elements in the elementary equality calculations. Where a proposed replacement X → Y is considered, H := Y−X. No condition on H is supplied merely by introducing this abbreviation.
• Sel(X) := X̃−X*, the selector expression. Sel names only this displayed operator throughout the record.
• D(X) := e₀(∂₀X)−e₁(∂₁X)−e₂(∂₂X)−e₃(∂₃X)−e₄(∂₄X)−e₅(∂₅X)−e₆(∂₆X).
• L(X) := h(D(X))B, F(X) := X e₀ Sel(X), and R(X) := L(X)−F(X). R is a written residual, not an additional axiom.
• Q(X,H) := X e₀ Sel(H) + H e₀ Sel(X) + H e₀ Sel(H), with the summands in the displayed order.
• k := h⁻¹ is the scalar inverse expression permitted by the input's explicit nonvoid guard on h; hk=u. It is used only for the derivative-contraction equivalence in part 5. No inverse of a general multivector is written.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, and with a declared nonvoid condition excluding equality with void, for every supplied pair of written elements X,H the derived full-expression identities below hold. The input separately supplies ∂_μ(X+H)=∂_μX+∂_μH for each of the seven labels; that rule is used, not derived anew.

Sel(X+H) = Sel(X)+Sel(H),
D(X+H) = D(X)+D(H),
L(X+H) = L(X)+L(H),
F(X+H) = F(X)+Q(X,H),
R(X+H) = R(X)+L(H)−Q(X,H).                       (1)

No term of Q is discarded in (1). In particular its last summand is present for an unrestricted H. The asserted axiom for Ψ is equivalent to R(Ψ)=v.

For a proposed replacement Y with H=Y−X, the exact conditions at the displayed consumers are

Sel(Y)=Sel(X)   iff Sel(H)=v;
∂_μY=∂_μX       iff ∂_μH=v;
F(Y)=F(X)       iff Q(X,H)=v;
R(Y)=R(X)       iff L(H)=Q(X,H).                 (2)

Under the additional, explicitly retained selector-null hypothesis Sel(H)=v, these become

F(Y) = F(X)+H e₀ Sel(X),
R(Y) = R(X)+h(D(H))B−H e₀ Sel(X),                 (3)

and therefore

F(Y)=F(X)       iff H e₀ Sel(X)=v;
R(Y)=R(X)       iff h(D(H))B=H e₀ Sel(X).          (4)

Under the same selector-null hypothesis, preserving the two sides separately means precisely

L(Y)=L(X) and F(Y)=F(X)
    iff D(H)=v and H e₀ Sel(X)=v.                 (5)

The written products used to undo the declared B satisfy

BB=−u,    B(−B)=u.                               (B0)

These identities use the generator rules and arithmetic, not an extension of scalar inversion to B.

If R(X)=v is also established or retained as a hypothesis, then R(Y)=v holds exactly when the residual-preservation condition in (2), or in (4) under the selector-null hypothesis, holds.

For an actually established identity Ψ=A+H, the asserted equation for Ψ gives the unrestricted condition

R(A) = Q(A,H)−L(H),
R(A)=v         iff L(H)=Q(A,H).                 (2Ψ)

Under the additional hypothesis Sel(H)=v, its right-hand equality is h(D(H))B=H e₀ Sel(A).

For the concrete written pair X=e₀, H=e₁, the selector is preserved while the source changes by h e₁, which is nonvoid. Their seven derivative outputs vanish separately. These statements do not identify either expression with Ψ.

The forcing

1. The elementary identities used

V01-PB-0001(1)–(9) supplies the two-sided additive identities and inverses, both additive cancellation rules, inverse uniqueness, commutation of addition, void absorption and sign/product arithmetic. Its exported statements (10)–(13) supply right unity, double negation, negation of a sum, and the equivalences

U−V=v iff U=V,
U+Z_add=U iff Z_add=v.                              (6)

These statements retain their parent's equality scope; no proof-only lemma is being added to that interface. In particular, with U=L(Ψ) and V=F(Ψ), the first equivalence establishes that the asserted L(Ψ)=F(Ψ) is exactly R(Ψ)=v.

2. The additive interfaces

By the two involution addition rules, negation of a sum, and additive commutation,

Sel(X+H)
  = (X̃+H̃)−(X*+H*)
  = (X̃+H̃)+((-X*)+(-H*))
  = (X̃+(-X*))+(H̃+(-H*))
  = Sel(X)+Sel(H).                                  (7)

There is no appeal to component independence or to a selector eigenvalue classification.

Derivative additivity is supplied for each of the seven labels. Multiplying each instance by its corresponding generator gives, by rule 5,

e_μ(∂_μ(X+H)) = e_μ(∂_μX)+e_μ(∂_μH).

For a subtracted component, the displayed negation-of-a-sum identity supplies its two negative summands. Thus the entire fixed expression is

D(X+H)
  = [e₀∂₀X+e₀∂₀H]
    +[-e₁∂₁X-e₁∂₁H]+[-e₂∂₂X-e₂∂₂H]
    +[-e₃∂₃X-e₃∂₃H]+[-e₄∂₄X-e₄∂₄H]
    +[-e₅∂₅X-e₅∂₅H]+[-e₆∂₆X-e₆∂₆H]
  = [e₀∂₀X-e₁∂₁X-e₂∂₂X-e₃∂₃X-e₄∂₄X-e₅∂₅X-e₆∂₆X]
    +[e₀∂₀H-e₁∂₁H-e₂∂₂H-e₃∂₃H-e₄∂₄H-e₅∂₅H-e₆∂₆H]
  = D(X)+D(H).                                  (8)

The regrouping is a fixed finite use of the proved additive laws; no product factor has been exchanged. Rules 5 and 6 then give

L(X+H)=h(D(X)+D(H))B
      =(hD(X)+hD(H))B
      =hD(X)B+hD(H)B=L(X)+L(H).                 (9)

3. Expand the complete nonlinear source

Use (7), then distribute without reordering any product:

F(X+H)
  = (X+H)e₀(Sel(X)+Sel(H))
  = (Xe₀+He₀)(Sel(X)+Sel(H))
  = [Xe₀Sel(X)+He₀Sel(X)] + [Xe₀Sel(H)+He₀Sel(H)]
  = F(X)+[Xe₀Sel(H)+He₀Sel(X)+He₀Sel(H)]
  = F(X)+Q(X,H).                                (10)

The penultimate line only reorders additive summands using PB-0001. In particular He₀Sel(X) has not become Sel(X)e₀H or Xe₀H.

From (9), (10) and the additive identities,

R(X+H)
  = [L(X)+L(H)]−[F(X)+Q(X,H)]
  = [L(X)−F(X)]+[L(H)−Q(X,H)]
  = R(X)+L(H)−Q(X,H).                           (11)

The same expression can also be grouped as

R(X+H) = R(X)+[L(H)−H e₀ Sel(H)]
         −[X e₀ Sel(H)+H e₀ Sel(X)]
       = R(X)+R(H)−[X e₀ Sel(H)+H e₀ Sel(X)].

The bracket contains the two mixed products. Exchanging X and H only exchanges its additive summands, so this mixed expression is symmetric. The unrestricted term H e₀ Sel(H) is retained inside R(H), not discarded.

This is a finite exact change identity. H is arbitrary in its written scope, not small. No expansion parameter, truncation, limiting process, fitted form or numerical trial occurs.

4. Derive the preservation conditions in both directions

For H:=Y−X, association, additive commutation, rule 3 and rule 2 give

X+H = X+(Y+(-X)) = (X+(-X))+Y = v+Y = Y.

Under the stated equality premise, substitute the established X+H=Y inside the operator arguments in (7)–(11): Sel, each ∂_μ, D, L, F and R retain the same supplied operations and every actual hypothesis.

For any displayed equality U'=U+Z_add, if U'=U, compare U+Z_add=U+v and cancel U to obtain Z_add=v. Conversely Z_add=v gives U'=U+v=U. Apply this written calculation to the selector and each derivative, to (10), and to (11). For the residual, use (6) to replace L(H)−Q(X,H)=v by L(H)=Q(X,H). This proves all of (2), not just sufficient conditions.

Now retain Sel(H)=v. Absorption gives

Xe₀Sel(H)=Xe₀v=v,
He₀Sel(H)=He₀v=v,
Q(X,H)=v+He₀Sel(X)+v=He₀Sel(X).

Substituting this proved equality into (10)–(11) yields (3). Applying the same cancellation calculation yields both equivalences in (4).

If R(X)=v, equation (11) instead reads R(Y)=L(H)−Q(X,H). By (6), R(Y)=v exactly when L(H)=Q(X,H). The asserted axiom concerns Ψ; this conditional step does not assert its equation for every written X.

5. Preserve the two sides separately

From (9), L(Y)=L(X) exactly when h(D(H))B=v.

The generator square and distinct-generator relation supply the required undoing factor for B. Inverse uniqueness gives e₂e₁=−(e₁e₂); then rule 4 and PB-0001's product sign law give

BB=(e₁e₂)(e₁e₂)
  =e₁(e₂e₁)e₂
  =e₁(-(e₁e₂))e₂
  =-((e₁e₁)(e₂e₂))
  =-((-u)(-u))=-u.

Hence B(-B)=-(BB)=u. This proves (B0). These are products of written elements, not an application of scalar-inverse notation to B.

The scalar k was written under the input's existing guard on h. Rule 8 gives kh=hk=u. Therefore

h(D(H))B=v
  => (h(D(H))B)(-B)=v(-B)=v
  => h(D(H))=v
  => k(h(D(H)))=kv=v
  => (kh)D(H)=v
  => D(H)=v.

Conversely, D(H)=v and absorption give h(D(H))B=v. This establishes the exact derivative-contraction equivalence. Combine it with the source equivalence in (4) to obtain (5).

No claim that every derivative component of H vanishes follows from D(H)=v. If a consumer needs the individual derivative outputs, the seven conditions in (2), one for each supplied label, give their exact preservation test.

6. Apply the result to the actual axiom without choosing a new field

Given an actually written retained expression A, put H:=Ψ−A. Then the identity calculation in part 4 gives Ψ=A+H. Any other established identity with this same form can be substituted in the same way. No selector-null hypothesis is used yet.

Equation (11) and the asserted axiom R(Ψ)=v give

v = R(A)+L(H)−Q(A,H).                            (12)

Append Q(A,H) to (12) and regroup; then append −L(H) and regroup. The two-sided additive identities give

R(A)=Q(A,H)−L(H).

Necessity: if R(A)=v, substitution in (12) gives L(H)−Q(A,H)=v; apply (6) to obtain L(H)=Q(A,H). Sufficiency: if L(H)=Q(A,H), substitution in (12) gives v=R(A)+v=R(A). Thus, without any selector restriction,

R(A)=v  iff  h(D(H))B
                = A e₀ Sel(H)+H e₀ Sel(A)+H e₀ Sel(H).    (13)

This proves (2Ψ). Now, if the written derivation also establishes Sel(H)=v, part 4 gives Q(A,H)=H e₀ Sel(A). Substitution in (12)–(13) yields

v = R(A)+h(D(H))B−H e₀ Sel(A),
R(A)=v  iff  h(D(H))B=H e₀ Sel(A).               (14)

No A or H has been selected by trial. These identities tell a consumer what it owes for whatever actual decomposition it has established. Omitting H from the selector is licensed by Sel(H)=v. Omitting H from the equation requires the further displayed balance. Omitting H from each side separately requires the two conditions in (5).

7. A written instance that distinguishes the interfaces

From the generator images and the additive identities,

Sel(e₀)=e₀-(-e₀)=e₀+e₀=he₀,
Sel(e₁)=e₁-e₁=v,
Sel(e₀+e₁)=he₀+v=he₀.

The full sources, with association but no factor interchange, are

F(e₀)=e₀e₀(he₀)=u(he₀)=he₀,
F(e₀+e₁)=(e₀+e₁)e₀(he₀)
         =h(e₀+e₁)(e₀e₀)
         =h(e₀+e₁)=he₀+he₁.

The middle line moves only the scalar h. Their difference is he₁.

This difference is nonvoid. If he₁=v, multiply on the right by -e₁:

h(e₁(-e₁))=v(-e₁)=v,
e₁(-e₁)=-(e₁e₁)=-(-u)=u,
hence h=hu=v,

contradicting the input's declared nonvoid condition. Thus the same selector output coexists with different full sources. All seven derivative outputs of both expressions are v by the generator derivative and addition rules. Their L expressions are consequently v. This evaluates already supplied expressions; neither is identified with Ψ.

What is not forced

The result applies to the supplied expressions and operations under the equality and nonvoid reading stated with the claim. A,H remain free expressions in a generic decomposition. A later construction may instantiate them, but is not presupposed here. The actual-field equivalence additionally uses the asserted R(Ψ)=v; the generic identities do not supply that premise for another X.

Each preservation condition concerns its named output. Source preservation in (2) concerns the complete Q. Residual preservation in (4) is the displayed balance between two changes; (5) gives the different requirement for preserving each side individually. Individual derivative preservation uses the seven separate conditions in (2), while the contracted expression has its own condition. No splitting of a collected sum into individually vanishing contributions has been proved.

Part 5 uses the existing guard on h to write k, and proves the products that undo B. The nonvoid source example also uses the declared condition on h. Those uses remain obligations of the displayed arguments. Other multiplicative factors require their own cancellation grounds.

Equality of residuals is a comparison of the residual expressions. Under the retained base-solution premise it also characterizes preservation of a true equation. No classification of all fields or of all possible later consumers follows from these specified interfaces.

What it means

A simplification is judged at the place that uses its result. The selector, the source, and the complete equation ask different questions of the same written object. Selector-nullness licenses one substitution in the selector factor; it does not erase the object's other appearances.

The full change identity keeps those appearances together. It shows exactly when the derivative-side change and the source-side change preserve the residual by balancing, and exactly when each side is separately unchanged. The work no longer needs a verbal judgement that a term “does not matter”: it can show the equality that the actual consumer needs.

Written coefficient recovery supplies finite collection and equality comparison (V01-PB-0004)

LAYER: 3 FORCED BY: V01-PB-0001, V01-PB-0002, V01-PB-0003

Declares

All expressions below use the input's operations. They add no scalar identity, field restriction, or nonvoid condition.

• u := unity, v := void, h := u+u, and k := h⁻¹, written under the input's declared nonvoid guard on h. Thus hk=u, and scalar commutation gives kh=u.

• G is the supplied ordered index list 0,1,2,3,4,5,6. 𝓑 is the list of its index subsets constructed in §1. I,J,K denote entries of that list, not scalar numbers. All sums below abbreviate actual finite, bracketed sums in that list's order. At a place with no summands the displayed accumulator is explicitly v.

• X,Y are supplied written elements. A presentation X=Σ_{I∈S} a_I e_I has a supplied sublist S of 𝓑 with no repeated index and supplied scalar coefficients a_I. When writing the same presentation on 𝓑, put a_I=v at an omitted index. Section 1 proves that this extension leaves X unchanged.

• ε_m retains PB-0002's abbreviation for the supplied generator-square value, so e_m e_m=ε_m.

• A sign is one of the two written scalar expressions u and −u. ρ(U) retains PB-0003's specified finite accumulation of −u on the ordered sublist U, with ρ(∅)=u.

• λ_m(I):=ρ(I∖{m}). θ(I):=u if 0 is absent from I, and θ(I):=−u if 0 is present.

• C_m(X):=ε_m e_m X e_m is the written conjugation expression. For a target J, define the normalized expressions

  𝒫_{J,m}(X) := k(X+λ_m(J) C_m(X)),
  𝒫_{J,*}(X) := k(X+θ(J) X*).
  

  The label * on the latter expression identifies its use of star. It is not a further involution or a numerical index.

• 𝓔_J(X) is the result of the following eight displayed applications, in this order:

  Z_in := X;
  Z_0 := 𝒫_{J,0}(Z_in);  Z_1 := 𝒫_{J,1}(Z_0);
  Z_2 := 𝒫_{J,2}(Z_1);   Z_3 := 𝒫_{J,3}(Z_2);
  Z_4 := 𝒫_{J,4}(Z_3);   Z_5 := 𝒫_{J,5}(Z_4);
  Z_6 := 𝒫_{J,6}(Z_5);   𝓔_J(X) := 𝒫_{J,*}(Z_6).
  

• b_J retains PB-0002's definition b_J:=σ_J ẽ_J and its proved interface e_J b_J=b_J e_J=u. This notation is not inversion of a nonscalar.

• 𝓒_J(X):=𝓔_J(X)b_J. This is initially a written element. The forcing proves that it equals a scalar expression and recovers the presented coefficient.

• For a prescribed finite occurrence list (w_r,K_r) from the supplied presentations, their unary-operation or fixed-generator distributions, their paired-input expansions, or the prescribed concatenations of those lists, V_r and A_K^{[r]} denote the expression-prefix and coefficient accumulators of §1. The labels in, old, new, and last denote its specified stages. Each occurrence has an actual scalar expression w_r and a supplied blade index K_r before the collection is performed. The superscript [r] is a position label, not exponentiation. Other capital letters in the local identities name written prefixes, suffixes, or elements; lowercase free coefficients are scalar expressions.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, and with a declared nonvoid condition excluding equality with void:

The supplied comparison expressions have the blade actions

C_m(e_I)=λ_m(I)e_I,    e_I*=θ(I)e_I.                         (C0)

Their signs are given by the declared finite calculations. For supplied written X,Y and scalar c, the same conjugation expressions satisfy

C_m(v)=v,
C_m(X+Y)=C_m(X)+C_m(Y),    C_m(cX)=cC_m(X),
C_m(X)=Σ_{I∈𝓑} λ_m(I)𝓒_I(X)e_I,
𝓒_J(C_m(X))=λ_m(J)𝓒_J(X).                               (C0a)

For coefficient collection and recovery:

1. For every supplied presentation X=Σ_{I∈S} a_I e_I over distinct blades, the displayed expression 𝓒_J(X) equals a_J when J is in S and equals v when J is omitted. Consequently

   X = Σ_{J∈𝓑} 𝓒_J(X)e_J.                                      (C1)
   

   The coefficients are recovered by operations on X, not assumed from its spelling. Two supplied presentations of the same element therefore have equal coefficients at each index after the harmless v extension.

2. For every prescribed occurrence list of the stated finite kind, the explicit accumulation in §1 gives

   Σ_r w_r e_{K_r} = Σ_{K∈𝓑} A_K^{[last]}e_K,
   𝓒_K(Σ_r w_r e_{K_r}) = A_K^{[last]},                         (C2)
   

   where A_K starts as v and is changed to A_K+w_r precisely at an occurrence with K_r=K. The displayed rule applications establish the collection; the occurrence list merely specifies where to apply them.

3. For every supplied X,Y and scalar c,

   𝓒_J(v)=v,
   𝓒_J(X+Y)=𝓒_J(X)+𝓒_J(Y),
   𝓒_J(−X)=−𝓒_J(X),
   𝓒_J(cX)=c𝓒_J(X),
   𝓒_J(X−Y)=𝓒_J(X)−𝓒_J(Y).                                  (C3)
   

4. The comparison and elimination criteria are exact in both directions:

   X=Y iff 𝓒_J(X)=𝓒_J(Y) for every J in 𝓑;
   X=v iff 𝓒_J(X)=v for every J in 𝓑.                         (C4)
   

   In particular: a supplied sum over pairwise distinct blades can equal v only if every supplied coefficient equals v. Conversely those coefficient equalities make that sum v.

The forcing

1. The finite listings and the actual collection step

PB-0001 supplies the two-sided additive identities, inverse uniqueness, commutation, absorption, and product sign laws in (1)–(9), and right unity, double negation, negation of a sum and elementary equality criteria in (10)–(13). These are its stated interfaces, used before coefficient recovery.

Construct 𝓑 without a cardinality premise. Start with the list containing the empty index. At the index 0, write each current entry first without and then with 0; do the same at the next index 1, and then at 2,3,4,5,6. At each of these seven specified stages the two entries have different membership in that stage's new label. Deleting that label recovers their unique preceding entry. Thus that stage adds no duplicate and represents exactly the choices for the labels already encountered. Following these seven displayed stages gives every subset of G exactly once. It supplies a fixed order for presentations and their sublists. The empty index names the empty blade u.

For addition or a unary operation the occurrences are indexed by a sublist of 𝓑. For a product expansion the occurrences are indexed by a sublist of the explicitly paired list: visit each entry of the first sublist and, beside it, each entry of the second sublist in order. A concatenation is likewise prescribed before the arithmetic. This includes the seven lists obtained by applying each declared derivative and left-multiplying by its specified generator, as used by PB-0008. That sentence identifies an occurrence-list form; no later coefficient result is used here. No arbitrary-arity induction axiom, trial-selected list, or limit is used.

Here is the collection calculation at every actual occurrence. For the chosen finite list set

V_in:=v,       A_K^{[in]}:=v for each K in 𝓑.

The sum Σ_K A_K^{[in]}e_K is v: each term is ve_K=v by absorption, and appending each next v leaves the accumulated v unchanged. This is the starting written equality, including the case that the occurrence list has no entries.

Suppose the equality at the preceding position has been written as

V_old = Σ_K A_K^{[old]}e_K.

The next occurrence is the actual pair (w,J). Define V_new:=V_old+w e_J; change only the J accumulator, to A_J^{[new]}:=A_J^{[old]}+w. Let P and Q denote the two finite blocks before and after the J term, using v for an empty block. Associativity exposes that term. The complete local change is

V_new
  = ((P+A_J^{[old]}e_J)+Q)+w e_J
  = P+((A_J^{[old]}e_J+Q)+w e_J)
  = P+(A_J^{[old]}e_J+(Q+w e_J))
  = P+(A_J^{[old]}e_J+(w e_J+Q))
  = P+((A_J^{[old]}e_J+w e_J)+Q)
  = P+((A_J^{[old]}+w)e_J+Q)
  = Σ_K A_K^{[new]}e_K.                                      (1)

The first, second, third, fifth and last regroupings use additive associativity; the fourth uses additive commutation; the penultimate combination is distributivity (a+b)e_J=ae_J+be_J in its equality direction. Empty P or Q is handled by the two-sided identity already proved. Append this displayed finite chain at each position of the prescribed occurrence list. Each next equality uses the actual preceding equality, and after the last prescribed position its left side is the original occurrence sum. This proves the first assertion of (C2). Merely saying that the list is finite would not supply (1).

For a two-input difference the required negation is also explicit. For a prefix U followed by the actual summand y_J e_J,

−(U+y_J e_J)=−U−(y_J e_J)=−U+(−y_J)e_J.

Starting with −v=v and applying this identity down the second input's list writes the negated summands. Concatenate them after the first input's summands; use (1). At each paired index the final coefficient combination is a_J+(−y_J)=a_J−y_J. The explicit blade-undoing expression b_J is a different object and is not used in this collection step.

For a scalar c, expand one prefix at a time using

c(U+w e_J)=cU+(cw)e_J,

by distributivity and multiplicative associativity. This gives a displayed route for distributing c over the finite list. Rebracket a sum into prefixes using rule 1 before applying this route. A term v e_J equals v, so inserting such a term into a presentation changes nothing. Hence extending a sublist presentation with v coefficients on omitted indices is justified. Every supplied written element has a finite blade presentation by the input; collection (1), when repetitions occur, and this v extension give a full-list presentation without assuming uniqueness. Those v coefficients complete the presentation without changing X.

2. The signs and generator conjugations

The four products of signs are

uu=u,       u(−u)=−u,       (−u)u=−u,       (−u)(−u)=u.          (2)

The first two use left unity; the last two are PB-0001's sign results. Thus any of the actual finite sign accumulations reduces to u or −u, and each sign s satisfies ss=u.

The accumulator defining ρ has one −u factor for each listed member. Adding or removing a specified member changes it by one −u factor; scalar commutation permits moving that factor to the front. Therefore

m absent from I:  λ_m(I)=ρ(I),
m present in I:  λ_m(I)=(−u)ρ(I).                             (3)

For the latter identity, ρ(I)=(−u)ρ(I∖{m}); multiply by −u and use (2).

PB-0003's stated contextual exchange permits moving e_m through the increasing word e_I. When m is absent, pass its leading occurrence across every member of I. When m is present, pass it across the members before that occurrence; the two copies then stand together. Let the right copy continue across the remaining members: the intermediate written word is unchanged by which identical occurrence is named. In either case one exchange is made for each member of I except m itself. Their actual finite contextual equalities give

e_m e_I=λ_m(I)e_I e_m.

For I empty this is e_m u=u e_m; no exchange is asserted. Multiply on the right by e_m and on the left by ε_m. The square rule and ε_mε_m=u give

C_m(e_I)=ε_m e_m e_I e_m=λ_m(I)e_I.                           (4)

This is a written calculation on the word; no independence is being used.

Star on e_I is obtained by applying its product rule down that word. All factors except an occurrence of e₀ are fixed; an occurrence of e₀ contributes one −u. The empty word is fixed because it is scalar. Hence

e_I*=θ(I)e_I.                                                (5)

For star at v, substitute the established equality v=u+(−u) inside its argument. Its supplied addition rule and scalar fixing give v*=(u+(−u))*=u+(−u)=v, the last equality by rule 3. For negation, substitute inside star the established equality X+(−X)=v; the addition rule and v*=v give X*+(−X)*=v. Inverse uniqueness yields (−X)*=−(X*). These substitutions use the equality premise stated with the claim. The conjugation definition, absorption, the two distributivities and scalar centrality give

C_m(v)=ε_m e_m v e_m=v,
C_m(X+Y)=ε_m[e_mX+e_mY]e_m
         =ε_m[(e_mX)e_m+(e_mY)e_m]
         =C_m(X)+C_m(Y),
C_m(cX)=ε_m e_m(cX)e_m
        =c(ε_m e_m X e_m)=cC_m(X).

These prove the first three laws of (C0a). The actual prefix distributions needed below are

C_m(U+a e_I)
  =ε_m e_m(U+a e_I)e_m
  =ε_m(e_mU+e_m(a e_I))e_m
  =ε_m[(e_mU)e_m+(e_m(a e_I))e_m]
  =ε_m(e_mU)e_m+ε_m(e_m(a e_I))e_m
  =C_m(U)+aλ_m(I)e_I;

(U+a e_I)*=U*+(a e_I)*=U*+aθ(I)e_I.                          (5a)

The first chain uses each distributivity, association, scalar centrality, and (4); the second uses the supplied star laws, scalar fixing, and (5). Their starting inputs are v, which C_m sends to v by absorption and star sends to v as just proved. Appending these chains down the actual sum proves the required whole-sum expansions. Scalar factors carry through C_m by the displayed centrality step and through star because star fixes them.

3. The signs separate every pair of distinct blade indices

Fix I≠J. The finite sign reductions make ρ(I),ρ(J) either the same sign or opposite signs.

If they are the same, choose a label m lying in exactly one of I,J. Such a label occurs in their actual differing membership lists. Equation (3) gives

λ_m(I)λ_m(J)=−u.                                             (6)

If the signs are opposite and some m lies in both sets, (3) contributes two −u factors, so (6) again holds. If some m lies in neither set, (3) contributes neither factor and (6) again holds. In the remaining case every label of G lies in exactly one of the two sets. In particular 0 does, and therefore

θ(I)θ(J)=−u.                                                (7)

These cases exhaust the membership relations of the two supplied subsets. Thus a different blade is separated by at least one of the seven conjugation signs or the star sign. No numerical parity formula is used.

4. Apply a normalized expression without assuming its coefficient law

For one actual scalar term a_I e_I, (4) and scalar commutation give

C_m(a_I e_I)=a_I λ_m(I)e_I.

Using (1)'s exposed pairing and the supplied distributivities,

𝒫_{J,m}(a_I e_I)
  =k(a_I e_I+λ_m(J)a_I λ_m(I)e_I)
  =k((a_I+λ_m(J)a_I λ_m(I))e_I)
  =k(u+λ_m(J)λ_m(I))a_I e_I.                                 (8)

If the two signs agree, their product is u, so the multiplier is kh=u. If they disagree, the product is −u, so the multiplier is k(u+(−u))=kv=v. In the first case the exact output is a_I e_I; in the second it is v. For the star expression, the same calculation with (5) gives

𝒫_{J,*}(a_I e_I)=k(u+θ(J)θ(I))a_I e_I,                        (9)

with those same two outputs.

For a whole supplied presentation, distribute C_m or star over its actual prefix sums, as established in §2. Distribute the outer scalar and additions using §1; then pair the two occurrences of each blade using the complete local chain (1). This derives the coefficient factor in (8) or (9) for every presented I. It does not infer that law from the closure clause. Each expression also sends v to v by absorption and star(v)=v.

5. Recover one blade term, then its scalar coefficient

Apply the eight explicitly declared expressions in their specified order to the full X. After each application, equations (8)–(9) and the finite collection chain give a presented sum whose term at each original index is exactly its former term or v. The J term is retained at every step: each of its compared signs agrees with itself. Every I≠J is made v by one of the steps, by §3. Subsequent steps send that v to v. Therefore, using the actual finite sum and removing its v summands by identity,

𝓔_J(X)=a_J e_J  when J is represented;
𝓔_J(X)=v        when J is not represented.                    (10)

PB-0002 supplies the written element b_J with e_J b_J=u. Multiplying (10) on the right by it gives

𝓒_J(X)=(a_J e_J)b_J=a_J(e_J b_J)=a_Ju=a_J,

or 𝓒_J(X)=vb_J=v for an omitted J. This establishes scalarness and exact coefficient recovery. No scalar a_J is cancelled, divided, assumed nonvoid, or required to have any positivity property.

The expression 𝓒_J(X) depends on X and the fixed word J, not on a presentation chosen to calculate it. If another supplied presentation of X has coefficient a'_J, the same derivation gives 𝓒_J(X)=a'_J. Transitivity then yields a_J=a'_J. Consequently no two presentations assign unequal coefficients to the same blade. Replacing the coefficients of the supplied presentation by their recovered equal expressions proves reconstruction (C1).

6. Collection, elementary coefficient laws, and equality

The whole-presentation application in (C0a) now follows at the recovered coefficients. The prefix chain (5a), starting at C_m(v)=v and appended once for each actual summand of the presentation, gives

C_m(X)=Σ_{I∈𝓑} a_I λ_m(I)e_I.

Section 5 gives a_I=𝓒_I(X); scalar commutation puts λ_m(I) first. Applying (C1)'s recovery to this displayed presentation gives

𝓒_J(C_m(X))=λ_m(J)a_J=λ_m(J)𝓒_J(X).

Thus every line of (C0a) is established on the same full finite presentation, including its void coefficients.

Applying the recovery result to the collected equality from §1 proves the second assertion of (C2). Thus the accumulation laws are laws of the recovered coefficients, not merely a possible presentation chosen by the author.

For supplied X,Y, use their presentations extended on the same 𝓑. The explicit collection in §1 applied to their concatenation gives coefficient x_J+y_J; its negation calculation gives −x_J, and its scalar distribution gives cx_J. Since x_J=𝓒_J(X) and y_J=𝓒_J(Y), recovery gives (C3). The all-v presentation proves 𝓒_J(v)=v. For the difference, distribution and pairing give x_Je_J+(−y_J)e_J=(x_J−y_J)e_J at each index only after the two contributions have actually been brought together by (1).

If X=Y, substitute inside the full written recovery expression; every operation is defined at that equality's retained hypotheses. Hence 𝓒_J(X)=𝓒_J(Y). Conversely, if those finitely indexed coefficient equalities are supplied, substitute them term by term in the two reconstructed sums (C1); the same bracketing and order then give X=Y. Take Y=v for the second equivalence of (C4).

For the original distinct-blade implication, begin with its actual presentation. The hypothesis X=v gives

a_J=𝓒_J(X)=𝓒_J(v)=v

at every represented J. Conversely substituting all the supplied a_J=v equalities makes every term and the finite sum v. This completes both directions without using coefficient independence as a premise.

For example, the recovery applied to the supplied expression e₀+e₁ returns u at the two corresponding singleton indices and v at every other index, as the general argument proves. It is not a choice of Ψ.

What is not forced

This is an interface for the specified finite presentations, occurrence lists, collection and recovered coefficients, under the equality/nonvoid reading stated with its claims. Applying recovery inside an established equality retains that equality's actual hypotheses.

The displayed recovery uses the input's nonvoid guard on h to write k=h⁻¹. Each normalized comparison then uses kh=u or kv=v. No further step in this recovery argument requires a separate sign-distinctness premise. The same guard's use to establish nonvoid selector outputs belongs to PB-0008's support argument. A version without the guard needed to write k is not covered by this proof; this is not an impossibility claim about every other recovery route.

Coefficient recovery establishes neither scalar order nor positivity nor square-sum definiteness. Those would require their own written grounds.

The interface establishes the coefficient identities, not a decision procedure for arbitrary scalar identities or the consistency of added coefficient declarations. A recovered v is the output of 𝓒_J. Distinguishing an absent contribution from contributing terms that collect to v remains a question about the actual occurrence list and its written collection; equal outputs do not identify their separate operands.

The blade exchanges and undoing products come from PB-0003 and PB-0002. The operation-specific coefficient laws and complete equation are supplied later by PB-0008. The eight displayed comparisons give one sufficient recovery construction; no minimality or physical cost is attached to their number.

What it means

The separation construction does more than rule out a hidden coefficient. It supplies an actual expression that recovers the coefficient, making the meaning of coefficient independent of a chosen presentation. Before that recovery is used, the collection proof shows exactly how contributing terms are combined.

The two responsibilities fit together: distribution and regrouping produce the coefficient expression; recovery proves that it is the coefficient of the resulting element. Naming the result a written element supplies neither responsibility on its own. No information about a still-present operand is discarded merely because one coefficient or one derived result vanishes.

Blade cancellation fixes the sign of every two-blade product (V01-PB-0005)

LAYER: 3 FORCED BY: V01-PB-0001, V01-PB-0002, V01-PB-0003

Declares

• u := unity, v := void. A sign is one of u and −u.
• I,J,M are supplied index subsets of the seven generator labels; j,r are supplied generator labels. I△J is the subset of labels present in exactly one of I,J. This is an index operation, not scalar subtraction.
• b_I is PB-0002's written undoing expression σ_I ẽ_I.
• ε_m retains PB-0002's generator-square value; ρ(U) retains PB-0003's finite sign accumulation on the actual ordered sublist U, with ρ(∅)=u. No result of coefficient recovery is used here.
• For an index M and next generator j, M_< and M_> list M's members below and above j, in increasing order. Define δ(M,j):=ρ(M_>) when j is absent, and δ(M,j):=ρ(M_>)ε_j when j is present.
• χ(I,J) is the scalar accumulated by starting with s_in=u and M_in=I, then, at each successive label j of the actual increasing J listing, changing the scalar to s_old δ(M,j) and the index to M△{j}. This names the prescribed construction whose product identity is proved below; it is not a presumed multiplication table or a power operation.
• X,Y are supplied written elements. s,s₁,s₂,s_in,s_old are the specified sign expressions in the reductions. P,Q,Q_pre,V are actual word prefixes or suffixes; an empty word is u. Stage names identify written positions, not scalar counts.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, the following hold:

1.  X e_I = Y e_I implies X=Y;
    e_I X = e_I Y implies X=Y.

2.  X e_I = v implies X=v;
    e_I X = v implies X=v.

3.  If two written exchange routes take the same original word of distinct
    generators to s₁ e_I and s₂ e_I, where e_I is its increasing-order
    blade, then s₁=s₂.

4.  For every supplied I,J,
        e_I e_J = χ(I,J)e_{I△J},
    with χ(I,J) a sign, χ(I,∅)=u and χ(∅,J)=u.
    Any other licensed reduction of that same original product to the
    same output blade has exactly that sign.

The fixed blade factor in claims 1 and 2 can be cancelled without any scalar nonvoid guard. Claim 4 includes the overlap cases, where actual repeated occurrences are removed by their generator-square identities.

The forcing

1. Cancel the specified blade by its displayed undoing

PB-0002(3) supplies e_I b_I=b_I e_I=u. PB-0001(10) supplies right unity. If X e_I=Y e_I, substitute this equality inside the right product by b_I:

X = Xu = X(e_I b_I) = (X e_I)b_I
  = (Y e_I)b_I = Y(e_I b_I) = Yu = Y.

For the other side, if e_I X=e_I Y,

X = uX = (b_I e_I)X = b_I(e_I X)
  = b_I(e_I Y) = (b_I e_I)Y = uY = Y.

Association is rule 4; the two unity forms are rule 7 and PB-0001(10). These are the two implications in claim 1, with the same fixed blade retained.

PB-0001(6) gives v e_I=e_I v=v. A hypothesis X e_I=v therefore reads X e_I=v e_I; claim 1 yields X=v. Likewise e_I X=v=e_I v yields X=v. This proves claim 2.

2. Compare two routes from the same distinct-generator word

PB-0003(3) supplies a signed increasing-order blade for that actual word, including its empty case. If two routes give s₁ e_I and s₂ e_I, transitivity of their equalities to the original word gives s₁ e_I=s₂ e_I. Each scalar can be written as itself times e_∅=u by PB-0001(10), so it is a written element to which claim 1 applies. Hence s₁=s₂, proving claim 3.

3. Construct a product of two blades, retaining repeated occurrences

PB-0003(2) supplies each contextual exchange below; PB-0001(7)–(9) supplies its sign arithmetic. The four sign products are explicitly

uu=u,    u(−u)=−u,    (−u)u=−u,    (−u)(−u)=u.

The first two use rule 7; the last two use PB-0001(7) and (9). Thus every specified accumulation of these signs stays one of the same two expressions.

First take an increasing blade e_M and one appended generator e_j. The sublists M_< and M_> inherit their order from M. Let P and Q be their increasing generator words, using u for an empty word.

If j is absent from M, e_M e_j=P Q e_j. Pass the appended e_j leftwards through Q, in Q's fixed reverse order. The actual single step, with Q_pre the unmoved prefix, e_r its final factor, V the already moved suffix, and s the accumulated sign, is

s P Q_pre e_r e_j V
   =s P Q_pre (e_r e_j)V
   =s P Q_pre (−(e_j e_r))V
   =(s(−u))P Q_pre e_j(e_r V).

Association supplies the first and last brackets; PB-0003 supplies the middle exchange because r>j; PB-0001 carries its negation to the scalar. The next input is the last expression just reached, with the suffix enlarged by that actual e_r. There is one such step for each member of the supplied M_> listing, ending when that listing has been used. The accumulated coefficient is precisely ρ(M_>), and the output is ρ(M_>)P e_j Q, whose word is the increasing blade of M∪{j}. No step count supplies an algebraic sign.

If j is present, e_M e_j=P e_j Q e_j. Apply the same contextual exchanges through Q, with the earlier e_j retained in the left prefix. The resulting adjacent equal generators are now removed by their actual square relation:

e_M e_j=ρ(M_>)P e_j e_j Q
       =ρ(M_>)ε_j P Q
       =ρ(M_>)ε_j e_{M∖{j}}.                                 (1)

The middle equality uses the declared generator square and scalar centrality. A square is actually removed; it is not evaluated through a missing power convention. When Q is empty, the exchange input already has the required shape with sign seed u. When M is empty, the absent case is u e_j=e_j; the two-sided unity identities supply its brackets.

Now take the increasing listing of J. Start with the coefficient s_in=u and index M_in=I, so

e_I u = s_in e_{M_in}.

Append the first generator of J, then its next, through its last, in J's specified order. If the preceding product is s_old e_M, the cases just written give the next equality

(s_old e_M)e_j
   = (s_old δ(M,j)) e_{M△{j}},

δ(M,j)=ρ(M_>)            when j is absent,
δ(M,j)=ρ(M_>)ε_j         when j is present.                    (2)

Every factor δ(M,j) is a sign by the four sign products. At each stage multiply the scalar by that displayed δ and replace M by M△{j}. The equality at the next stage follows from (1)–(2) inside the actual preceding product. Because J contains each of its labels exactly once, each of those memberships is toggled once and every other membership is left alone. The final index is exactly I△J. Define χ(I,J) to be the resulting scalar sign. This proves the product identity of claim 4. If J is empty, its seed equality is already the result: χ(I,∅)=u. If I is empty, the appended increasing J has no exchange or repeated factor, so χ(∅,J)=u.

This is a definite finite chain for each pair of supplied index lists, derived before any computational verification. It uses no list length as a scalar, no parity exponent, and no recurrence assumed valid at an unprovided arity.

4. The full product's sign is fixed by that product

If another licensed reduction of e_I e_J gives s e_{I△J}, the two reductions of that one original product give

χ(I,J)e_{I△J}=s e_{I△J}.

Claim 1 cancels the same output blade and yields χ(I,J)=s. This proves the remaining assertion in claim 4. The reasoning does not compare two different original words merely because their final index sets agree.

What is not forced

The result concerns a fixed blade factor and actual products of two supplied blades. It supplies both their constructive reduction and the equality of signs obtained by different licensed routes from that same product.

A sum over different blades needs a separate collection and coefficient recovery argument; that is PB-0004's interface. Cancelling one specified blade does not extract the separate coefficients of a sum or cancel an arbitrary written factor. Each product law here remains an identity of the full written expressions, independent of any equation asserted for Ψ.

The sign is given by the finite insertion construction, with its actual empty case. No numerical count exponent is needed, and no general consistency claim for additional coefficient declarations is made.

What it means

One written undoing does two jobs. It permits cancellation at a known blade, and it establishes that distinct reduction routes of one product cannot leave different signs at that same output blade. The full two-blade law then combines actual exchanges with actual square removals.

This result is available before coefficient recovery. A later argument that needs only a blade product need not import the stronger machinery for recovering coefficients from sums. The coefficient-level consumer combines this law with that separate recovery interface in PB-0008.

The composite involution determines the selector-recoverable part and its exact remainder (V01-PB-0007)

LAYER: 3 FORCED BY: V01-PB-0001, V01-PB-0006

Declares

Let u := unity, v := void, h := u+u, and k := h⁻¹. The input itself supplies the nonvoid guard on h, so hk=u; scalar commutation also gives kh=u. No other scalar inverse is introduced. Put

Sel(X) := X̃−X*,
T(X) := (X̃)*.

For each supplied written X, the part expressions are defined by

H_X := k(X+T(X)),
A_X := k(X−T(X)),
K_A(X) := e₀Ã_X.

The argument in K_A(X) identifies the input whose A part is used; K_A may abbreviate K_A(X) only with that input specified. These definitions abbreviate permitted expressions. Their recombination, sign properties and uniqueness are proved below; no such property is added as a hypothesis on X.

Sel retains the operator definition of V01-PB-0006. T names only the displayed composite in this record, not an identification with any historical object also called T. D, L, F and R retain their definitions and scopes from V01-PB-0006. All other letters stand for supplied written elements or specified coefficients/finite word prefixes. Subscripts on A and H identify which input expression they describe; they are not derivatives or numerical labels.

“Selector-null” means precisely Sel(X)=v, a property of a whole written element. “Selector-silent” is used for a blade with that property. In the local map calculations, M denotes either of the two supplied operations, reversion or star; c is a supplied scalar. No other map is introduced by that notation.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, and with a declared nonvoid condition excluding equality with void, every supplied written X satisfies the following identities. For M equal to either reversion or star, and each supplied scalar c,

M(v)=v,    M(−X)=−M(X),    M(cX)=cM(X).           (M0)

The composite also fixes scalars and carries their factors: T(c)=c and T(cX)=cT(X). In addition,

(X̃)̃=X,    (X*)*=X,    (X̃)*=(X*)̃,
T(T(X))=X,
T(U+V)=T(U)+T(V),    T(UV)=T(V)T(U),
Sel(X)=v  iff T(X)=X.

The operation identities also give, for the same supplied scalar c,

(T(X))̃=X*,    (T(X))*=X̃,
Sel(v)=v,    Sel(cX)=cSel(X).                       (M1)

The expressions

H_X=k(X+T(X)),    A_X=k(X−T(X))

are the unique pair with

X=A_X+H_X,    T(H_X)=H_X,    T(A_X)=-A_X.

They satisfy

H_(H_X)=H_X,    A_(A_X)=A_X,
A_(H_X)=v,      H_(A_X)=v,
Sel(X)=hÃ_X,   A_X=k(Sel(X))̃,
Sel(Y)=Sel(X) iff A_Y=A_X.

For every supplied written X,Y, the selector-recoverable part satisfies

A_(X+Y)=A_X+A_Y.                                  (SF1)

The same-selector characterization is equivalently stated as follows: for supplied expressions Y and K, the expressions with the same selector output as X are precisely those expressible as Y=A_X+K with T(K)=K. For supplied U,V with Sel(U)=Sel(V)=v,

Sel(UV)=v iff VU=UV.

The complete source and its actual-field application are

F(X)=F(A_X)+H_Xe₀Sel(A_X)=hXe₀Ã_X=hXK_A(X),
R(A_Ψ)=v iff L(H_Ψ)=H_Ψe₀Sel(Ψ),

where the last equivalence uses the asserted equation R(Ψ)=v. Preserving the left and right sides individually upon retaining A_Ψ is equivalent to

D(H_Ψ)=v and H_Ψe₀Sel(Ψ)=v.

The supplied e₁,e₂,B and e₀+e₁ instances are calculated in the forcing. No coefficient-support equivalence is part of the claim.

The forcing

1. The elementary map identities

PB-0001's exported statements (1)–(13) supply two-sided additive identity and inverse, additive cancellation, inverse uniqueness, additive commutation, void absorption, sign/product laws, right unity, double negation, negation of a sum, and the void-difference criterion. Those parent claims are used here at their stated scope.

For either of the two supplied additive operations M (reversion or star), substitute the established equality v=v+v inside M's argument, then use its own addition rule:

M(v)=M(v)+M(v).

Comparing with M(v)=M(v)+v and cancelling gives M(v)=v. Substituting X+(-X)=v inside M's argument and using its addition rule gives M(X)+M(-X)=v; inverse uniqueness gives M(-X)=-M(X). Each assertion uses the corresponding declared operation, not an unprovided general family of maps.

For every supplied scalar c, the product rules, scalar fixing and centrality give

(cX)̃ = X̃c = cX̃,
(cX)* = cX*.

These are the identities exported as (M0). Applying the two scalar-factor identities successively gives T(cX)=cT(X); scalar fixing twice gives T(c)=c. For Sel the scalar and void steps are explicitly

Sel(v)=ṽ−v*=v+(−v)=v+v=v,
Sel(cX)=cX̃−cX*=cX̃+(−(cX*))
        =cX̃+c(−X*)=c(X̃−X*)=cSel(X).

The void calculation uses M0, PB-0001(11) and rule 2. The scalar calculation uses M0, PB-0001(8), and rule 5. These prove the last two M1 identities. Sel is additive by the two addition rules and the cited negation/regrouping identities.

PB-0001(13) supplies the exact elementary equivalence

U−V=v  iff U=V.                                    (A)

Its two directions are used below at the same equality scope.

2. Reversion and star on the actual finite words

Here the fact that the two operations are involutions, and that they commute, is derived rather than supplied by their names.

On the empty word u, both operations fix u, so applying either twice or composing in either order gives u. On a spatial generator eᵢ, both fix the generator. On e₀, the supplied images and the negation rule give

(e₀*)*=(-e₀)*=-(e₀*)=-(-e₀)=e₀,
(ẽ₀)̃=e₀,
(ẽ₀)*=-e₀=(e₀*)̃.

For any generator g from the supplied listing, these establish

(g̃)̃=g,    (g*)*=g,    (g̃)*=(g*)̃,
(g*)̃=g*.

Now take one actual increasing generator word. Start with its empty prefix w=u. Append its own next generator g, in the order of its actual finite listing. If the three identities below have been obtained for the preceding prefix w, the supplied product rules give the next prefix identities by these displayed steps:

((wg)̃)̃ = (g̃w̃)̃ = (w̃)̃(g̃)̃ = wg;

((wg)*)* = (w*g*)* = (w*)*(g*)* = wg;

((wg)̃)* = (g̃w̃)* = g*(w̃)*,
((wg)*)̃ = (w*g*)̃ = (g*)̃(w*)̃ = g*(w*)̃.

The last two right-hand sides are equal by the preceding prefix's commutation identity. This gives the same three identities for wg. For the chosen blade, apply this step at each of its actual generator occurrences, ending at that blade. Its listing is a subset of the seven supplied indices; no unbounded word class, trial selection or arbitrary-arity induction principle is invoked. The empty blade uses the starting calculation and needs no appending step.

For a scalar-coefficient term c e_I, section 1 carries c through both operations and both composites, so the same identities hold. For a supplied written element X=Σ_I c_I e_I, distribute each operation over its actual finite sum. Applying the established equality to each written summand gives

(X̃)̃=X,    (X*)*=X,    (X̃)*=(X*)̃.                (B)

Only the displayed representation and its finite distribution are used. No equality of coefficients is inferred from equality of elements; a second possible representation is not excluded by this proof.

3. The composite operation and the selector

The product rules directly give

T(UV)=(ṼŨ)*=T(V)T(U).                            (C)

Thus T reverses product order; it is not being treated as an order-preserving operation. It is additive and fixes scalars by section 1. From (B),

(T(X))̃=((X̃)*)̃=((X̃)̃)*=X*,
(T(X))*=((X̃)*)*=X̃,
T(T(X))=((T(X))̃)*=(X*)*=X.

In particular,

Sel(T(X))=X*−X̃=-Sel(X).                           (D)

The selector-null condition has an equivalent statement that uses T alone:

Sel(X)=v  iff T(X)=X.                             (E)

For the forward direction, (A) gives X̃=X*; applying star and (B) gives (X̃)*=X, hence T(X)=X. Conversely, apply star to T(X)=X. Using (T(X))*=X̃ gives X̃=X*, hence Sel(X)=v by (A). No use of k is made in this equivalence.

4. The complete decomposition characterized by the two signs

For each supplied written X, define

H_X := k(X+T(X)),
A_X := k(X−T(X)).                                 (F)

These are already permitted expressions. They impose no new equation or guard on X. Because T fixes k and T(T(X))=X,

T(H_X)=k(T(X)+X)=H_X,
T(A_X)=k(T(X)−X)=-A_X.                            (G)

The additive identities give their exact recombination:

A_X+H_X
  =k[(X−T(X))+(X+T(X))]
  =k(X+X)
  =k(hX)=(kh)X=X.                                (H)

By (E), Sel(H_X)=v, and additivity then gives Sel(A_X)=Sel(X).

To prove uniqueness at the claimed scope, let an actual second decomposition satisfy

X=A'+H',    T(H')=H',    T(A')=-A'.

Applying T gives T(X)=-A'+H'. Adding and subtracting these two written equalities and regrouping produces

X+T(X)=H'+H'=hH',
X−T(X)=A'+A'=hA'.

Multiplication by k gives H'=H_X and A'=A_X. Conversely (F)–(H) provide these identities and both sign properties. Consequently the formulas supply exactly the decompositions with these two sign conditions, not just one example among them.

Uniqueness is relative to those conditions. A generic identity X=A+H without them is not asserted to be this decomposition. The proof establishes a construction from the supplied operations, not a claim of uniqueness without those two sign conditions.

The repeated applications are also explicit. At T(H_X)=H_X, formula (F) gives

H_(H_X)=k(hH_X)=H_X,    A_(H_X)=k(H_X−H_X)=v.

At T(A_X)=-A_X, the same formula gives

A_(A_X)=k(hA_X)=A_X,    H_(A_X)=k(A_X+(-A_X))=v.

For the sum X+Y, use T's addition law, negate its two summands with PB-0001(12), and regroup the X terms beside each other and then the Y terms:

A_(X+Y)=k[(X+Y)−(T(X)+T(Y))]
        =k[(X−T(X))+(Y−T(Y))]
        =A_X+A_Y.

The last step is rule 5. This proves (SF1).

Thus applying either part-taking expression again preserves that part and removes the other in the stated algebraic sense. No projection theorem is imported to obtain these identities.

5. Exactly what equal selector output determines

From T(A_X)=-A_X, apply star and use (B) to obtain

Ã_X=-A_X*.

Therefore

Sel(X)=Sel(A_X)=Ã_X−A_X*=Ã_X+Ã_X=hÃ_X.          (I)

Conversely, directly from the selector definition and (B),

(Sel(X))̃=X−(X*)̃=X−T(X),
A_X=k(Sel(X))̃.                                    (J)

For any supplied X and Y, these identities prove both directions of

Sel(Y)=Sel(X)  iff A_Y=A_X.                        (K)

The forward direction uses (J); the reverse uses (I). The selector determines A_X by an explicit recovery expression. It does not, through that recovery expression, specify H_X.

A complete way to state the remainder, without naming an unprovided collection of all possible fields, is the following pair of implications for supplied expressions:

If Sel(Y)=Sel(X), then Y=A_X+H_Y and T(H_Y)=H_Y.

If Y=A_X+K and T(K)=K, then Sel(Y)=Sel(X).          (L)

The first follows from (H), (K) and (G). For the second, (E) gives Sel(K)=v, so additivity gives Sel(Y)=Sel(A_X)+v=Sel(X). Hence the precise same-selector freedom consists of the supplied T-fixed remainder; no additional coordinate, representative search or coefficient extraction is needed.

For the already supplied expressions e₀ and e₁,

T(e₀)=-e₀,    T(e₁)=e₁.

Formula (F) therefore gives

A_(e₀+e₁)=e₀,    H_(e₀+e₁)=e₁.

This is an algebraic instance of the construction. The pair is not here designated the physical field.

6. Product silence is exactly commutation of silent operands

Let supplied U,V satisfy Sel(U)=v and Sel(V)=v. By (E), T(U)=U and T(V)=V. Using (C),

T(UV)=T(V)T(U)=VU.

Apply (E) to UV itself. It gives the exact two-direction result

Sel(UV)=v  iff VU=UV.                             (M)

The condition VU=UV is also equivalent to V*U*=U*V*: apply star to either equality and use (B). This comparison retains the two selector-null operand hypotheses in (M); it supplies no statement about products without them.

The declared B=e₁e₂ supplies a concrete distinction. Each of e₁ and e₂ is selector-silent. Inverse uniqueness applied to e₁e₂+e₂e₁=v gives e₂e₁=-B, so

B̃=-B,    B*=B,    Sel(B)=(-B)+(-B)=-hB.

For the nonvoid check, use PB-0006's exported (B0):

BB=−u,    B(−B)=u.

Its complete local generator calculation supplies these products without a general blade-product dependency.

If -hB=v, right-multiplication by -B gives -h=v; negation gives h=v, contradicting the declared condition. Thus the product of these two selector-silent blades is not selector-null. The general positive characterization remains (M); the example does not replace it.

7. The source still uses the remainder

Use the L,F,R definitions and complete preservation identity in V01-PB-0006. Substituting (H) and Sel(H_X)=v into the full source, without discarding its first factor, gives

F(X)=(A_X+H_X)e₀Sel(A_X)
    =F(A_X)+H_Xe₀Sel(A_X)
    =hX e₀Ã_X=hXK_A(X).                          (N)

Thus recovering A_X from the selector does not recover the source independently of the remainder. For the supplied example X=e₀+e₁, the retained contribution is

H_Xe₀Sel(A_X)=e₁e₀(he₀)=he₁.

PB-0006's claimed concrete example establishes that he₁ is nonvoid under the same guard. The retained contribution is that exact expression, so its nonvoidness is inherited here, not proved again.

For the distinguished Ψ, equation (H) supplies Ψ=A_Ψ+H_Ψ. Equation (E) at (G) gives Sel(H_Ψ)=v, and (I) gives Sel(A_Ψ)=Sel(Ψ). The axiom supplies R(Ψ)=v. PB-0006 (2Ψ), at this decomposition with Sel(H_Ψ)=v, therefore gives

R(A_Ψ)=v  iff L(H_Ψ)=H_Ψe₀Sel(Ψ),                 (O)

because Sel(A_Ψ)=Sel(Ψ). For preservation of the two sides individually, apply PB-0006 (5) at its X=A_Ψ and Y=Ψ. Its difference is H=Ψ−A_Ψ=H_Ψ by (H) and additive arithmetic, and its selector-null hypothesis is the established Sel(H_Ψ)=v. Substituting Sel(A_Ψ)=Sel(Ψ) into that equivalence gives

L(Ψ)=L(A_Ψ) and F(Ψ)=F(A_Ψ)
    iff D(H_Ψ)=v and H_Ψe₀Sel(Ψ)=v.

The existence and uniqueness of the split do not supply either further equality. Nor do they supply individual derivative vanishing or a classification of solutions.

What is not forced

The supplied operations, equality reading and actual hypotheses are the scope of the claims. The decomposition and uniqueness use the inverse k under h's existing nonvoid guard. The sign characterizations of the two parts, not a generic equation X=A+H, determine the unique pair claimed here.

The generator-word identities, decomposition and same-selector characterization are element-level results: they are proved on each supplied presentation without inferring equality of its coefficients. The coefficient-support identification is a separate consequence established in PB-0008 (P3), whose collection and recovery premises are not premises of this record.

The generic PB-0006 theorem remains generic. Here the constructed A_Ψ,H_Ψ supply an identity to which that theorem applies. The actual-field conclusion uses R(Ψ)=v and retains the further balance L(H_Ψ)=H_Ψe₀Sel(Ψ). Separate-side preservation has its separately displayed two conditions. The constructed split alone establishes none of those additional equalities or individual derivative vanishing.

What it means

Every written X has the constructed decomposition X=A_X+H_X: A_X is recoverable from Sel(X), and H_X is the T-fixed remainder. For supplied U,V with Sel(U)=Sel(V)=v, their product is selector-null exactly when VU=UV.

For Ψ, the asserted R(Ψ)=v gives the balance R(A_Ψ)=v iff L(H_Ψ)=H_Ψe₀Sel(Ψ). Preserving the two sides individually is equivalent, under this constructed decomposition, to D(H_Ψ)=v and H_Ψe₀Sel(Ψ)=v. Recovering A_Ψ does not discharge these separate conditions; the full source still contains H_Ψ.

The recovered coefficients carry the written-element operations and exactly characterize selector-null support (V01-PB-0008)

LAYER: 4 FORCED BY: V01-PB-0001, V01-PB-0002, V01-PB-0003, V01-PB-0004, V01-PB-0005, V01-PB-0006, V01-PB-0007

Declares

The input, the free written elements, and all retained hypotheses have the same scope as the cited records. No new coefficient identity, nonvoid guard, derivative rule, or restriction of Ψ is introduced.

• u,v,h,k, the finite index lists G,𝓑, the star sign θ(I) and the recovery expression 𝓒_I(X) retain their definitions from PB-0004. The generator-square signs ε_m and blade-undoing expressions b_I retain PB-0002's definitions. PB-0004's written collection/recovery theorem is used before any coefficient comparison.

• For supplied written X,Y, put x_I:=𝓒_I(X), y_I:=𝓒_I(Y). These retain PB-0004's recovered coefficients, not independent new field variables. Its claim (C1) supplies their full-list presentations.

• I△J and χ(I,J) retain PB-0005's symmetric-difference index and the sign of its specified finite insertion construction, including the empty-list cases. They are not newly chosen multiplication data.

• η(I) is the sign produced when the reversed generator word of e_I is put into increasing order by the contextual exchanges of PB-0003, using the ordered destination construction of §3 below; η(∅)=u.

• τ(I):=η(I)θ(I) and w_I:=η(I)−θ(I) are scalar expressions. 𝓑_H is the sublist of 𝓑 with τ(I)=u, and 𝓑_A is its sublist with τ(I)=−u, with membership decided by the prescribed finite sign calculation. These subscripts refer to the sign-characterized parts, not physical interpretations or cardinalities.

• T, Sel, A_X, and H_X are precisely PB-0007's expressions. In particular T(X)=(X̃)*, A_X=k(X−T(X)), and H_X=k(X+T(X)).

• D,L,F,R retain PB-0006's complete expressions. β:={1,2} denotes the index set of the declared B, so e_β=B. A singleton {μ} denotes the supplied generator's blade, including {0}.

• All restricted sums in this record mean prescribed accumulations: start with v and append the displayed scalar contribution, in the displayed finite input-list order, precisely when its index condition holds. In a sum over I,J the I position is visited first and then J; in a sum over μ,I the order is G followed by 𝓑. A target with no contributing occurrences has the unchanged accumulator v. This construction does not label an absence as a new object.

• For a target K define the written scalar accumulations

  p_K(X,Y) := Σ_{I,J : I△J=K} x_I y_J χ(I,J),
  
  ℓ_K(X) := h Σ_{μ,I : ({μ}△I)△β=K}
                ε_μ (∂_μ x_I) χ({μ},I) χ({μ}△I,β),
  
  f_K(X) := Σ_{I,J : (I△{0})△J=K}
                x_I x_J w_J χ(I,{0}) χ(I△{0},J).
  

  Their equality with the relevant recovered coefficients is proved, not assumed by these definitions. Every pair occurrence is retained, including diagonal occurrences. All sums use the given finite lists rather than an assumed scalar repetition count.

• μ denotes a member of G; I,J,K denote members of 𝓑. a is a supplied scalar expression. U,V,W denote the actual finite sum or generator-word blocks at their uses. g denotes the actual next generator in §4. a_old,c_next and b_old,b_new in §7 are the unweighted and weighted scalar accumulations specified there. Empty word blocks are u; empty sums are v. Stage subscripts locate a specified written position, not a power or scalar count.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, and with a declared nonvoid condition excluding equality with void, the following hold for all supplied written X,Y and each K in 𝓑.

Whole-blade interface. For every actual blade in the derived finite listing,

ẽ_I=η(I)e_I,    e_I*=θ(I)e_I,    T(e_I)=τ(I)e_I,
Sel(e_I)=w_Ie_I.                                             (P0)

The signs are the declared written sign products, with τ(I)=η(I)θ(I). At the empty index all three signs are u and all three images are u. The reversion sign is independent of the permitted ordering route from the same original reversed word to the same increasing word. This does not compare signs of different original words.

The product coefficient law. PB-0005(4) supplies e_I e_J=χ(I,J)e_{I△J} and its sign's route independence. Combining that blade law with collection and recovery gives

𝓒_K(XY)=p_K(X,Y).                                               (P1)

The separate blade law includes repeated-generator and empty-list cases; the coefficient result here retains every paired input occurrence.

Together with PB-0004's additive/scalar laws, the involution and derivative operations satisfy

𝓒_K(X̃)=η(K)x_K,
𝓒_K(X*)=θ(K)x_K,
𝓒_K(T(X))=τ(K)x_K,
𝓒_K(Sel(X))=w_K x_K,
𝓒_K(∂_μ X)=∂_μ x_K.                                           (P2)

The complete selector-support characterization. The two sign lists 𝓑_H,𝓑_A partition the derived index listing. The former is precisely the selector-silent blade list. The latter has w_I equal to h or −h. Therefore

Sel(X)=v
  iff x_I=v for every I in 𝓑_A
  iff X has a presentation using only blades in 𝓑_H.           (P3)

The parts already constructed in PB-0007 have the exact collected presentations

H_X=Σ_{I∈𝓑_H} x_I e_I,
A_X=Σ_{I∈𝓑_A} x_I e_I.                                        (P4)

A supplied X need not have nonvoid coefficients at every index of either list. No nonvoidness test on arbitrary x_I is required to define these two finite lists.

The complete equation's coefficients. Each singleton undoing from PB-0002 is the signed generator in the contraction, giving

b_{{μ}}=ε_μe_μ,    D(X)=Σ_μ b_{{μ}}(∂_μX).              (P5u)

For every supplied X,

𝓒_K(L(X))=ℓ_K(X),
𝓒_K(F(X))=f_K(X),
𝓒_K(R(X))=ℓ_K(X)−f_K(X).                                      (P5)

Consequently

R(X)=v iff ℓ_K(X)=f_K(X) for every K in 𝓑.                     (P6)

For X=Ψ this is an exact equivalent writing of the asserted equation. No side condition or coupling term has been dropped, and no independent scalar solution family has been asserted.

The forcing

1. The blade product supplied before coefficient recovery

PB-0005(4), under the same equality scope, establishes

e_I e_J=χ(I,J)e_{I△J},
χ(I,∅)=u,    χ(∅,J)=u.                                      (1)

That parent constructs each sign by the actual exchanges and generator-square removals and proves its route independence using its blade-cancellation result. It requires no coefficient recovery. Here that already established whole-product equality is substituted into each written occurrence before collection.

2. From paired input occurrences to the product's actual coefficient

Use the reconstructed presentations of X,Y from PB-0004. Distribution must precede collection. For one scalar-blade term in the first input and a prefix V followed by the next term of the second input,

(x_I e_I)(V+y_J e_J)
  =(x_I e_I)V+(x_I e_I)(y_J e_J)
  =(x_I e_I)V+(x_I y_J)(e_I e_J).

The first equality is rule 5. The second uses rule 4 and moves only the scalar y_J through e_I. Apply this identity at every prescribed J position, starting from (x_I e_I)v=v. It gives that entire row of products. Then use the left-prefix distribution

(U+x_I e_I)Y=UY+(x_I e_I)Y

at each prescribed I position, beginning at vY=v. This gives every ordered pair once, with no omitted diagonal or mixed pair. Substitution of the parent blade law (1) gives the occurrence sum

XY=Σ_{I,J} (x_I y_J χ(I,J)) e_{I△J}.                           (4)

The occurrence index is the paired input list, not just the output-blade list. Apply PB-0004's explicit collection chain at each occurrence in (4). The target K accumulator receives exactly x_I y_J χ(I,J) when I△J=K, and otherwise is unchanged. Its final expression is p_K(X,Y) as declared. The recovery part of that theorem now establishes 𝓒_K(XY)=p_K(X,Y). This completes (P1); closure alone was not used to guess the coefficient law.

3. The two involution signs on a blade

Star on a blade is PB-0004's exported (C0). PB-0002(0) supplies reversion as the exact reversed generator word, including ẽ_∅=u.

For a nonempty reversed word ẽ_I, list the destination labels in increasing order. For the label belonging first, perform the contextual exchange once with each factor to its left; then fix the second position in the same way, then the next, following I's finite listing. At each step a pair of distinct generator occurrences is exchanged and the remaining word is unchanged. Carry a scalar initialized at u; each actual exchange multiplies it by −u. This is exactly the stated construction of PB-0003 applied to that reversed word. Its final scalar is η(I), a sign, and its final word is e_I. Hence

ẽ_I=η(I)e_I,
e_I*=θ(I)e_I,
T(e_I)=η(I)θ(I)e_I=τ(I)e_I.                                  (5)

For the composite image, apply star to the first equality: scalar fixing and the product rule give (η(I)e_I)*=η(I)θ(I)e_I=τ(I)e_I. At I=∅, the reversed word and increasing word are both u. No exchange occurs, so η(∅)=u; PB-0004 defines θ(∅)=u, and (C0) gives e_∅=u. Their product is u. Both involutions fix this scalar, and T(u)=(ũ)=u. These are all the empty cases of (P0).

No grade-parity formula or coefficient comparison was needed. PB-0005(3), applied to that same original reversed word and its increasing destination, establishes agreement of η(I) with every other licensed ordering route. Subtract its two involution images to obtain Sel(e_I)=[η(I)−θ(I)]e_I=w_Ie_I. This identity and (5) prove the whole-blade interface (P0) before its coefficient uses.

For a scalar term a e_I,

(a e_I)̃=ẽ_I a=η(I)a e_I,
(a e_I)*=a e_I*=θ(I)a e_I.

Under the stated equality premise, substitute X's established reconstructed presentation inside reversion and star. For an actual prefix U and its next scalar-blade term, the supplied addition rule gives (U+a e_I)̃=Ũ+(a e_I)̃, and the corresponding star identity. Starting with each operation applied to v, which PB-0007's exported (M0) establishes equal to v, append these identities down the supplied presentation. The output has one term at each original blade, with the coefficient just displayed. PB-0004 recovery therefore gives the first two identities in (P2). Apply these same identities in sequence to obtain the T coefficient. Subtract the two displayed whole-element sums with PB-0004's actual difference-collection chain to obtain

Sel(X)=Σ_I (η(I)−θ(I))x_I e_I=Σ_I w_I x_I e_I.                 (6)

Recovery gives its asserted coefficient. Subtraction here acts on scalar coefficients only after its two summands have actually been paired at each blade.

4. The individual derivative coefficient law

For each supplied derivative, first substitute the established equality u=e₀e₀ inside its argument; then the product rule gives

∂_μu=(∂_μe₀)e₀+e₀(∂_μe₀)=v.

Substituting v=v+v inside that same derivative's argument and using its addition rule gives ∂_μv+∂_μv=∂_μv; comparison with ∂_μv+v=∂_μv and additive cancellation gives ∂_μv=v.

At a word prefix W whose derivative has been proved v, appending the actual next generator g gives

∂_μ(Wg)=(∂_μW)g+W(∂_μg)=vg+Wv=v.

The starting prefix is u, just handled. Applying that displayed rule down the blade's actual generator listing proves ∂_μe_I=v. For a scalar coefficient a,

∂_μ(a e_I)=(∂_μa)e_I+a(∂_μe_I)=(∂_μa)e_I.

For the full sum, the actual prefix step is

∂_μ(U+a e_I)=∂_μU+(∂_μa)e_I.

Its start is ∂_μv=v. The established reconstructed presentation equals X; substituting that equality inside ∂_μ under the stated premise identifies the last derivative prefix with ∂_μX. All ∂_μa are scalar by the input. Recovery now gives 𝓒_I(∂_μX)=∂_μx_I, the last identity of (P2). This is the derivative coefficient law in (P2).

5. Derive the selector's complete support condition

The actual sign calculations place η(I),θ(I) in the two signs u,−u. Here their distinctness is used: u=−u would imply h=u+u=u+(−u)=v, contrary to the input's nonvoid condition. Their four combinations give

η=u,  θ=u:    τ=u,   w=v;
η=−u, θ=−u:   τ=u,   w=v;
η=u,  θ=−u:   τ=−u,  w=h;
η=−u, θ=u:    τ=−u,  w=−h.                                  (7)

The last equality uses negation of a sum. The third row uses double negation from PB-0001(11), so u−(−u)=u+u=h; the other steps are sign products or self-subtraction. This is an exhaustive computation on two specified signs, not an imported spectrum theorem. Apply this membership rule to each entry of the already derived 𝓑. Every entry goes into exactly one of 𝓑_H and 𝓑_A, preserving the list's order. No numerical count of either list is needed for completeness.

If I lies in 𝓑_H, (P0) and the first two rows of (7) give Sel(e_I)=w_Ie_I=v. If I lies in 𝓑_A, w_I is h or −h. In that case Sel(e_I) cannot be v: PB-0005(2) applied to w_Ie_I=v would give w_I=v. Since w_I is h or −h, that equality gives h=v directly or by negation, contrary to the guard. Thus 𝓑_H is precisely the selector-silent blade list.

Suppose Sel(X)=v. Equation (6) is the derived collected form of the selector output, so PB-0004 gives w_I x_I=v for every I. At an index in 𝓑_A there are two cases. If w_I=h, multiply by k to obtain x_I=v. If w_I=−h, the product sign law gives −(h x_I)=v; negate both sides to get hx_I=v, and multiply by k. No inverse or nonvoidness is assumed of x_I. Thus all the 𝓑_A coefficients are v.

Conversely, suppose those coefficients are v. In (6) a 𝓑_H term has multiplier v by (7); a 𝓑_A term has coefficient x_I=v by hypothesis. Absorption makes each written summand v. The finite additive identity chain then gives Sel(X)=v. This proves the first equivalence of (P3).

For its last equivalence, substitute the proved active coefficient equalities into the reconstructed presentation X=Σ_I x_Ie_I; remove only the resulting v terms to get a presentation on 𝓑_H. Conversely a supplied presentation confined to 𝓑_H has v recovery at every omitted index, by PB-0004, so the previous implication applies. The claim is about which coefficient expressions must vanish, not about deciding in advance which individual x_I are nonvoid.

6. Identify the previously derived parts with their coefficients

PB-0007 supplies the expressions A_X,H_X and their sign-characterized uniqueness. Apply the already derived elementary coefficient laws to those exact expressions, using (P2):

𝓒_I(H_X)=k(x_I+τ(I)x_I),
𝓒_I(A_X)=k(x_I−τ(I)x_I).                                     (8)

If I is in 𝓑_H, τ(I)=u; the first expression is khx_I=x_I and the second is v. If I is in 𝓑_A, τ(I)=−u; the first is v and the second is khx_I=x_I. The reconstruction identity (C1) from PB-0004 therefore gives both of (P4). This supplies the coefficient-support interface that PB-0007 explicitly left unproved, without changing the meaning of its previously defined parts.

It does not remove H_X from the source. PB-0007's existing identity F(X)=F(A_X)+H_Xe₀Sel(A_X) remains unchanged. The coefficient description just proved cannot discharge the further equality needed to discard its second term.

7. Every term of the left side is transported and collected

PB-0006 defines the seven-term D and L(X)=hD(X)B. By §4,

D(X)=Σ_{μ,I} ε_μ(∂_μ x_I)(e_μ e_I).

This display is obtained from the seven explicit derivative sums by first distributing each left generator over its finite I prefix list and then concatenating the seven lists. The local distribution is e_μ(U+a e_I)=e_μU+a(e_μe_I), with the scalar moved by centrality. Negative derivative contributions are multiplication by −u by PB-0001, hence ε_μ is exactly the sign in D.

For the singleton index {μ}, PB-0002(0)'s reversion is e_μ and its blade–reverse scalar is e_μe_μ=ε_μ. Its already defined undoing expression is therefore b_{{μ}}=ε_μe_μ. Directly,

e_μ b_{{μ}}=ε_μ(e_μe_μ)=ε_με_μ=u,
b_{{μ}} e_μ=ε_μ(e_μe_μ)=u.

Thus the same contraction can be written

D(X)=Σ_μ b_{{μ}}(∂_μX).

This proves (P5u). The sign in D is exactly the sign supplying each generator's written undoing. This uses no new inverse operation or guard. It does not make the derivative contributions independently vanishing, and the full transported coefficient calculation below is unchanged.

The two actual blade products in a left-side contribution are

(e_μe_I)B
  =χ({μ},I)e_{{μ}△I}e_β
  =χ({μ},I)χ({μ}△I,β)e_{({μ}△I)△β}.                         (9)

Multiplying the displayed finite D sum on the right by B uses (U+V)B=UB+VB on each actual prefix. Multiplying on the left by h uses the corresponding scalar prefix distribution. No commutation of B through the derivative output is asserted. The B product is a paired-input expansion whose second input is the one-term presentation B=u e_β; (9) supplies the actual output blade and scalar at each pair. The h product multiplies each scalar coefficient in that prescribed list by h. Concatenating these seven fixed-generator distributions is therefore an occurrence list of the finite kind specified in PB-0004 (C2), with every blade index and scalar coefficient supplied before collection. Substitute (9) in that list and apply (C2). For a target K its scalar accumulator initially has the form of a sum of contributions h c_next. Here c_next denotes precisely the displayed unweighted contribution

ε_μ(∂_μx_I)χ({μ},I)χ({μ}△I,β)

at a qualifying occurrence. To connect this accumulated sum to the declared ℓ_K, start both the unweighted accumulator and the weighted accumulator at v. Their equality v=hv is PB-0001(6). If the preceding weighted scalar b_old equals h a_old, appending that occurrence gives

b_new=b_old+h c_next
     =h a_old+h c_next
     =h(a_old+c_next).

The last equality is rule 5 in its equality direction. At a noncontributing occurrence neither accumulator changes. Follow these identities over the exact same supplied list; they establish that the final weighted sum is h times the unweighted sum, including the case with no contributions. That expression is ℓ_K(X) as defined. Recovery (C2) therefore gives 𝓒_K(L(X))=ℓ_K(X).

8. Every term of the nonlinear source is transported and collected

Use (6), not an assumed coefficient law for the selector. The two finite presentations give

F(X)=(Σ_I x_Ie_I)e₀(Σ_J w_Jx_Je_J).

For its distribution, first right-multiply each X prefix by e₀ using (U+V)e₀=Ue₀+Ve₀. Then distribute the second finite sum against each resulting term using the two prefix calculations of §2. The full raw paired occurrence is

x_I x_J w_J (e_I e₀ e_J).

Only scalars have moved. The two blade reductions are

e_I e₀ e_J
  =χ(I,{0})e_{I△{0}}e_J
  =χ(I,{0})χ(I△{0},J)e_{(I△{0})△J}.                         (10)

Consequently the exact source occurrence sum is

F(X)=Σ_{I,J} x_Ix_Jw_Jχ(I,{0})χ(I△{0},J)e_{(I△{0})△J}.

Apply the written collection step at every prescribed pair occurrence. Its K accumulator is f_K(X), proving the second equality of (P5). In particular the first factor's I list has not been restricted to 𝓑_A. A multiplier w_J=v can remove its particular J contribution; it does not eliminate a selector-null component in the separate first factor. Both self and mixed products remain in the complete occurrence sum.

9. The residual equivalence closes the operation interface

PB-0004's derived difference law gives

𝓒_K(R(X))=𝓒_K(L(X))−𝓒_K(F(X))=ℓ_K(X)−f_K(X).

Its exact extraction criterion makes R(X)=v equivalent to these differences being v for every K. PB-0001(13) supplies both directions of U−V=v iff U=V. Applying it to each displayed scalar difference gives (P6).

For X=Ψ the original input supplies L(Ψ)=F(Ψ), equivalently R(Ψ)=v. Conversely, if all the displayed coefficient equalities are established for Ψ, reconstructing the residual gives its original whole-element equation. Every occurrence, derivative, and scalar condition still belongs to that same Ψ. This is an exact finite rewriting of the full nonlinear equation, not a separately chosen scalar model.

What is not forced

The operation laws and equivalences hold for supplied written elements under the equality and nonvoid reading stated with the claim. The nonvoid condition on h and its permitted inverse k enter through PB-0004's explicit recovery construction and through the selector-support proof here. Removing that guard would remove premises of these particular arguments; no assertion about every conceivable alternative recovery construction is made.

The finite sign predicates specify the two complete sublists of the supplied blade index. Numerical cardinalities and graded counts require their own written counting argument. A supplied expression can have void coefficients at any members of either list; membership is a property of the blade, not a test of an arbitrary field coefficient.

The output of collection is the scalar at its particular blade. It does not identify every contributing term with that scalar or with void. (P6) is an exact simultaneous rewriting of the full equation: every source pair and all seven generator-weighted derivative lists remain. Different derivative contributions can reach the same blade and must be collected together there. It does not supply a solution of those simultaneous equations. No scalar order or positivity is established here. The parts retain PB-0007's source-preservation condition.

Coefficient identities are used where established, with their actual hypotheses. The only scalar inverse introduced is k under h's existing guard; cancellation at a specified blade uses PB-0002/PB-0005's written products. No decision procedure for arbitrary scalar equalities or general factor cancellation is contained in these coefficient interfaces.

What it means

The coefficient interface now reaches the operations the equation actually uses. Each input occurrence is distributed, reduced to its stated blade, and collected by a displayed arithmetic step. Recovery then identifies the collected scalar as the coefficient of the resulting element.

That makes the selector-support statement a theorem rather than a glossary equivalence. Selector-nullness says exactly that the coefficients on the derived nonsilent list vanish; the involution-defined parts retain precisely the corresponding two coefficient lists. The whole source nevertheless retains its full first factor, and the contracted derivative retains contributions that can meet at the same output blade.

A vanishing coefficient is an exact relation at that collected output. It is neither permission to erase every contributing operand nor evidence that nothing was ever contributed. The complete occurrence calculation and the complete nonlinear equation remain available together.

The exact mixed source recovers the whole written input (V02-PB-0014)

LAYER: 4 FORCED BY: V01-PB-0001, V01-PB-0006, V01-PB-0007

Declares

All expressions use the supplied operations and the equality scope of the parents. No equation is imposed on a supplied written input merely by naming it. The distinguished field is still Ψ, whose asserted equation remains the one in 04.

• u=unity, v=void, h=u+u, and k=h⁻¹ retain 04's meanings. The only inverse used here is k under h's existing nonvoid guard.

• Sel(X)=X̃−X*, D, L=hDB, F(X)=Xe₀Sel(X), R=L−F, and Q(X,Y) retain exactly the expressions of PB-0006. The constructed A_X,H_X and composite T(X)=(X̃)* retain PB-0007's meanings.

• For supplied written X,Y define the mixed-source expression

  Mix(X,Y) := X e₀ Sel(Y) + Y e₀ Sel(X).
  

  This names the two mixed summands already retained in PB-0006. It is not an additional term in the axiom or a new differential operation.

• a,b below are supplied scalar expressions. They need not have void derivatives. X,Y,Z denote supplied written elements, not necessarily expressions satisfying R=v.

What is forced

For every supplied written input and actual hypotheses retained,

Mix(X,Y)=Mix(Y,X),
Mix(X+Y,Z)=Mix(X,Z)+Mix(Y,Z),
Mix(X,Y+Z)=Mix(X,Y)+Mix(X,Z),
Mix(aX,bY)=ab Mix(X,Y),
Mix(v,Y)=Mix(X,v)=v.                                      (MixOps)

The complete two-input expansion and diagonal identities are

F(X+Y)=F(X)+F(Y)+Mix(X,Y),
Q(X,Y)=Mix(X,Y)+F(Y),
Mix(X,X)=hF(X),       F(X)=k Mix(X,X),
F(aX)=aaF(X),
R(X+Y)=R(X)+R(Y)−Mix(X,Y).                                (M2)

PB-0006 already supplies the full finite-change equality. MixOps and M2 make its mixed term an explicitly reusable two-input expression and expose its exact scalar laws; they do not discard its self-source remainder.

The two specified second arguments u and e₀ give

Mix(X,u)=e₀Sel(X),
Mix(X,e₀)=hX+Sel(X),
Sel(X)=e₀ Mix(X,u),
X=k[Mix(X,e₀)−e₀ Mix(X,u)].                               (M3)

Consequently,

Mix(X,u)=v and Mix(X,e₀)=v  iff X=v.                      (M4)

More generally, for supplied X,Y,

Mix(X,u)=Mix(Y,u) and Mix(X,e₀)=Mix(Y,e₀)
    iff X=Y.                                             (M5)

The first output alone gives precisely the already constructed selector-recoverable part:

A_X=k(e₀ Mix(X,u))̃.                                      (M6)

Under the additional selector-null condition, the second output is

Sel(X)=v  implies  Mix(X,e₀)=hX.                          (M7)

Thus no nonvoid supplied written input can have both outputs void. This is an element-level reconstruction result and needs no coefficient-support restriction, coefficient inverse, or scalar square root.

The forcing

1. The inherited mixed summands and their exact algebraic laws

PB-0001 supplies the two-sided additive identities, additive commutation and cancellation, sign/product laws, absorption, right unity, and the void-difference criterion. PB-0006(1) supplies Sel's additive law; PB-0007(M0,M1) supplies the two operations' scalar laws and Sel's remaining identities, including Sel(v)=v and Sel(aX)=aSel(X).

Exchanging X and Y in the definition of Mix exchanges its two additive summands. Their commutation therefore gives

Mix(Y,X)=Y e₀ Sel(X)+X e₀ Sel(Y)=Mix(X,Y).

For the first input's addition, expand in the given product order:

Mix(X+Y,Z)
  =(X+Y)e₀Sel(Z)+Z e₀Sel(X+Y)
  =X e₀Sel(Z)+Y e₀Sel(Z)
      +Z e₀Sel(X)+Z e₀Sel(Y)
  =Mix(X,Z)+Mix(Y,Z).

The second input's addition follows by the just-proved symmetry and this displayed identity, or by the same four written terms. No nonscalar factors are commuted.

For the two actual scalar inputs, Sel's scalar law and scalar centrality give

Mix(aX,bY)
  =(aX)e₀(b Sel(Y))+(bY)e₀(a Sel(X))
  =ab X e₀Sel(Y)+ba Y e₀Sel(X)
  =ab[X e₀Sel(Y)+Y e₀Sel(X)]
  =ab Mix(X,Y).

The equality ba=ab is an instance of the supplied scalar centrality. There is no derivative in this calculation, so no derivative condition on a or b is required. Substitution of v, the identity Sel(v)=v, and absorption give both void-input identities. This proves MixOps.

2. The complete two-input source and residual expansions

The exact finite change in PB-0006 is

F(X+Y)=F(X)+Q(X,Y),
Q(X,Y)=X e₀Sel(Y)+Y e₀Sel(X)+Y e₀Sel(Y).

By the definitions of Mix and F, the second equality is precisely

Q(X,Y)=Mix(X,Y)+F(Y).

Substitution and additive association give the first two identities of M2. This uses the previously proved full expansion; it is not a second expansion with one term suppressed.

At equal inputs, the definition gives

Mix(X,X)=X e₀Sel(X)+X e₀Sel(X)
        =F(X)+F(X)
        =(u+u)F(X)=hF(X).

The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. Multiplying by the existing k, associating, and using this product gives F(X)=k Mix(X,X). Scalar homogeneity can be written without any division:

F(aX)=(aX)e₀Sel(aX)
      =(aX)e₀(aSel(X))
      =aa X e₀Sel(X)=aaF(X).

Finally PB-0006's derivative additivity and the complete source expansion just obtained give, with all terms retained,

R(X+Y)
  =L(X)+L(Y)−[F(X)+F(Y)+Mix(X,Y)]
  =[L(X)−F(X)]+[L(Y)−F(Y)]−Mix(X,Y)
  =R(X)+R(Y)−Mix(X,Y).

This is the residual form already visible in that parent's complete change proof, now exported using the single named Mix expression. It proves M2.

3. Substitute the two specified written second arguments

The operations fix the scalar u, so

Sel(u)=u−u=v.

The supplied generator images and double negation give

Sel(e₀)=e₀−(−e₀)=e₀+e₀=he₀.

Consequently the two substitutions are

Mix(X,u)
  =X e₀Sel(u)+u e₀Sel(X)
  =X e₀v+e₀Sel(X)
  =e₀Sel(X),

Mix(X,e₀)
  =X e₀(he₀)+e₀e₀Sel(X)
  =hX(e₀e₀)+uSel(X)
  =hX+Sel(X).

The second computation moves only the scalar h. It uses the supplied e₀e₀=u, not an inverse of the written input. Multiplying the first output on the left by e₀ gives

e₀Mix(X,u)=e₀e₀Sel(X)=Sel(X).

Subtract it from the second output, use additive inverse and association, and then multiply by k:

Mix(X,e₀)−e₀Mix(X,u)=hX+Sel(X)−Sel(X)=hX,
k[Mix(X,e₀)−e₀Mix(X,u)]=(kh)X=X.

This proves every equality in M3 by permitted written products.

4. Reconstruction, equal outputs, and the retained remainder

If both outputs in M4 are v, M3 gives X=k(v−e₀v)=v by absorption and the additive identities. Conversely, substituting X=v in MixOps gives both outputs v. This proves both directions of M4.

If X and Y have the equal pair of outputs in M5, substitute those two equalities in the recovery formula M3 for X. The resulting written expression is exactly M3's recovery formula for Y, so X=Y. Conversely, substitution of X=Y into the two Mix expressions gives equal outputs. No equality of separately assumed coefficient presentations is needed.

PB-0007's selector recovery is A_X=k(Sel(X))̃. Substitution of M3's selector recovery gives M6. That parent also gives the exact remaining part H_X=X−A_X. M3 and M6 therefore recover the full pair of constructed parts as well as X itself; no part is suppressed in this recovery.

Under Sel(X)=v, M3's second output becomes hX+v=hX, proving M7. If this output were v, multiplication by k would force X=v. Thus a nonvoid selector-null input is not lost by the second algebraic output. That conclusion uses the actual undoing of h only.

What is not forced

Mix is a symmetric algebraic two-input expression. The derivative contraction L is additive but is not asserted to be linear over arbitrary supplied scalars; their derivative terms belong to their full product rule. MixOps' scalar law supplies no permission to discard those terms in L.

M3 uses the two entire written outputs. It does not replace them by two scalars. No equation R(e₀)=v is used or asserted. The axiom asserts its equation for Ψ; other supplied expressions are not solutions merely because they can be substituted in F or Mix.

M4 does not assert that Mix(X,Y)=Mix(X,Y′) gives Y=Y′ for a fixed X. M5 concerns the specified pair of output maps together. It does not assert that the two specified outputs are the unique or minimal possible reconstruction.

Selector-nullness, self-source vanishing, equality of a specified mixed output, and absence from every mixed output are different statements. M3–M7 give their displayed exact relationships without identifying them. In particular no cancellation of an arbitrary written element is used to infer Sel(X)=v from F(X)=v.

What it means

The source’s two-input dependence retains enough information to recover the whole supplied expression. Substitution of unity as the second input recovers exactly the selector information; substitution of e₀ then supplies what is needed to reconstruct the remainder. Both outputs are complete written elements, and their displayed combination returns the original input. Selector-nullness therefore does not mean absence from the mixed source. The result is an algebraic reconstruction available before imposing an equation or a support condition on an input.

The complete change of the source and residual, with both composite signs (V07-PB-0042)

LAYER: 4 FORCED BY: V01-PB-0001, V01-PB-0006, V01-PB-0007

Declares

X and Δ are supplied full written elements; E and Z are actual supplied elements at their conditional use. Sel,D,L,F,R,Q,h,k retain V01-PB-0006's definitions. T,A and K_A(X)=e₀Ã_X retain V01-PB-0007's definitions. The names u,v abbreviate 04's unity and void; B is its ordered e₁e₂. The inverse k is used under the axiom's existing nonvoid guard on h.

The sign σ ranges over the actual two-expression listing u,−u. An element Δ is called right-null at a specified K when ΔK=v. In this record that factor is K_A(X) with the base X retained. This name abbreviates that ordered-product condition only.

What is forced

For every supplied X,Δ, without a sign or equation hypothesis,

F(X+Δ)−F(X)
  =Q(X,Δ)
  =hΔK_A(X)+h(X+Δ)K_A(Δ),
R(X+Δ)−R(X)
  =h(DΔ)B−hΔK_A(X)−h(X+Δ)K_A(Δ).                 (FC1)

If T(Δ)=σΔ, with σ=u or σ=−u, then

Sel(Δ)=(u−σ)Δ̃,
A_Δ=k(u−σ)Δ,        K_A(Δ)=k(u−σ)e₀Δ̃,
F(X+Δ)−F(X)
  =hΔK_A(X)+(u−σ)(X+Δ)e₀Δ̃,
R(X+Δ)−R(X)
  =h(DΔ)B−hΔK_A(X)−(u−σ)(X+Δ)e₀Δ̃.              (FC2)

Under both retained hypotheses T(Δ)=σΔ and ΔK_A(X)=v,

F(X+Δ)−F(X)=(u−σ)(X+Δ)e₀Δ̃,
R(X+Δ)−R(X)=h(DΔ)B−(u−σ)(X+Δ)e₀Δ̃.              (FC3)

Under T(Δ)=Δ alone, A_(X+Δ)=A_X. Under both T(Δ)=Δ and ΔK_A(X)=v, the source and residual comparisons are

F(X+Δ)=F(X),
R(X+Δ)−R(X)=h(DΔ)B,
R(X+Δ)=R(X)  iff DΔ=v.                              (FC4)

Still under both hypotheses, if also R(X)=v, then

R(X+Δ)=v  iff DΔ=v.                                 (FC4s)

Under both T(Δ)=−Δ and ΔK_A(X)=v, the negative-sign member is

F(X+Δ)−F(X)=h(X+Δ)e₀Δ̃,
F(X+Δ)=F(X)  iff (X+Δ)e₀Δ̃=v,
R(X+Δ)=R(X)  iff (DΔ)B=(X+Δ)e₀Δ̃.                (FC5)

Still under both negative-sign hypotheses, if also R(X)=v, then

R(X+Δ)=v  iff (DΔ)B=(X+Δ)e₀Δ̃.                     (FC5s)

For either σ, an actual representation Δ=T(E)ZE, with T(Z)=σZ and EK_A(X)=v, supplies both hypotheses of FC3. (FC6)

The forcing

1. Apply the full change identity at its actual arguments

For any supplied Y, V01-PB-0007 gives Sel(Y)=hÃ_Y. Its definition of K_A(Y), association and scalar commutation therefore give, for any supplied P,

e₀Sel(Y)=e₀(hÃ_Y)=h(e₀Ã_Y)=hK_A(Y),
Pe₀Sel(Y)=P[hK_A(Y)]=hPK_A(Y).

Apply V01-PB-0006(1) at its free pair (X,H)=(X,Δ). There is no selector-null or base-equation hypothesis at that parent identity:

F(X+Δ)=F(X)+Q(X,Δ),
R(X+Δ)=R(X)+L(Δ)−Q(X,Δ).

Add the negative of the old output on each side and use V01-PB-0001's additive laws. The results are the complete differences Q(X,Δ) and L(Δ)−Q(X,Δ). Now substitute the displayed selector-factor identity at each of the three already present ordered terms of Q:

Q(X,Δ)
  =Xe₀Sel(Δ)+Δe₀Sel(X)+Δe₀Sel(Δ)
  =hXK_A(Δ)+hΔK_A(X)+hΔK_A(Δ)
  =hΔK_A(X)+h(X+Δ)K_A(Δ).

The last step reorders additive summands and reverses distribution for the two terms with the same right factor. It exchanges no written product operands. This proves the source expressions in FC1. Substitute the same Q into the parent's residual identity and use the definition L(Δ)=h(DΔ)B. This gives FC1's residual expression. The source expansion is the one proved in V01-PB-0006; here its three terms are expressed through the specified source factor.

2. Retain the sign while resolving the selector factor

Keep T(Δ)=σΔ. V01-PB-0007(M1) gives (T(Δ))̃=Δ*. Applying its M0 scalar rule to the retained equality gives

Δ*=(T(Δ))̃=(σΔ)̃=σΔ̃,
Sel(Δ)=Δ̃−Δ*=(u−σ)Δ̃.

The definition of A and V01-PB-0001's sign and distribution laws give

A_Δ=k[Δ−T(Δ)]=k[Δ−σΔ]=k(u−σ)Δ.

Reversion fixes the scalar factor, so the actual K_A definition gives

K_A(Δ)=e₀Ã_Δ=k(u−σ)e₀Δ̃.

Substitute this identity into FC1 and move only its scalar factor. 04's guarded inverse gives hk=u, and rule 8 gives kh=hk=u. Using hk=u yields exactly the source and residual identities in FC2. If also ΔK_A(X)=v, its h multiple is void by absorption, giving FC3. The ordered product (X+Δ)e₀Δ̃ remains unchanged.

3. Evaluate the complete comparison at the positive sign

For σ=u, self-subtraction gives u−σ=v. FC2 gives Sel(Δ)=v and A_Δ=v. V01-PB-0007(SF1) therefore gives

A_(X+Δ)=A_X+A_Δ=A_X+v=A_X.

This step uses only T(Δ)=Δ. Retain also ΔK_A(X)=v. FC3 at σ=u, with u−σ=v and absorption, gives directly

F(X+Δ)−F(X)=v,
R(X+Δ)−R(X)=h(DΔ)B.

PB-0001's difference criterion gives F(X+Δ)=F(X) and makes equality of the residuals equivalent to h(DΔ)B=v.

For the reverse implication in the required derivative criterion, DΔ=v gives h(DΔ)B=v by absorption. For its forward implication, multiply h(DΔ)B=v on the left by k and on the right by −B. PB-0006(B0) and the existing scalar inverse give

k[h(DΔ)B](−B)=(kh)(DΔ)[B(−B)]=(DΔ)u=DΔ,
kv(−B)=v.

Thus the two residuals are equal exactly when DΔ=v. Substituting the additional retained premise R(X)=v proves FC4s in both directions.

4. Retain the whole negative-sign source

For σ=−u, V01-PB-0001(11) gives u−σ=u−(−u)=h. Under the two FC3 hypotheses its source and residual identities become

F(X+Δ)−F(X)=h(X+Δ)e₀Δ̃,
R(X+Δ)−R(X)=h[(DΔ)B−(X+Δ)e₀Δ̃].

PB-0001(13) makes equality of the sources equivalent to the voidness of the first difference. If that difference is void, multiply by k to obtain (X+Δ)e₀Δ̃=v; the converse follows by multiplication by h and absorption. This proves FC5's source criterion.

For the residual criterion, the same difference rule and multiplication by k make equality of the residuals equivalent to

(DΔ)B−(X+Δ)e₀Δ̃=v.

PB-0001(13) then gives (DΔ)B=(X+Δ)e₀Δ̃. Conversely this equality makes the bracket void, hence the residuals equal. Substitution of R(X)=v proves FC5s. Distribution, with the order retained, gives

(X+Δ)e₀Δ̃=Xe₀Δ̃+Δe₀Δ̃.

This identifies the last contribution as a written part of the full product; it makes no separate nonvoid assertion about that contribution.

5. Apply the comparisons to actual represented differences

Retain the actual hypotheses of FC6. V01-PB-0007's product reversal, involution law and scalar rule give

T(T(E)ZE)
  =T(E)T(Z)T(T(E))
  =T(E)(σZ)E=σT(E)ZE=σΔ.

Association and the retained right-null condition give

ΔK_A(X)=T(E)Z[EK_A(X)]=T(E)Zv=v.

Thus FC3 applies, and FC4 applies when σ=u. The proofs of FC3–FC5 used only their displayed hypotheses on Δ, not E or Z. An actual difference for which those hypotheses have been established directly has the same comparisons. No representation is inferred from them.

What is not forced

FC1 is unrestricted. FC2 needs the displayed T sign; FC3 additionally needs right-nullness at the specified K_A(X). Each solution comparison also retains R(X)=v. The source and residual criteria use the guarded scalar inverse to undo h, and PB-0006(B0) supplies the actual product undoing in FC4. FC6 applies only to an actual supplied representation with its stated hypotheses; it asserts no existence of E or Z. None of these claims removes Δe₀Δ̃ from the negative-sign member.

What it means

The complete change identity supplies one comparison before either sign restriction. The selector sign determines which terms survive in that comparison, and right-nullness removes one specified product. Equal sources and equal residuals then have their own exact conditions. The negative-sign comparison retains the source contribution of the difference itself.

The positive-sign comparison is the selector-null comparison of V01-PB-0006(3)–(4) at Y=X+Δ: T(Δ)=Δ makes Sel(Δ)=v, and the additional right-null condition removes Δe₀Sel(X)=hΔK_A(X). Here FC3 already supplies the same comparison by evaluating u−σ=v. Its residual equality criterion needs both conditions; the solution criterion FC4s additionally needs R(X)=v. FC5s carries the corresponding negative-sign conditions and the same base-equation premise.

The complete source pairing fixes its sharp output-support boundary and an unsourced derivative balance (V01-PB-0009)

LAYER: 5 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0005, V01-PB-0006, V01-PB-0007, V01-PB-0008

Declares

All definitions abbreviate already permitted written expressions or finite lists. Equality is used symmetrically, transitively, and by substitution in written expressions with the actual hypotheses retained. The declared nonvoid condition on h excludes equality with void.

• u=unity, v=void, h=u+u, k=h⁻¹ under the input's existing guard; hk=kh=u.

• G, 𝓑, e_I, χ(I,J), η(I), θ(I), τ(I), w_I, 𝓒_I, ρ, λ_m, ε_m, β, f_K, ℓ_K, Sel, T, D, L, F, R retain their cited parent definitions. I,J,K are entries of 𝓑 and μ is a member of the supplied G listing. In particular τ(I)=η(I)θ(I), w_I=η(I)−θ(I), T(X)=(X̃)*, F(X)=Xe₀Sel(X), L(X)=hD(X)B.

• x_I=𝓒_I(X), for any supplied written X. These are its recovered coefficients, not newly independent scalar variables.

• I_s={1,2,3,4,5,6}, I_all=G, I_perp={3,4,5,6}. Put Ω_s=e_{I_s} and Ω_all=e_{I_all}. The words are fixed by the axiom's order. These names supply no physical volume or geometric interpretation.

• For a target K put C={0}△K, where △ is the already derived symmetric-difference index operation. The letter C here denotes this local index only, not PB-0004's subscripted conjugation expression C_m.

• s(I,J):=χ(I,{0})χ(I△{0},J).

• A distinct-pair list at K contains (I,J) with J=I△C, taking only the member I that occurs earlier than J in the already fixed finite 𝓑 listing. This is list order, not an order on coefficients. It is constructed from all matching ordered pairs, not by a field-support restriction.

• A sign is one of u,−u. Every product of signs below is evaluated by the four products supplied by PB-0001. No count is used as a scalar exponent.

• X_K and s_K^w name the witness and its coefficient sign constructed in §4. Local m is the specified generator label, and q is the sign χ({m},C) there. Neither is chosen by trial.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, and with a declared nonvoid condition excluding equality with void, the following hold for every supplied written X. The inverse k is used only under the input's existing nonvoid guard on h.

Paired source interface. For I△{0}△J=K,

e_I e₀ e_J = s(I,J)e_K,
s(J,I) = s(I,J)η(K)η(I)η(J).                         (S1)

At K={0}, the full source coefficient is the diagonal accumulation

f_{ {0} }(X) = Σ_I x_I x_I w_I s(I,I).                 (S2)

At every other K, it is the accumulation over the declared distinct-pair list of

x_I x_J s(I,J)[w_J+η(K)η(I)η(J)w_I]
  = x_I x_J s(I,J)η(J)
      [(u−τ(J))+η(K)(u−τ(I))].                      (S3)

The bracketed weight in the first line of (S3), before multiplication by x_Ix_J s(I,J), is exactly

τ(I)=u,  τ(J)=u:    v;
τ(I)=u,  τ(J)=−u:   hη(J);
τ(I)=−u, τ(J)=u:    hη(K)η(J);
τ(I)=−u, τ(J)=−u:   hη(J)(u+η(K)).                  (S4)

The last case is hhη(J) when η(K)=u and v when η(K)=−u. All original ordered occurrences are retained by (S2)–(S4); diagonal terms are not doubled.

Complete-word and complementary-sign interface.

XΩ_all=Ω_all X,
Ω̃_s=−Ω_s,      Ω_s*=Ω_s,
Ω̃_all=−Ω_all,  Ω_all*=−Ω_all,  T(Ω_all)=Ω_all,
η(I_s)=−u, η(I_all)=−u, θ(I_s)=u, θ(I_all)=−u,
τ(I_all)=u,   τ(I△I_all)=τ(I).                       (S5)

Sharp source-output support.

f_{I_s}(X)=𝓒_{I_s}(F(X))=v.                         (G1)

For each K other than I_s, §4 gives a definite constructed X_K, using only unity and supplied generators, for which

f_K(X_K)=h s_K^w,   s_K^w a specified sign,   h s_K^w≠v.     (S6)

For K distinct from {0} and I_s, the same construction has the following two forms, with C={0}△K:

τ(C)=−u:  X_K=u+e_C;
τ(C)=u:   X_K=e_I+e_J, I≠J, τ(J)=−τ(I),
          I={m}, J={m}△C with m prescribed in §4b.       (S6p)

The same witnesses have the following specified signs, with the same C, I and J as in S6p:

τ(C)=−u:                    s_K^w=η(C)s(∅,C);
τ(C)=u, τ(I)=u, τ(J)=−u:    s_K^w=η(J)s(I,J);
τ(C)=u, τ(I)=−u, τ(J)=u:    s_K^w=η(I)s(J,I).       (S6s)

The superscript is a name label, not a power. No claim of independently assignable simultaneous source coefficients follows.

Unrestricted derivative coefficient. For every supplied X, with x_I=𝓒_I(X),

ℓ_{I_s}(X)=𝓒_{I_s}(L(X))
  =h[∂₀x_{03456}+∂₁x_{13456}+∂₂x_{23456}
      −∂₃x_{456}+∂₄x_{356}−∂₅x_{346}+∂₆x_{345}].     (G0)

No equation on X is required. Under the existing guard on h, the left side is v exactly when the whole bracket is v. This is one equality of collected expressions, not seven separately vanishing derivatives.

For every supplied X, the same collected expression is also its residual coefficient:

𝓒_{I_s}(R(X))=ℓ_{I_s}(X),                            (G0r)

with ℓ_{I_s}(X) equal to h times the complete seven-term bracket in G0. Thus the voidness of that one residual coefficient is exactly the voidness of that bracket, without a full-equation premise.

Actual-field consequence. Put x_I=𝓒_I(Ψ). Then

∂₀x_{03456}+∂₁x_{13456}+∂₂x_{23456}
  −∂₃x_{456}+∂₄x_{356}−∂₅x_{346}+∂₆x_{345}=v.       (G2)

The concatenated subscripts stand for the listed index sets, not scalar numbers. More generally (G2), with x_I=𝓒_I(X), follows under the retained hypothesis R(X)=v. It is one consequence, not a sufficient replacement for the complete equation.

The forcing

1. Pair the complete source, not a selected field

For I△{0}△J=K, apply PB-0005(4) first at (I,{0}), then at (I△{0},J). Association and movement of only the scalar sign give

e_Ie₀e_J
  =[χ(I,{0})e_{I△{0}}]e_J
  =χ(I,{0})χ(I△{0},J)e_{(I△{0})△J}
  =s(I,J)e_K.

This proves S1's first line before either reversal uses it.

PB-0008(P5) gives, including all input coefficients,

f_K(X)=𝓒_K(F(X))
  =Σ_{I,J : I△{0}△J=K} x_I x_J w_J s(I,J).

Fix K and put C={0}△K. The index condition is exactly J=I△C: for each label, the two memberships in I,C specify whether it is in J. Applying the same toggle again recovers I. Thus at C=∅ the pairs are precisely I=J; at nonempty C there is no fixed pair and every matching ordered pair belongs to exactly one distinct unordered pair. Selecting its earlier member in 𝓑 gives one representative without loss or duplication.

For the diagonal target K={0}, keep the original formula

f_{ {0} }(X)=Σ_I x_I x_I w_I s(I,I).

It must not be doubled by the distinct-pair argument. At each other K, use commutation of scalar coefficients and additive associativity/commutation to combine the two actual contributions:

x_Ix_J w_J s(I,J)+x_Jx_I w_I s(J,I)
  =x_Ix_J[s(I,J)w_J+s(J,I)w_I].

Use PB-0008(P0) at I,J,K, including any empty index. The reversion product rule supplies the relative sign without another multiplication table. Reverse the actual e_Ie₀e_J product in two ways:

(e_Ie₀e_J)̃ = η(J)η(I)e_Je₀e_I
            =η(I)η(J)s(J,I)e_K,
(s(I,J)e_K)̃=s(I,J)η(K)e_K.

PB-0005 cancels the same final blade. Multiply the resulting scalar equality by η(I)η(J), whose self-product is u. Hence

s(J,I)=s(I,J)η(K)η(I)η(J).

The paired contribution is therefore exactly

x_Ix_J s(I,J)[w_J+η(K)η(I)η(J)w_I]                 (P)

or, since w_I=η(I)(u−τ(I)),

x_Ix_J s(I,J)η(J)
   [(u−τ(J))+η(K)(u−τ(I))].                       (P')

Here w_I=η(I)(u−τ(I)) follows by distributing and using η(I)η(I)=u in τ(I)=η(I)θ(I). No coefficient is cancelled or divided.

For each distinct pair, (P') gives the whole classification:

• Both τ signs u: the pair contributes v.
• τ(I)=u and τ(J)=−u: the weight in (P) is hη(J).
• τ(I)=−u and τ(J)=u: the weight is hη(K)η(J).
• Both τ signs −u: the weight is hη(J)(u+η(K)); this is hhη(J) if η(K)=u, and v if η(K)=−u.

These are four exhaustive algebraic cases on the two derived signs. Each paired formula retains the coefficient product x_Ix_J beside its weight. The claims make no separate nonvoidness assertion about their product.

The finite regrouping is licensed before using it: begin with PB-0008's exact ordered-pair list; take each earlier pair member in turn, move its reversed occurrence beside it by PB-0001's additive commutation/association, and use the displayed combination. Different pairs involve disjoint occurrence positions. After the prescribed pair list is exhausted every original occurrence has been used exactly once. An empty pair accumulator begins at v. This supplies the sum of (P), not merely a count of its terms.

2. The complete generator word is central and selector-null

Move a generator e_m through Ω_all by PB-0003's contextual exchange. Its own occurrence requires no exchange; it crosses each of the other six distinct generators once. At the identical occurrence, keep that same adjacent pair and continue moving the other named occurrence through the remaining suffix; no generator square is removed in this commutation calculation. The sign is the product of six copies of −u, which may be grouped as

((−u)(−u))((−u)(−u))((−u)(−u))=u.

Thus e_mΩ_all=Ω_all e_m for each of the seven labels. Moving through each actual blade word and then distributing through each actual finite scalar-coefficient presentation proves

XΩ_all=Ω_all X

for every supplied X. Scalars commute by rule 8. This is a finite generator-word calculation, not an imported structural theorem.

Reversal of Ω_s reaches the increasing word with sign −u. The destination procedure moves label 1 across 5 factors, then 2 across 4, then 3 across 3, then 4 across 2, then 5 across 1, and the last across none. These are the actual finite position lists. Their respective sign products are −u,u,−u,u,−u,u, whose product is −u. Therefore

Ω̃_s=−Ω_s,       Ω_s*=Ω_s.

For Ω_all, the corresponding successive crossing lists have lengths 6,5,4,3,2,1,0. Their sign products are u,−u,u,−u,u,−u,u, giving −u. Star contributes the one negative from e₀. Consequently

Ω̃_all=−Ω_all,    Ω_all*=−Ω_all,
T(Ω_all)=Ω_all.

Counts in this paragraph locate finite lists of exchanges. Each sign is obtained by the specified multiplication of −u factors, pairing neighboring factors when available. No numerical exponent, repeated differentiation or extra coefficient declaration occurs.

3. Complementary source inputs cancel at Ω_s

For K=I_s, the pair index in §1 is C=I_all. Thus J=I△I_all is exactly the complementary blade index. There are no diagonal occurrences.

We first prove that complementary blades have the same T sign. PB-0005 gives

e_I Ω_all=χ(I,I_all)e_J.

Apply T, using PB-0007's reversed product order and scalar fixing, PB-0008(P0) at I, and the centrality and T(Ω_all)=Ω_all just established:

T(e_IΩ_all)=T(Ω_all)T(e_I)
           =Ω_all τ(I)e_I
           =τ(I)e_IΩ_all.

Applying T to the other presentation, using PB-0008(P0) at J, gives χ(I,I_all)τ(J)e_J. Substitution, cancellation of e_J by PB-0005, and multiplication by the sign χ(I,I_all) give

τ(J)=τ(I).

At K=I_s, η(K)=−u by §2. Equation (P') then has the bracket

(u−τ(J))−(u−τ(I))=v.

Every complementary pair therefore contributes v, independently of the supplied coefficients. Summing the actual complete pair list gives f_{I_s}(X)=v. This proves (G1).

The cancellation includes the cases where both individual contributions are nonvoid. It is not an empty occurrence list. For example the complementary indices I={0}, J=I_s give

w_{I_s}=−h,  e₀e₀e_{I_s}=e_{I_s},
w_{ {0} }=h, e_{I_s}e₀e₀=e_{I_s}.

For the supplied X=e₀+Ω_s their two I_s coefficients are −h and h. Their sum is v although neither equals v. This example follows the proved general pairing; it is not its basis and is not offered as Ψ.

4. Positive completeness: every other output index has a derived witness

At K={0}, take the supplied X_K=e₀. The actual generator images and arithmetic give

Sel(e₀)=e₀−(−e₀)=he₀,
F(e₀)=(e₀e₀)(he₀)=he₀.

PB-0004's recovery gives its {0} coefficient h, which is nonvoid under the input's existing guard; here s_K^w=u.

For all other K except I_s, C={0}△K is a nonempty proper subset of G. We construct a witness using the derived sign predicates; no search over supports or values occurs.

4a. C has τ(C)=−u

Take X_K=u+e_C. These are supplied expressions with coefficient u, not new scalar declarations. Its only self-pair outputs are at {0}, because I△{0}△I={0}. Its cross-pair output is K. Since Sel(u)=v and w_C=hη(C), the K coefficient is

hη(C)s(∅,C).

Here s_K^w=η(C)s(∅,C), a sign. This is nonvoid: equality with v, followed by multiplication by that sign, would imply h=v. Hence this derived witness establishes the required nonvoid K coefficient.

4b. C has τ(C)=u

A nonempty proper C anticommutes with at least one supplied generator. This can be located without a trial:

• If ρ(C)=u, choose the first label m in C. Its one −u factor can be carried to the front of the finite scalar accumulation, giving ρ(C)=(−u)ρ(C∖{m}). Multiply by −u and use (−u)(−u)=u. Thus λ_m(C)=ρ(C∖{m})=(−u)ρ(C)=−u.
• If ρ(C)=−u, choose the first label m outside C. That label exists because C is proper. Then C∖{m}=C and λ_m(C)=ρ(C)=−u directly from its definition.

Both choices are from the fixed generator/list order; neither is an order or nonvoid test on an input coefficient. In either case PB-0004's exported comparison law gives ε_m e_m e_C e_m=−e_C. Multiply on the right by e_m, use e_me_m=ε_m, move only the scalar ε_m, and use ε_mε_m=u. The resulting equality is

e_m e_C=−e_C e_m.

Put I={m} and J={m}△C. PB-0005 gives e_m e_C=χ({m},C)e_J. If q=χ({m},C), q is a sign with qq=u, and therefore

e_J=q e_m e_C.

PB-0007 gives the reversed product order of T. PB-0008(P0) at C and I={m} gives T(e_C)=e_C and T(e_m)=τ(I)e_m. Hence

T(e_J)=q T(e_C)T(e_m)
      =qτ(I)e_Ce_m
      =−qτ(I)e_me_C
      =−τ(I)e_J.

Comparing with T(e_J)=τ(J)e_J from PB-0008(P0) at J and cancelling that same blade gives τ(J)=−τ(I). Thus precisely one of I,J is in the selector-nonsilent list and the other is selector-silent. The signs are distinct here: u=−u would give h=u+u=u+(−u)=v by rule 3, contrary to the input's declared nonvoid guard on h.

Take X_K=e_I+e_J. The two indices are different because C is nonempty. Its two self-pair outputs are at {0}, not at K. Its two cross-pair outputs are at K. Exactly one of their selector weights is v; the other is h times a sign. Their sum at K is consequently h times a sign and is nonvoid by the argument of §4a. Explicitly, use

s_K^w=η(J)s(I,J) when τ(I)=u;
s_K^w=η(I)s(J,I) when τ(J)=u.

Exactly one case applies, as just proved. Together with §4a this establishes S6s. These are the weights of the actual surviving ordered occurrences, not a coefficient sign selected after evaluation.

This exhausts all derived K: {0}, I_s, and the remaining indices distinguished by τ(C). We have proved both that every source coefficient at I_s vanishes and that no other blade coefficient vanishes for all supplied X. There is no assertion about arbitrary simultaneous coefficient values, the full nonlinear image, or which witnesses satisfy R(X)=v.

5. The whole derivative coefficient before the equation premise

Start with arbitrary supplied X and x_I=𝓒_I(X). PB-0008(P5) gives

ℓ_{I_s}(X)=h Σ_{μ,I : ({μ}△I)△β=I_s}
   ε_μ(∂_μx_I)χ({μ},I)χ({μ}△I,β).

Its derivative index condition is

({μ}△I)△β=I_s,       β={1,2}.

Toggling β and {μ} gives exactly I={μ}△I_perp. There is one such input index for each declared derivative. The signed blade products are the following; the middle column contains the factor ε_μ that is already in D:

μ  I                    ε_μ e_μ e_I          (ε_μ e_μ e_I)B
0  {0,3,4,5,6}          e_{3456}             Ω_s
1  {1,3,4,5,6}          e_{3456}             Ω_s
2  {2,3,4,5,6}          e_{3456}             Ω_s
3  {4,5,6}              −e_{3456}            −Ω_s
4  {3,5,6}              e_{3456}             Ω_s
5  {3,4,6}              −e_{3456}            −Ω_s
6  {3,4,5}              e_{3456}             Ω_s.

Here e_{3456} abbreviates e_{ {3,4,5,6} }, and the other concatenated subscripts abbreviate their explicitly listed index sets, not scalar numbers.

Each row follows the word itself. In the first row e₀e₀=u. In the next two rows e_me_m=−u cancels the already present ε_m=−u. For μ=3 the word is already increasing and only ε_3 remains. For μ=4 the appended leading e₄ crosses e₃ once, cancelling ε_4; for μ=5 it crosses e₃,e₄, retaining ε_5; for μ=6 it crosses e₃,e₄,e₅, cancelling ε_6. Finally e_{3456}B=e_{3456}e₁e₂=Ω_s: each of e₁ and e₂ crosses the four-element list 3,4,5,6, with sign product u in each pass.

The two χ signs are precisely the two successive blade reductions of the original word (ε_μe_μe_I)B. PB-0005's route independence and cancellation of the common output blade identify their product, including ε_μ, with the sign of each row. The toggling equation leaves exactly one I in each of the seven μ rows and no additional input occurrences. Substitution into the complete accumulation therefore gives

ℓ_{I_s}(X)
  =h[∂₀x_{03456}+∂₁x_{13456}+∂₂x_{23456}
      −∂₃x_{456}+∂₄x_{356}−∂₅x_{346}+∂₆x_{345}].     (G0)

If this coefficient is v, multiplying by k gives a void whole bracket, using kh=u. If the whole bracket is v, multiplying by h gives the coefficient v by absorption. Neither implication requires R(X)=v.

For the same unrestricted X, PB-0008(P5) and G1 give

𝓒_{I_s}(R(X))=ℓ_{I_s}(X)−f_{I_s}(X)=ℓ_{I_s}(X)−v=ℓ_{I_s}(X).

This proves G0r and gives its two-way void criterion through the already proved h undoing. Now retain R(X)=v. PB-0008(P6) and (G1) give ℓ_{I_s}(X)=f_{I_s}(X)=v. Apply (G0) and its guarded equivalence:

∂₀x_{03456}+∂₁x_{13456}+∂₂x_{23456}
  −∂₃x_{456}+∂₄x_{356}−∂₅x_{346}+∂₆x_{345}=v.       (G2)

No constituent derivative has been set to v. This is the collected exact balance, not seven separate equations. No source term has been omitted as small: the complete paired source at this particular output was proved to cancel.

The input asserts R(Ψ)=v, so substituting Ψ for X supplies the displayed actual-field consequence with x_I=𝓒_I(Ψ). For any other X the stated equation premise remains. Conversely (G2) supplies only this one component of the complete equation, not the remaining component equalities.

What is not forced

The coefficient-support statement is about the complete operation F on supplied written expressions. Its positive witnesses have a nonvoid coefficient at their specified target; they are not asserted solutions of R(X)=v and do not supply arbitrary simultaneous values of all other source coefficients. No support restriction, independent scalar field variables, scalar order, positivity, or additional nonvoid declaration has been introduced.

The universal vanishing in (G1) is a relation at one recovered output coefficient. It does not make Ω_s an absent input, does not identify an input coefficient with void, and does not erase the nonvoid occurrences that cancel in the complete ordered sum. The exact pair weight is not a nonvoidness theorem about its product with arbitrary coefficients. S4 states the weight of each paired contribution; the coefficient is the complete accumulation of those contributions.

G0 identifies the collected left coefficient for arbitrary X; it does not assert its voidness. The actual-field consequence uses R(Ψ)=v and keeps the seven contributing derivatives in their single collected balance. It does not assert that those derivatives vanish separately, supply the other component equations, or solve the full system. The role of B in that derivative routing is retained.

The listings select written occurrences and signs, not arbitrary-arity scalar operations. The only scalar inverse used is k under the input's guard on h; cancellation of a blade uses the parent's proved undoing. This remains exactly the declared equality/nonvoid scope; it adds no new scalar primitive or guard.

What it means

The full source has two distinct kinds of routing: its e₀ coefficient is diagonal in the two input indices, while every other output collects paired off-diagonal contributions. Reversion relates the signs of the two ordered contributions, so their addition can be understood before any particular input is chosen.

The complete-generator complement has equal composite-involution signs. At the all-spatial output, the pair rule therefore cancels every contribution in its own pair. The boundary is sharp: for each output K other than I_s, the constructed generator-built input X_K reaches that output with a nonvoid coefficient. This is a statement about which individual outputs are possible, not a solution or surjectivity theorem for the nonlinear source.

When the asserted equation is used, that universally void source coefficient, cancelled pair by pair, obliges a specific derivative-side balance. The balance is obtained from the full equation and the declared ordered B, rather than by dropping terms or restricting the field. The separate relation between source reversion and the constructed parts is supplied next.

Each supplied derivative respects the two involutions and their constructed parts (V02-PB-0011)

LAYER: 5 FORCED BY: V01-PB-0001, V01-PB-0004, V01-PB-0007, V01-PB-0008

Declares

The scalar abbreviations u := unity, v := void, h := u+u, and k := h⁻¹ retain their 04 meanings. The inverse k uses only the input's existing nonvoid guard on h. The finite blade listing and θ(I) retain PB-0004's definitions; η(I) and τ(I) retain PB-0008's definitions. The signs are produced by their written finite constructions, not new scalar parameters.

T(W) := (W̃)*, A_W := k(W−T(W)), and H_W := k(W+T(W)) are exactly PB-0007's constructed parts. W is any supplied written element, a is any supplied scalar expression, and μ is one of the seven supplied derivative labels. Where a scalar inverse a⁻¹ is used, that expression and its actual nonvoid guard and identity aa⁻¹=u are retained hypotheses of that instance. No such inverse is asserted for an arbitrary a. All occurrences of a derivative below are the operations of 04.

G=(0,1,2,3,4,5,6) retains PB-0004's supplied generator-label list. D retains PB-0006's derivative-contraction definition; b_{{ν}} retains PB-0002's blade-undoing definition at the singleton {ν}. Each displayed sum over ν is the accumulation over this actual seven-position list, starting at v.

In the supplied finite presentation W=Σ_I a_I e_I, each a_I is its actual scalar-expression coefficient; this presentation need not be a recovered-coefficient presentation. The letters c,d in the finite sign calculation denote the actual derivative-constant scalar factors at that step. W₀ denotes an actual generator-word prefix; e_ν is its next generator when that prefix is extended.

What is forced

Under equality used symmetrically, transitively and by substitution with the actual hypotheses retained, and with a declared nonvoid condition excluding equality with void, each supplied derivative satisfies

∂_μ(−W) = −∂_μW,
∂_μu = ∂_μv = ∂_μh = ∂_μk = v.                    (D1)

For an actually supplied scalar inverse under its retained guard,

∂_μ(a⁻¹) = −a⁻¹(∂_μa)a⁻¹.                         (D2)

For every supplied written W the two involutions and the constructed parts have the exact compatibility identities

∂_μ(W̃) = (∂_μW)̃,
∂_μ(W*) = (∂_μW)*,
∂_μ(T(W)) = T(∂_μW),                              (D3)

∂_μ(A_W) = A_(∂_μW),
∂_μ(H_W) = H_(∂_μW).                              (D4)

Every actual blade and each of its derived signs has void derivatives:

∂_μe_I=v,    ∂_μη(I)=∂_μθ(I)=∂_μτ(I)=v.            (D6w)

For supplied full X,K, using the singleton undoing expressions b_{{ν}},

D(XK)=D(X)K+Σ_(ν∈G)b_{{ν}}X(∂_νK).                 (D7p)

For supplied full X,Y under the actual equality XY=v,

(∂_μX)Y+X(∂_μY)=v.                                  (D7z)

The sum in D7p has exactly the seven supplied derivative labels. Neither D7p nor D7z moves a generator through X.

Every statement applies to the full written element. It supplies no component restriction on the distinguished Ψ. D3–D4 concern each individual derivative; D7p separately keeps every generator factor of the full contraction.

The forcing

1. Additive inverse and the fixed scalar derivatives

PB-0001 supplies additive identity, inverse uniqueness, cancellation, product-sign rules, and void absorption; 04 rule 8 supplies scalar centrality. The scalar and blade derivative identities are established below from the generator rules and the product rule. First,

∂_μu = ∂_μ(e₀e₀)
      = (∂_μe₀)e₀+e₀(∂_μe₀)
      = ve₀+e₀v = v.

The two last terms are derived voids by absorption, using the asserted generator derivatives. Substitution of v=v+v into additivity gives

∂_μv = ∂_μv+∂_μv.

Compare with ∂_μv=∂_μv+v and cancel the common written summand to obtain ∂_μv=v. Differentiate the established additive identity W+(−W)=v:

∂_μW+∂_μ(−W)=v.

Inverse uniqueness gives ∂_μ(−W)=−∂_μW. The void here comes from differentiating that identity, not from any separate readout operation. Additivity now gives

∂_μh = ∂_μ(u+u)=v+v=v.                            (D-a)

2. Differentiating an actual scalar inverse

Retain the particular identity aa⁻¹=u together with its guard. 04's product rule and D1's unity derivative give

(∂_μa)a⁻¹+a∂_μ(a⁻¹)=v.

Multiply this whole equality on the left by the written scalar a⁻¹. Distribute and associate, and use only a⁻¹a=u (scalar centrality applied to the supplied inverse identity):

a⁻¹(∂_μa)a⁻¹+∂_μ(a⁻¹)=v.

The additive inverse law therefore yields D2. No multiplicative factor has been cancelled without its displayed undoing expression. Taking a=h and a⁻¹=k, whose guard is in 04, gives hk=u by the declared inversion and kh=hk=u by rule 8. Substituting D-a then gives

∂_μk = −k(∂_μh)k = −kvk = v.                      (D-b)

The last void is produced by absorption and negation of void. This establishes the remaining part of D1.

3. The finite blade signs are derivative-constant

PB-0008's blade signs η(I), θ(I), and τ(I) are obtained as the specified products of the two expressions u and −u. The first is derivative-constant by D1; the second is derivative-constant by D1's negation law. At a supplied product of two such derivative-constant expressions c,d, 04's product rule gives

∂_μ(cd)=(∂_μc)d+c(∂_μd)=vd+cv=v.

Apply this written calculation only at the actual products occurring in each sign's finite construction. Hence

∂_μη(I)=∂_μθ(I)=∂_μτ(I)=v.                        (D-c)

This conclusion concerns the signs fixed by the blade rules. It does not say that a general scalar coefficient of W is derivative-constant.

4. Reversion and star commute with each supplied derivative

First write the derivative of each blade from the supplied generator rules. The empty prefix is u, whose derivative is v by §1. At the next actual generator of a blade, if ∂_μW₀=v has been obtained, 04's product rule gives

∂_μ(W₀e_ν)=(∂_μW₀)e_ν+W₀(∂_μe_ν)
           =ve_ν+W₀v=v.

Following the actual finite generator listing therefore gives ∂_μe_I=v. Each void in this calculation is an asserted generator derivative or an output of the written absorption rule. No claim about an unprovided word or an infinite extension is used.

Use the input's own actual finite presentation W=Σ_I a_Ie_I. The same prefix product-rule calculation applies to the reversed word ẽ_I and the star word e_I*, whose actual generator factors are e_ν or −e_ν. Their derivatives are v by §1 and the supplied generator rules. Reversion and star fix the scalar expressions a_I and ∂_μa_I. Thus, using only the finite addition and product rules and scalar centrality,

∂_μ(W̃)=Σ_I (∂_μa_I)ẽ_I
         =(Σ_I (∂_μa_I)e_I)̃=(∂_μW)̃,
∂_μ(W*)=Σ_I (∂_μa_I)e_I*
         =(Σ_I (∂_μa_I)e_I)*=(∂_μW)*.

Each sum follows that same supplied finite presentation. The only removed products contain the explicitly derived void derivative of a fixed word. This proves the first two equalities of D3 without coefficient recovery.

Apply the two just-proved equalities successively to the actual composite T:

∂_μ(T(W))
  = ∂_μ((W̃)*)
  = (∂_μ(W̃))*
  = ((∂_μW)̃)*
  = T(∂_μW).

This proves the last equality of D3.

5. The part expressions commute with each supplied derivative

Use the actual definitions of A_W and H_W, not a supposition that operators commute. 04's product rule, D-b, the derivative addition/negation laws, and D3 give

∂_μ(A_W)
  = (∂_μk)(W−T(W))+k[∂_μW−∂_μ(T(W))]
  = k[∂_μW−T(∂_μW)]
  = A_(∂_μW),

∂_μ(H_W)
  = (∂_μk)(W+T(W))+k[∂_μW+∂_μ(T(W))]
  = k[∂_μW+T(∂_μW)]
  = H_(∂_μW).

In each line the first product is derived void by ∂_μk=v and absorption. The two final expressions are precisely the parent's definitions applied to ∂_μW. This proves D4.

6. Keep the complete derivative of an ordered product

PB-0008(P5u) gives D(Z)=Σ_ν b_{{ν}}∂_νZ for its seven supplied labels. Apply 04's product rule at each actual label and distribute its left factor:

D(XK)=Σ_ν [b_{{ν}}(∂_νX)K+b_{{ν}}X(∂_νK)].

Write the regrouping at its actual prefixes. Let P_pre and Q_pre be the preceding accumulations of b_{{ν}}∂_νX and b_{{ν}}X(∂_νK). At the empty prefix their values are v and v=vK+v. At the next actual label ν, additive association and commutation followed by right distribution give

(P_pre K+Q_pre)+[b_{{ν}}(∂_νX)K+b_{{ν}}X(∂_νK)]
  =[P_pre K+b_{{ν}}(∂_νX)K]+[Q_pre+b_{{ν}}X(∂_νK)]
  =(P_pre+b_{{ν}}∂_νX)K+[Q_pre+b_{{ν}}X(∂_νK)].

The two updated expressions are precisely the two prescribed accumulations. Apply this equality at the actual seven labels. Their final expressions are D(X) and the second sum in D7p, proving that claim without moving K through a factor. For D7z retain XY=v, apply ∂_μ to that whole equality, and use D1 and 04's product rule:

(∂_μX)Y+X(∂_μY)=∂_μ(XY)=∂_μv=v.

This is one equation on the two ordered terms, with the original product hypothesis retained.

What is not forced

The equation in 04 still concerns Ψ. No other W is declared to satisfy it by these operation identities. D2 retains an actual scalar inverse and its guard; it does not prove that a nonvoid expression is cancellable without such an inverse, nor introduce an inverse for a written element that is not scalar.

D3–D4 assert commutation with an individual ∂_μ. They assert no commutation with the full contraction D, which also contains left generator multiplications. For example, reversion reverses the order of those multiplications. Every such occurrence must be expanded with its own product rule.

The parts A_W and H_W remain complementary descriptions of W. D4 concerns their individual derivatives. Equations for their full residuals and their mixed source are separate expressions to which D4 supplies only the displayed operation identities.

What it means

The two-part split constructed from T is compatible with each of the seven supplied derivatives. The same part is obtained whether one first differentiates the whole expression or first forms that part and then differentiates it. This is a proved permission to move the individual derivative through the two involutions and their part formulas. The equation’s generator multiplications remain separate operations, so the full derivative contraction still has to be carried explicitly. The result connects the constructed parts to the individual derivatives without restricting Ψ.

The complete finite mixed accumulator controls composition (V02-PB-0015)

LAYER: 5 FORCED BY: V01-PB-0001, V01-PB-0006, V02-PB-0014

Declares

The scalar expressions u,v,h,k retain 04's meanings; Sel,D,L,F,R retain PB-0006's definitions. Mix(X,Y)=Xe₀Sel(Y)+Ye₀Sel(X) is exactly PB-0014's mixed-source expression. No new operator or equation is introduced.

Let an actual finite ordered list of supplied written elements be given. Its members are called W at their actual positions. A position label is a place in this supplied list, not an ordered scalar value or a premise about an infinite sequence. Nothing here declares the members separately to satisfy the equation.

Along that list define four finite written accumulators. Their initial values are

S_in=v,    U_in=v,    V_in=v,    M_in=v.

At its actual next member W, append the prescribed expressions

S_new=S_old+W,
U_new=U_old+F(W),
V_new=V_old+R(W),
M_new=M_old+Mix(S_old,W).                            (C0)

The final expressions are denoted S,U,V,M, respectively. Thus S is the actual written sum, U and V are the separate source and residual sums, and M is the complete accumulated mixed source. These are definitions of finite written expressions, not a new scalar domain or convergence rule.

When two positions p,q of the actual supplied list are compared, p<q means only that p precedes q in that list. The notation Σ_(p<q) Mix(W_p,W_q) below abbreviates the exact finite accumulation proved from C0, with the later position visited first and its preceding positions visited in their inherited order. It is not an independent summation operation.

In C4–C7, Z and E are further supplied written elements. E names the additional expression in the supplied total. Any hypothesis R(Z)=v or R(Z+W)=v is stated separately where used.

What is forced

For the actual supplied finite list, retaining every input occurrence,

M=Σ_(p<q) Mix(W_p,W_q),
F(S)=U+M,
R(S)=V−M.                                             (C1)

There is one term for each pair of distinct positions. Two positions containing equal written elements still contribute their term; a diagonal self-source F(W_p) belongs to U instead. No position is omitted because its blade support or output happens to coincide with another position.

The exact composition criterion is

R(S)=v iff V=M.                                       (C2)

Under the additional hypotheses R(W)=v for every member of this actual supplied list, it becomes

R(S)=v iff M=v.                                       (C3)

This condition is on the entire mixed accumulator. No assertion that each of its summands must be void is included.

For the further hypotheses on Z

R(Z)=v,
R(Z+W)=v for every member W of the actual supplied list, (C4)

the exact intermediate and final residuals are

R(Z+S)=−M,
R(Z+S+E)=L(E)−F(E)−Mix(Z+S,E)−M.                    (C5)

Consequently, under precisely C4, the proposed total satisfies the full equation if and only if

L(E)−Mix(Z+S,E)=M+F(E).                              (C6)

For the particular two-member list X,Y this is

L(E)−Mix(Z+X+Y,E)=Mix(X,Y)+F(E).                      (C7)

The self-source F(E) remains in C6 and C7. Neither equation asserts that such an E exists or selects one. They are exact necessary and sufficient conditions on a supplied proposed total under their displayed premises.

For the empty supplied list all four accumulators S,U,V,M are v, and

F(v)=L(v)=R(v)=v.                                      (C8)

The forcing

1. Initialize all expressions at the actual empty prefix

PB-0014 MixOps gives Mix(v,Y)=v, and M2 gives the two-input source and residual identities. The definition and absorption give F(v)=v e₀Sel(v)=v. For completeness, PB-0006's additivity also gives

L(v)=L(v+v)=L(v)+L(v).

Comparing with L(v)=L(v)+v and using PB-0001's additive cancellation gives L(v)=v. Hence

R(v)=L(v)−F(v)=v.

At the empty prefix all four accumulators in C0 are v, and therefore

F(S_in)=U_in+M_in,
R(S_in)=V_in−M_in.                                   (C8p)

These identities explicitly cover an empty supplied list as well.

2. Carry the exact source and residual equalities through each position

At the next actual member W, substitute PB-0014(M2)'s complete two-input expansion:

F(S_new)
  =F(S_old+W)
  =F(S_old)+F(W)+Mix(S_old,W).

If the previous written prefix equality is F(S_old)=U_old+M_old, substitution and additive regrouping give

F(S_new)
  =U_old+M_old+F(W)+Mix(S_old,W)
  =[U_old+F(W)]+[M_old+Mix(S_old,W)]
  =U_new+M_new.                                      (C9)

The entire residual calculation is likewise

R(S_new)
  =R(S_old+W)
  =R(S_old)+R(W)−Mix(S_old,W)
  =[V_old−M_old]+R(W)−Mix(S_old,W)
  =[V_old+R(W)]−[M_old+Mix(S_old,W)]
  =V_new−M_new.                                      (C10)

The third line substitutes the preceding written prefix equality. PB-0001's negation-of-a-sum and additive association/commutation justify the next line. There is no multiplication cancellation.

Start with C8p and attach C9–C10 at each of the supplied list's actual positions, in its specified order. The last written prefix is exactly S,U,V,M, and its two equalities give the last two parts of C1. This is a finite chain of the displayed local equalities for the supplied list, not an appeal to an unprovided infinite collection or induction axiom.

3. The mixed accumulator contains precisely all distinct-position pairs

Fix one actual next member W. The earlier written sum is the finite sum of its preceding actual members by C0. PB-0014 MixOps' additive law supplies the explicit prefix step, with S_pre naming this preceding sum,

Mix(S_pre+Y,W)=Mix(S_pre,W)+Mix(Y,W),                 (C11)

beginning with Mix(v,W)=v. Apply this displayed step to the actual earlier members in their order. It gives exactly one term Mix(Y,W) for each position preceding W and no other term.

Appending those terms at W is exactly the update of M in C0. Every pair of distinct positions has one later member; that pair enters at that later member's step. It did not enter earlier because the later member had not yet occurred, and it does not enter later because both members are then already in the old prefix. Thus each such pair enters once, including a pair whose two positions contain equal expressions.

No term Mix(W,W) from a single position is added at its own step: the update uses S_old, which precedes that position. Its unmixed self-source was separately appended to U as F(W). This proves the first equality of C1 and the stated occurrence accounting.

Mix is symmetric, so the named expression attached to a pair does not depend on which operand is written first. The accumulation itself uses the specified finite ordering; only proved additive commutation is needed to compare another finite writing of those same terms.

4. Obtain the complete composition criteria with the void sum collected

C1 gives R(S)=V−M. PB-0001's exact difference criterion therefore gives both directions of C2:

R(S)=v iff V−M=v iff V=M.

Now additionally retain R(W)=v at each actual member. Each update of V then has the form V_new=V_old+v=V_old. Beginning with v gives V=v at the final position. C1 becomes R(S)=−M. Its being void is equivalent to M=v by double negation and the void identities. This proves C3.

C3 states the condition M=v on the complete accumulator. This proof states no separate equation for an individual summand.

5. Retain the C4 conditions on Z in the complete calculation

Retain exactly C4. For each actual member W, PB-0014's residual identity and the two supplied void-residual hypotheses give

v=R(Z+W)=R(Z)+R(W)−Mix(Z,W)
         =R(W)−Mix(Z,W).

The difference criterion gives

R(W)=Mix(Z,W).                                      (C12)

This does not set R(W) to v. It states exactly what the two hypotheses in C4 imply for that increment.

Append C12 down the actual list. The defining updates of V and the second-input additive law for Mix give

V=Σ_W R(W)=Σ_W Mix(Z,W)=Mix(Z,S).                   (C13)

For the last equality the actual prefix calculation is Mix(Z,S_old+W)=Mix(Z,S_old)+Mix(Z,W), with starting value v; no extra summation rule is assumed. Use C1 and C13 in the full residual of Z+S:

R(Z+S)
  =R(Z)+R(S)−Mix(Z,S)
  =v+[V−M]−Mix(Z,S)
  =−M.                                             (C14)

Finally append the supplied proposed addition E, using the same exact two-input identity once more:

R(Z+S+E)
  =R(Z+S)+R(E)−Mix(Z+S,E)
  =−M+[L(E)−F(E)]−Mix(Z+S,E)
  =L(E)−F(E)−Mix(Z+S,E)−M.                         (C15)

This proves C5. Additive regrouping and the difference criterion show that C15 is void exactly when

L(E)−Mix(Z+S,E)=M+F(E),

which proves C6 in both directions. At the actual list X,Y, the definitions give S=X+Y and M=Mix(X,Y); substituting them proves C7 without a separate interaction construction.

What is not forced

C1–C2 hold for supplied written inputs without requiring their individual residuals to vanish. C3 has its individual-solution hypotheses, and C5–C7 have the different hypotheses C4 on Z. The axiom asserts its equation for Ψ only. A separately named element does not receive that equation automatically.

The vanishing of M is the vanishing of one complete finite expression. No no-cancellation principle, positivity, cancellation of arbitrary nonvoid factors, or splitting of that equality into termwise zero equalities is supplied. Pairwise void mixed terms are sufficient for M=v, but the exact result retained here is the full accumulated equality.

The entries of the list are not required to be distinct or to have disjoint blade supports. The calculation states the supplied sum's complete residual; it does not classify every solution or every decomposition of Ψ.

C6 retains F(E). It selects no E and asserts no inverse for the expression L(E)−Mix(Z+S,E). Neither completion equation asserts existence of E; its two directions test the supplied total under C4.

What it means

For a supplied finite list, the equation of its sum is determined by the individual residuals together with one complete mixed-source accumulator. That accumulator retains every pair of positions in one collected expression. Under C4 on Z and the listed sums Z+W, −M is precisely the residual of Z+S. The final displayed equation states what an additional written expression must satisfy to restore the full equation, including its own complete source F(E). It supplies an exact test of a proposed composition without choosing its constituents or asserting that a correcting expression exists.

The centered source and residual pairs retain both the finite change and its full quadratic remainder (V03-PB-0019)

LAYER: 5 FORCED BY: V01-PB-0001, V01-PB-0006, V02-PB-0014

Declares

The symbols u,v,h,k retain 04's meanings, with h=u+u, k=h⁻¹ under the input's existing nonvoid guard on h. Sel,D,L,F,R retain PB-0006's original definitions. In particular L(W)=h(D(W))B, F(W)=We₀Sel(W), and R(W)=L(W)−F(W). Mix retains V02-PB-0014's mixed-source definition.

W,C,E,K,U,V,V₁,V₂ below are supplied written elements. The letter K here names a supplied change of C, not a blade index or a coefficient operation. The new abbreviation is

Cmp_C(V) := L(V)−Mix(C,V).                                (C1)

It denotes exactly this written expression. The subscript C supplies its first fixed input; it does not denote a derivative or a limiting construction about a further input. No equation for C or V is imposed by the definition. The complete nonlinear residual remains R.

For a supplied pair U,V, define its constructed center and half-difference by

C:=k(U+V),              E:=k(U−V).                      (C2)

Where C and E instead begin as independently supplied written elements, the endpoints are explicitly defined as U:=C+E, V:=C−E. The two directions are related by the written recovery proof below. The word center means exactly the expression in C2. C',E' in the uniqueness proof name supplied alternative parts, under the two endpoint equations displayed at that step.

What is forced

For every supplied C and written V, the exact change expression satisfies

Cmp_C(V₁+V₂)=Cmp_C(V₁)+Cmp_C(V₂),
Cmp_C(−V)=−Cmp_C(V),            Cmp_C(v)=v,
Cmp_(C+K)(V)=Cmp_C(V)−Mix(K,V).                            (C3)

No derivative or scalar condition is needed for C3. These identities claim additivity, not arbitrary-scalar linearity.

Every supplied pair C,E has the complete centered identities

F(C+E)=F(C)+F(E)+Mix(C,E),
F(C−E)=F(C)+F(E)−Mix(C,E),                             (C4)

R(C+E)=R(C)+Cmp_C(E)−F(E),
R(C−E)=R(C)−Cmp_C(E)−F(E).                             (C5)

For a supplied U,V, the C,E in C2 uniquely recover it by

U=C+E,                  V=C−E.                       (C6)

Conversely C2 applied to the supplied endpoints C+E,C−E returns that same C,E. Hence C4–C5 equivalently give the full paired recoveries

Mix(C,E)=k[F(U)−F(V)],
F(C)+F(E)=k[F(U)+F(V)],                               (C7)

Cmp_C(E)=k[R(U)−R(V)],
R(C)−F(E)=k[R(U)+R(V)].                               (C8)

Their full-difference forms are

F(U)−F(V)=Mix(C,U−V),
R(U)−R(V)=Cmp_C(U−V),                                 (C9)

where C still means exactly k(U+V), not either endpoint separately.

The exact paired criteria, with those same constructed C,E, are

F(U)=F(V)  iff Mix(C,E)=v,
R(U)=R(V)  iff Cmp_C(E)=v,                              (C10)

R(U)=v and R(V)=v
    iff Cmp_C(E)=v and R(C)=F(E).                       (C11)

C11 retains both conditions. If its endpoint hypotheses hold, the additional assertion R(C)=v is equivalent to F(E)=v. The residual-equality criterion in C10 instead states the single Cmp condition. The criteria concern arbitrary supplied pairs and keep their entire inputs. They select no support, sector, coefficient inverse, scalar root, or numerical value.

The forcing

1. The change expression is additive

PB-0001 supplies the additive identities, additive commutation and cancellation, inverse uniqueness, sign laws, absorption, and the void-difference criterion. PB-0006 (1) supplies the additivity of L. V02-PB-0014 supplies the additive and scalar laws of Mix and the exact quadratic source identities. All are used at their stated scope.

Expand C1 at a supplied sum and then use these additive laws:

Cmp_C(V₁+V₂)
  =L(V₁)+L(V₂)−[Mix(C,V₁)+Mix(C,V₂)]
  =[L(V₁)−Mix(C,V₁)]+[L(V₂)−Mix(C,V₂)]
  =Cmp_C(V₁)+Cmp_C(V₂).                                  (C12)

Substitute v=v+v into the already supplied additivity of L: L(v)=L(v)+L(v). Compare this with L(v)=L(v)+v and cancel the common summand to obtain L(v)=v. Mix(C,v)=v by V02-PB-0014(MixOps). Also L(V)+L(−V)=L(V+(−V))=L(v)=v, so inverse uniqueness gives L(−V)=−L(V). Thus C1 gives Cmp_C(v)=v. Apply C12 to V+(−V)=v: it gives Cmp_C(V)+Cmp_C(−V)=v. Inverse uniqueness proves Cmp_C(−V)=−Cmp_C(V).

For the supplied change K in the first input, Mix additivity gives

Cmp_(C+K)(V)
  =L(V)−[Mix(C,V)+Mix(K,V)]
  =Cmp_C(V)−Mix(K,V).                                  (C13)

These are all of C3. They make no derivative assumption on C,K or V.

2. Expand both source endpoints, with the same quadratic term

V02-PB-0014(M2) gives the first equation of C4 directly. To obtain the second, apply it to the supplied pair C,−E:

F(C−E)=F(C)+F(−E)+Mix(C,−E).

The exact scalar homogeneity there gives F(−E)=(-u)(-u)F(E)=F(E), since (-u)(-u)=u by PB-0001's sign laws. Mix's scalar law gives Mix(C,−E)=−Mix(C,E). Therefore

F(C−E)=F(C)+F(E)−Mix(C,E).                            (C14)

Thus the mixed term changes sign between the two endpoints; the whole self-source F(E) appears with the same sign at both. It has not been approximated or set to void.

Additivity of L and its negation law give

L(C+E)=L(C)+L(E),
L(C−E)=L(C)−L(E).

Subtract each corresponding source identity from its L identity, preserving all terms:

R(C+E)
  =L(C)+L(E)−F(C)−F(E)−Mix(C,E)
  =[L(C)−F(C)]+[L(E)−Mix(C,E)]−F(E)
  =R(C)+Cmp_C(E)−F(E),

R(C−E)
  =L(C)−L(E)−F(C)−F(E)+Mix(C,E)
  =[L(C)−F(C)]−[L(E)−Mix(C,E)]−F(E)
  =R(C)−Cmp_C(E)−F(E).                                (C15)

This proves both equations C5. The signs of the retained F(E) agree.

3. The pair construction loses no information

The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. For the C,E actually defined by C2, distribute k and regroup:

C+E=k[(U+V)+(U−V)]=k(U+U)=khU=U,
C−E=k[(U+V)−(U−V)]=k(V+V)=khV=V.                   (C16)

Conversely, start with supplied C,E and the endpoints U=C+E,V=C−E. Their sum and difference are

U+V=(C+E)+(C−E)=hC,
U−V=(C+E)−(C−E)=hE.                                (C17)

Multiplying each by k returns C,E. Consequently any supplied pair C',E' reconstructing the same U,V by these two endpoint equations must satisfy C'=k(U+V) and E'=k(U−V) by the same calculations. That proves C6 and the stated uniqueness relative to the two displayed endpoint equations. No inverse of U,V,C or E is required.

4. Recover the complete source and residual balances

Use C16 to replace C+E,C−E in C4 by U,V. Subtract the two source identities. The two copies of F(C) and F(E) cancel pairwise, leaving

F(U)−F(V)=Mix(C,E)+Mix(C,E)=hMix(C,E).                (C18)

Adding the same two identities instead cancels just the mixed terms:

F(U)+F(V)=h[F(C)+F(E)].                              (C19)

Multiply C18 and C19 by k and use kh=u. These are exactly C7. The vanishing of the even source term has not been inferred from the odd comparison C18.

Perform the same two operations on C15. The subtraction cancels the two copies of R(C) and the two copies of −F(E), giving

R(U)−R(V)=hCmp_C(E).                                  (C20)

The addition cancels the two opposite Cmp terms, giving

R(U)+R(V)=h[R(C)−F(E)].                              (C21)

Multiplication by k yields C8. The residual at C and the self-source of E remain together in C21; neither has disappeared from the full pair of equations.

For C9 use the exact difference U−V=hE=E+E from C17. Mix additivity and C3 give respectively

Mix(C,U−V)=Mix(C,E+E)=hMix(C,E),
Cmp_C(U−V)=Cmp_C(E+E)=hCmp_C(E).                           (C22)

Comparing with C18 and C20 proves C9. In particular the residual difference uses the constructed center C. It is not the fixed-endpoint expression Cmp_U(U−V); C13 keeps their difference explicit whenever such a change of first input is made.

5. Equal outputs and the two-endpoint equation retain different conditions

If F(U)=F(V), their difference is v by PB-0001's criterion; the first equation C7 and absorption give Mix(C,E)=v. Conversely if that mixed expression is v, C18 gives F(U)−F(V)=v and the same criterion gives F(U)=F(V). This is the first equivalence in C10. The same two steps, using C8 and C20, prove the residual equivalence in C10.

Now additionally require both endpoint residuals themselves to be v. C8 gives

Cmp_C(E)=k(v−v)=v,
R(C)−F(E)=k(v+v)=v.

The difference criterion turns the second condition into R(C)=F(E). This proves necessity in C11. For sufficiency, retain both displayed conditions Cmp_C(E)=v and R(C)=F(E) and substitute them in the two complete equations C15:

R(U)=F(E)+v−F(E)=v,
R(V)=F(E)−v−F(E)=v.                                 (C23)

Thus both endpoint equations hold, proving the converse. No separate term of a collected expression has been forced to vanish.

Under those endpoint equations, the already proved equality R(C)=F(E) makes R(C)=v equivalent in both directions to F(E)=v by substitution. C10 states the criterion for equality of the two residuals; C11 states the two conditions for both residuals to be void.

What is not forced

Cmp_C is an exact named expression. C5 and C8 retain the complete F(E) term, so neither an infinitesimal perturbation nor a dropped quadratic remainder is used. C9 is a finite centered identity. It has no limit, smallness parameter, convergence assumption, or numerical evaluation. The supplied derivative operations have their own exact consequences in the derivative records; this construction asserts none merely because its name or form resembles a differential expression.

Cmp_C is additive in its second input. No arbitrary-scalar linearity is asserted: L contains the supplied derivatives, whose product rule must be retained whenever a supplied scalar multiplies that input. The complete input pair has an invertible algebraic description C2, C6 under h's existing guard; this provides no inverse for Cmp_C, no existence or uniqueness theorem for its kernel, and no solving rule for the additional condition R(C)=F(E).

C10 distinguishes equal source outputs from equal residual outputs. C11 further distinguishes equal residual outputs from two vanishing ones. The center of two supplied expressions satisfying the equation need not itself be declared a solution; its exact residual is F(E). No universal assertion of a nonvoid F(E), or existence of such a pair of solutions, is used to make this conditional statement.

No coefficient, derivative or component of Ψ has been selected. The source guard is still the single unconditional nonvoid guard on h.

What it means

Every supplied pair of inputs has a unique description by its center and complete half-difference. The source and residual comparisons then separate into the part that reverses sign between the endpoints and the part that stays the same. With its first input C=k(U+V), Cmp gives the complete residual difference R(U)−R(V)=Cmp_C(U−V). The common balance still contains the self-source of that change. For E=k(U−V), both endpoints satisfy the equation exactly when Cmp_C(E)=v and R(C)=F(E). This gives one reconstructable description of two complete inputs without imposing an equation on either input or replacing the quadratic equation by its difference alone.

The constructed remainder has sharp mixed-output support and a reversion-resolved equation (V01-PB-0010)

LAYER: 6 FORCED BY: V01-PB-0001, V01-PB-0002, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V01-PB-0008, V01-PB-0009

Declares

All expressions below use the existing operations. There is no new hypothesis on the distinguished Ψ and no trial-selected family.

• u, v, h, k, G, 𝓑, η, τ, 𝓑_H, 𝓑_A, f_K, ℓ_K and 𝓒_K retain their parent meanings. k is written only under h's existing nonvoid guard, with kh=hk=u.
• For each supplied written X, x_I=𝓒_I(X); T(X)=(X̃)*, A_X=k(X−T(X)), H_X=k(X+T(X)), and Sel, D, L, F, R are exactly PB-0006/PB-0007's expressions. The abbreviated A,H in a calculation mean these constructed A_X,H_X, not independently chosen parts.
• I_s, Ω_s, I_all, Ω_all and s(I,J) retain PB-0009's definitions. Its X_K and s_K^w are the constructed witness and its coefficient sign, not field variables or a scalar power.
• For each target K in §4, C_index={0}△K names PB-0009's local witness index C. It is an index set, not a field component or PB-0004's conjugation expression.
• For an index K, q_K=𝓒_K(A_Xe₀Ã_X), c_K=𝓒_K(H_Xe₀Ã_X), and r_K:=𝓒_K(R(X)) name specified recovered coefficients. Other local capital letters denote supplied written elements; empty sums have accumulator v. Restricted sums follow the same finite 𝓑 order and collection as the parents.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, and with a declared nonvoid condition excluding equality with void, for every supplied X and its constructed parts A=A_X,H=H_X, with r_K=ℓ_K(X)−f_K(X) supplied by PB-0008(P5):

Ω_sΩ_s=−u,       Ω_s e₀=e₀Ω_s,
Sel(Ω_s)=−hΩ_s≠v,       F(Ω_s)=he₀≠v;                 (R1)

F(X)=hAe₀Ã+hHe₀Ã,
(Ae₀Ã)̃=Ae₀Ã,        (He₀Ã)̃=Ae₀H̃;               (R3–R4)

F(X)−F(X)̃=h(He₀Ã−Ae₀H̃),
k(F(X)−F(X)̃)=He₀Ã−Ae₀H̃;                            (R5)

F(X)=F(X)̃  iff  He₀Ã=Ae₀H̃;                          (R6)

η(K)=−u  implies  q_K=v and f_K(X)=h c_K.             (R7)

For every supplied X, separately from that η condition,

c_{ {0} }=v,    c_{I_s}=v.

At both {0} and I_s, the mixed coefficient's qualifying occurrence list, with I in 𝓑_H and J in 𝓑_A from the PB-0008 P4 presentations, is empty. These are exactly the two output positions where c_K is void for every supplied X. For every other K, the existing PB-0009 witness and its specified sign give the whole-expression identity

H_(X_K)e₀Ã_(X_K)=s_K^w e_K,
𝓒_K(H_(X_K)e₀Ã_(X_K))=s_K^w≠v.                      (R7w)

This completes individual mixed-output support on the whole blade index. It does not assert independently assignable simultaneous mixed coefficients or that a witness is a solution.

For any supplied X, the displayed relation

D(X)B+B(D(X))̃=H_Xe₀Ã_X−A_Xe₀H̃_X                   (R8)

holds if and only if R(X)=R(X)̃, equivalently if and only if

ℓ_K(X)=f_K(X) for every K with η(K)=−u.              (R8e)

Its companion relation

D(X)B−B(D(X))̃
  =hA_Xe₀Ã_X+H_Xe₀Ã_X+A_Xe₀H̃_X                 (R9)

holds if and only if R(X)=−R(X)̃, equivalently if and only if

ℓ_K(X)=f_K(X) for every K with η(K)=u.               (R9e)

R8e and R9e together give exactly all coefficient equations of PB-0008 P6. The conjunction of R8 and R9 is therefore equivalent to the complete equation R(X)=v. Each holds for Ψ because the axiom supplies that equation. R8 alone does not supply the reversion-positive coefficient conditions: at the supplied X=e₀, R8 holds but ℓ_{ {0} }(e₀)=v differs from f_{ {0} }(e₀)=h≠v. The balance written as PB-0009(G2) is equivalent, by G0's h-equivalence, to the I_s component of R8: the unrestricted G0 identifies its left coefficient, and G1 makes its source coefficient void. This use does not assume the full equation on X.

The actual e₀ diagnostic has the complete identities

D(e₀)=L(e₀)=v,    F(e₀)=he₀,
R(e₀)=−he₀,       −he₀≠v.                            (R10)

All reversions of D(X) apply to that whole written expression; no commutation with a derivative is presumed. The two relations concern the same constructed A_X,H_X; their claims retain those constructed parts. H_X=v is sufficient but not necessary for R6; the supplied X=u+e₀ below has H_X=u≠v and a reversion-fixed source.

The forcing

1. The all-spatial output and input are different questions

Use the equality/nonvoid premises stated with the claims. The actual supplied Ω_s is an input to Sel and F; 𝓒_{I_s}(F(X)) is instead a recovered output coefficient of F. The following calculation establishes the first two outputs without changing the universal coefficient identity of PB-0009.

Here is the explicit calculation. PB-0009 (S5) establishes

Ω̃_s=−Ω_s,    Ω_s*=Ω_s.

04's product-reversion rule gives the actual word e₁e₂e₃e₄e₅e₆e₆e₅e₄e₃e₂e₁. Associate and replace its adjacent e₆e₆, then e₅e₅, e₄e₄, e₃e₃, e₂e₂ and e₁e₁ by their declared squares −u, moving only those scalars by rule 8. Associating the six resulting scalar factors into adjacent pairs gives

Ω_s Ω̃_s = ((−u)(−u))((−u)(−u))((−u)(−u)) = u.

Substitute Ω̃_s=−Ω_s. The product-negation identity gives −(Ω_sΩ_s)=u; negate both sides and use double negation:

Ω_sΩ_s=−u.

Moving e₀ through the actual six distinct factors of Ω_s contributes the same six signs, whose product is u. Thus

Ω_s e₀=e₀ Ω_s,
Sel(Ω_s)=−Ω_s−Ω_s=−hΩ_s,
F(Ω_s)=Ω_s e₀(−hΩ_s)
       =−h e₀(Ω_sΩ_s)
       =−h e₀(−u)
       =h e₀.                                           (R1)

These steps use only the displayed identities, contextual exchanges, association, scalar commutation, and product-negation laws. The result is nonvoid: he₀=v would give h=v upon right multiplication by e₀, since e₀e₀=u, contradicting the existing guard. Likewise −hΩ_s is nonvoid by right multiplication by its written undoing from PB-0002.

This proves R1 for the supplied Ω_s. It asserts no equation R(Ω_s)=v.

The converse distinction is already explicit in PB-0006: Sel(e₁)=v, yet

F(e₀+e₁)−F(e₀)=he₁,

which that record proves nonvoid. An input the selector omits can still affect the full source in the other factor. In addition, PB-0009 §3 already shows two nonvoid contributions reaching the Ω_s output and cancelling. “No net output there” is not “no occurrence ever reached there.”

2. Reversion of the complete source with the constructed parts

For an unrestricted supplied written X, retain PB-0007's actual constructed parts:

T(X)=(X̃)*,
A_X=k(X−T(X)),    H_X=k(X+T(X)).

For the rest of this section only, abbreviate A=A_X, H=H_X. These are not free replacements for the constructed parts and not a chosen field family. PB-0007 establishes

X=A+H,    T(A)=−A,    T(H)=H,
Sel(H)=v,    Sel(X)=h Ã,
F(X)=h X e₀ Ã.                                      (R2)

Distribution gives the complete identity

F(X)=h A e₀ Ã + h H e₀ Ã.                           (R3)

The first contribution is fixed by reversion. Using the supplied product rule, ẽ₀=e₀, and PB-0007's proved double-reversion identity,

(A e₀ Ã)̃ = (Ã)̃ ẽ₀ Ã = A e₀ Ã.                  (R4)

Reversion of the second contribution gives

(H e₀ Ã)̃ = A e₀ H̃.

Reversion fixes h; moving only this scalar back to the left, (R3) therefore gives

F(X)̃ = h A e₀ Ã + h A e₀ H̃.

Subtract this equality from (R3). Use negation of a sum, additive commutation and association, and cancel only the two equal h A e₀ Ã contributions. Distribute h in the equality direction to obtain

F(X)−F(X)̃ = h(H e₀ Ã−A e₀ H̃),
k(F(X)−F(X)̃) = H e₀ Ã−A e₀ H̃.                    (R5)

The second line uses kh=u, association, and right/left unity; it introduces no inverse of an input coefficient or written element. These identities hold for every supplied X with its constructed parts. They are an exact organization of the full nonlinear source, not separate imposed equations on the parts.

Equation (R5) says exactly how the remainder that disappears from the selector can supply the source's departure from reversion symmetry. In particular,

F(X)=F(X)̃  iff  H e₀ Ã=A e₀ H̃.                    (R6)

For necessity, equality of F and its reversion makes the left side of R5 void, so the displayed product difference is void and PB-0001's difference criterion gives equality. For sufficiency, equality of the products makes their difference void, then the first line of R5 and the same criterion give F=F̃. This condition does not assert H=v. If H=v, PB-0007 M0 gives H̃=v and PB-0001 absorption makes both products v, proving sufficiency. The exact condition remains the displayed equality.

The sufficient condition H=v is not necessary. Take the supplied X=u+e₀. Since T(u)=u and T(e₀)=−e₀, the part construction gives H_X=u and A_X=e₀. Then

H_Xe₀Ã_X=u e₀e₀=u,
A_Xe₀H̃_X=e₀e₀u=u,
F(X)=h(u+e₀)=F(X)̃.

Here H_X=u≠v: otherwise h=u+u=v, contradicting the existing guard. The same X=u+e₀ is PB-0009 S6p's witness at K=∅. This is a diagnostic for R6, not a solution declaration.

To connect this element-level statement to blade output coefficients, take a supplied index K with η(K)=−u, and put

q_K=𝓒_K(A e₀ Ã).

PB-0008(P2) and R4 give q_K=−q_K. Add q_K to both sides and use the additive inverse identity and scalar distribution:

h q_K=v.

Multiplication by k gives q_K=v. Apply PB-0004(C3)'s coefficient addition/scalar laws to R3:

η(K)=−u  =>  𝓒_K(F(X))=h 𝓒_K(H_X e₀ Ã_X).         (R7)

Thus every reversion-negated output coefficient of F comes entirely from the mixed contribution between the two already constructed parts. This neither guarantees that such an output is nonvoid nor confines the full source to those outputs. For η(K)=u, R3 remains the complete expression with both contributions.

The connection does not assign physical meanings to A,H or reversion. It supplies an exact relationship a later interpretation must preserve.

3. The two exceptional mixed outputs have empty occurrence lists

Apply PB-0008's exported product law P1 twice, its reversion law P2, and its exact part presentations P4. PB-0009 S1 supplies the resulting three-factor sign. Retaining all occurrences and their prescribed collection gives, for every K,

c_K=𝓒_K(H_Xe₀Ã_X)
   =Σ_{I∈𝓑_H,J∈𝓑_A : I△{0}△J=K}
         x_Ix_Jη(J)s(I,J).                              (M1)

At K={0}, the routing equation requires I=J. The two lists have opposite τ memberships, so no pair qualifies. At K=I_s, the equation requires J=I△I_all. PB-0009 S5 gives τ(J)=τ(I), while the H–A memberships require τ(I)=u and τ(J)=−u. Equality of those signs would give h=v, contrary to the existing guard. No pair qualifies here either.

Both accumulators therefore remain their initial v. This proves the two universal coefficient identities stated with R7 and the claimed empty occurrence lists. The emptiness concerns the displayed 𝓑_H×𝓑_A index conditions in M1. Writing either part on the full 𝓑 with v coefficients instead would introduce padded occurrences with v factors; that different presentation does not alter these coefficients or the absence of qualifying pairs in M1. At I_s, the η-negative implication of R7 also recovers f_{I_s}(X)=v. The empty mixed list and the cancelling complementary occurrences of PB-0009 are different expansions with their own actual operands; the equality of the final void values is unchanged.

4. Positive completeness over the full mixed-output index

Let K differ from {0} and I_s, and use exactly the witness X_K and sign s_K^w already constructed in PB-0009 S6 and S6p. The declared C_index is then nonempty and proper.

PB-0008(P0) supplies the whole T and reversion images at the actual indices of each witness. The two displayed input indices are distinct, so PB-0004(C1) recovers coefficient u at each and v at all other indices. We now apply the k-defined parts to those complete inputs; the operands of the mixed expression are not chosen independently.

If τ(C_index)=−u, X_K=u+e_{C_index}. Scalar fixing, P0 and additivity give

T(X_K)=u−e_{C_index},
H_(X_K)=k[(u+e_{C_index})+(u−e_{C_index})]
       =k[(u+u)+(e_{C_index}−e_{C_index})]=khu=u,
A_(X_K)=k[(u+e_{C_index})−(u−e_{C_index})]
       =k[(u−u)+(e_{C_index}+e_{C_index})]=khe_{C_index}=e_{C_index}.

These equalities use additive commutation, double negation, distribution and kh=u. P0 then gives the reversion of the actual A part. Thus

H_(X_K)e₀Ã_(X_K)
  =η(C_index)e₀e_{C_index}
  =η(C_index)s(∅,C_index)e_K
  =s_K^w e_K.

If τ(C_index)=u, PB-0009(S6p) supplies distinct I,J with τ(J)=−τ(I) and X_K=e_I+e_J. Both sign cases must be retained. For τ(I)=u, τ(J)=−u, P0 and the actual part definitions give

T(X_K)=e_I−e_J,
H_(X_K)=k[(e_I+e_J)+(e_I−e_J)]
       =k[(e_I+e_I)+(e_J−e_J)]=khe_I=e_I,
A_(X_K)=k[(e_I+e_J)−(e_I−e_J)]
       =k[(e_I−e_I)+(e_J+e_J)]=khe_J=e_J,
H_(X_K)e₀Ã_(X_K)=η(J)e_Ie₀e_J=η(J)s(I,J)e_K.

For τ(I)=−u, τ(J)=u, the complete calculation instead is

T(X_K)=−e_I+e_J,
H_(X_K)=k[(e_I+e_J)+(−e_I+e_J)]
       =k[(e_I−e_I)+(e_J+e_J)]=khe_J=e_J,
A_(X_K)=k[(e_I+e_J)−(−e_I+e_J)]
       =k[(e_I+e_I)+(e_J−e_J)]=khe_I=e_I,
H_(X_K)e₀Ã_(X_K)=η(I)e_Je₀e_I=η(I)s(J,I)e_K.

Each last equality uses PB-0009(S1) with its actual ordered arguments. The corresponding scalar is exactly the sign s_K^w in PB-0009(S6s), with its stated sign case. There is one blade in each constructed part, so these are the whole mixed products. PB-0004(C1) applied to each final signed-blade expression recovers s_K^w at K and v at every other index. Thus R7w holds on the whole stated index, including η(K)=u. If s_K^w were v, absorption would make h s_K^w=v, contrary to PB-0009(S6). The two empty lists and these witnesses complete the individual support claim. They do not supply arbitrary simultaneous coefficients or select Ψ.

5. The exact reversion-resolved equation

Reversion of B=e₁e₂ and the single generator exchange give B̃=−B. Reversing L(X)=hD(X)B with its actual order yields

L(X)̃=−hB(D(X))̃,
L(X)−L(X)̃=h[D(X)B+B(D(X))̃].

R5 holds without a base-equation premise. Subtract it from this derivative identity, using R=L−F and the additive laws:

R(X)−R(X)̃
  =h{D(X)B+B(D(X))̃−[H_Xe₀Ã_X−A_Xe₀H̃_X]}.       (E1)

The actual inverse k and PB-0001's difference criterion make the bracket void exactly when R(X)=R(X)̃. That proves R8's first equivalence without asserting R(X)=v for arbitrary X.

For the coefficient equivalence, PB-0008 P2 and P5 and PB-0004 C4 give

R(X)=R(X)̃ iff r_K=η(K)r_K for every K.

For η(K)=u that equality is automatic. For η(K)=−u it gives h r_K=v, hence r_K=v by k; the converse is immediate substitution. Since r_K=ℓ_K−f_K, PB-0001's difference criterion gives R8e.

For the same arbitrary X, PB-0009(G0) supplies, without R(X)=v,

ℓ_{I_s}(X)=h[∂₀x_{03456}+∂₁x_{13456}+∂₂x_{23456}
               −∂₃x_{456}+∂₄x_{356}−∂₅x_{346}+∂₆x_{345}].

PB-0009(S5) gives η(I_s)=−u and G1 gives f_{I_s}(X)=v. Consequently the I_s member of R8e is exactly ℓ_{I_s}(X)=v. Multiplying by k makes the displayed whole bracket v; conversely its voidness makes ℓ_{I_s}(X)=hv=v. This proves that the balance displayed as G2 follows already from R8. It does not import G2's stronger full-equation premise, and it does not identify that single coefficient condition with all of R8. No derivative summand is set separately to v.

The additive companion of E1 follows by adding the reversion instead of subtracting. R3 and R4 give

F(X)+F(X)̃
  =h[hA_Xe₀Ã_X+H_Xe₀Ã_X+A_Xe₀H̃_X].

Thus

R(X)+R(X)̃
  =h{D(X)B−B(D(X))̃
      −[hA_Xe₀Ã_X+H_Xe₀Ã_X+A_Xe₀H̃_X]}.         (E2)

Again k and the additive inverse criterion show that R9 holds exactly when R(X)=−R(X)̃. For its coefficient characterization, PB-0008 P2 and PB-0004 C3–C4 give

R(X)=−R(X)̃ iff r_K=−η(K)r_K for every K.

At η(K)=u this is r_K=−r_K. Adding r_K gives h r_K=v, and the existing k gives r_K=v. Conversely r_K=v makes that equality true. At η(K)=−u the equality is automatic by double negation. Hence, using r_K=ℓ_K−f_K and PB-0001's difference criterion, R9 holds exactly when ℓ_K=f_K for every K with η(K)=u. This proves R9e in both directions. The signs η(K) lie in u,−u, which are distinct under h's nonvoid guard; R8e and R9e therefore account for all of PB-0008 P6 without omitting or duplicating an index condition.

If R(X)=v, both E1 and E2 have void left sides, hence both R8 and R9 hold. Conversely, if R8 and R9 hold, E1 and E2 give R−R̃=v and R+R̃=v. Adding these equalities gives hR=v, and multiplication by k yields R=v. This proves their joint equivalence to the full equation. Only the asserted R(Ψ)=v is used to specialize the result to Ψ.

The R8 insufficiency claim uses the supplied generator e₀. Compute the entire diagnostic locally from its supplied images and the actual parts:

Sel(e₀)=ẽ₀−e₀*=e₀−(−e₀)=he₀,
F(e₀)=e₀e₀(he₀)=he₀,
T(e₀)=(ẽ₀)*=−e₀,
A_(e₀)=k[e₀−(−e₀)]=khe₀=e₀,
H_(e₀)=k[e₀+(−e₀)]=kv=v.

Each declared derivative kills this generator:

∂₀e₀=v,  ∂₁e₀=v,  ∂₂e₀=v,  ∂₃e₀=v,
∂₄e₀=v,  ∂₅e₀=v,  ∂₆e₀=v.

Insert all seven equalities into the definition, then use absorption:

D(e₀)=e₀v−e₁v−e₂v−e₃v−e₄v−e₅v−e₆v=v,
L(e₀)=hvB=v,    R(e₀)=v−he₀=−he₀,
R(e₀)̃=(−he₀)̃=−hẽ₀=−he₀=R(e₀).

The scalar reversion step is PB-0007(M0). The established equivalence therefore gives R8. PB-0004(C1,C3) recovers the complete coefficient lists of these displayed outputs, and PB-0008(P5) identifies their names:

ℓ_K(e₀)=v                          at every K,
f_{ {0} }(e₀)=h,    r_{ {0} }(e₀)=−h,
f_K(e₀)=v,          r_K(e₀)=v      at every K≠{0}.

The reversed singleton has no exchange, so η({0})=u. The existing guard makes h≠v; also −h≠v, since negating −h=v would give h=v. Thus the {0} equality required by R9e fails, and recovery makes R(e₀) nonvoid. This establishes precisely R8-alone insufficiency. It supplies no additional conclusion about R9 alone. This generator has not been designated Ψ or asserted to solve its equation.

What is not forced

The constructed parts are exactly PB-0007's A_X,H_X. The complete source retains both its A–A and H–A contributions. The diagnostic X=u+e₀ proves that H_X=v is not necessary for R6, but does not construct a solution with that active part.

At {0} and I_s the mixed-product qualifying lists from the 𝓑_H×𝓑_A presentation are empty. In the full source, the I_s coefficient is universally void by cancellation, including nonvoid complementary occurrences; its {0} coefficient remains reachable through A–A. Equal final void values do not identify the operands of these different expansions or erase them elsewhere. R1 keeps the corresponding input behavior separate.

R7w proves the full mixed expression's sharp individual output support, with pure signed-blade witnesses. It does not characterize the full nonlinear image, permit arbitrary simultaneous coefficients, assert physical field solutions, or establish positivity. The witnesses are the supplied expressions from the preceding construction, not selected fields.

R8 is exactly the reversion-negative coefficient part of the equation; R9 is exactly its reversion-positive coefficient part. Their conjunction is equivalent to R(X)=v. The complete e₀ diagnostic proves that R8 alone is insufficient. R9-alone insufficiency or sufficiency is not asserted here. Both relations concern the same X and its constructed parts; their claims retain those constructed parts. Reversion of D(X) remains whole; no derivative/reversion interchange is used.

The only scalar inverse introduced is k under the input's nonvoid guard. Blade undoing is the parents' displayed product. The source, coefficients and residual retain the same equality scope and all actual hypotheses. No source-support family restricts Ψ.

What it means

The selector's omission of a constructed remainder and a universally void source-output coefficient are different questions. The complete source makes both precise. Its A–A contribution is fixed by reversion; every reversion-negative source coefficient comes from the H–A contribution. The mixed contribution itself has exactly two universally void output positions, each with an empty qualifying occurrence list in the derived H–A sublists. Every other mixed position has a constructed input whose whole mixed expression is a nonvoid signed blade at that position.

The full-index mixed-support statement extends beyond R7's reversion-negative coefficient relation without changing the complete source's one exceptional output. A position can be unavailable to the mixed contribution yet available through A–A. Conversely, the all-spatial blade is an active input even though its own output position is universally void by paired cancellation. Silence remains relative to the specified operation and placement.

The derivative/source relation also has an exact place within the whole equation. R8 expresses precisely its reversion-negative coefficient conditions; G2 is equivalent, by G0's h-equivalence, to its I_s component. R9 expresses precisely its reversion-positive coefficient conditions. Together the two account for every coefficient equation and reconstruct the full equation. They do not split one input into autonomous systems. An interpretive account must preserve that relation rather than treat the selector-null part as absent or independent.

The two involutions have one complete four-part refinement (V02-PB-0012)

LAYER: 6 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V01-PB-0008, V02-PB-0011

Declares

All expressions use the existing written-element operations. No coefficient of the distinguished Ψ is set to void, and no new scalar identity or inverse guard is supplied.

• u:=unity, v:=void, h:=u+u, and k:=h⁻¹ under the input's existing guard.

• W and Z are supplied written elements; c is a supplied scalar expression, without a derivative-constancy or inverse condition. W_{r,t} names a supplied candidate part only where its displayed sign identities and recombination are retained. μ is one of the seven supplied derivative labels. Reversion and star retain 04's definitions; T(W)=(W̃)*, A_W=k(W−T(W)), and H_W=k(W+T(W)) retain PB-0007's definitions.

• r,t,p,q each denote one of the two explicitly written scalar signs u,−u. The order of the four sign pairs is (u,u),(u,−u),(−u,u),(−u,−u).

• The four part expressions are

  Π_{r,t}(W):=kk[W+rW̃+tW*+rtT(W)].                         (J1)
  

  Subscripts on Π name this pair of signs; they do not name additional inputs.

• Π_{++},Π_{+-},Π_{-+},Π_{--} abbreviate those four expressions in the stated order. A displayed sum over (r,t) means just the four terms in that actual list.

• 𝓑,θ(I),𝓒_I retain PB-0004's definitions; 𝓑_H,𝓑_A,η(I),τ(I)=η(I)θ(I) retain PB-0008's definitions. Put x_I(W):=𝓒_I(W) and let 𝓑_{r,t} be the sublist with η(I)=r and θ(I)=t, retaining 𝓑's order.

• 𝒫_{J,*}(W)=k(W+θ(J)W*) retains PB-0004's star expression. Its two sign cases below use J=∅ and J={0}. Put P_r^~(W):=k(W+rW̃) for the reversion pair. These expressions impose no equation on W.

• Sel(W):=W̃−W* and D retain PB-0006's definitions. Neither is redefined here.

What is forced

Under equality used symmetrically, transitively and by substitution in written expressions with the actual hypotheses retained, and with the declared nonvoid condition excluding equality with void, the following hold for every supplied W:

For each of the four specified sign pairs, the complete part map preserves the displayed operations on arbitrary supplied inputs:

Π_{r,t}(W+Z)=Π_{r,t}(W)+Π_{r,t}(Z),
Π_{r,t}(cW)=cΠ_{r,t}(W),
Π_{r,t}(v)=v,
Π_{r,t}(−W)=−Π_{r,t}(W),
Π_{r,t}(W−Z)=Π_{r,t}(W)−Π_{r,t}(Z).                    (J1a)

These are exact finite operation laws, without an equation or support condition on W,Z. The sign identities are

(Π_{r,t}(W))̃=rΠ_{r,t}(W),
(Π_{r,t}(W))*=tΠ_{r,t}(W),
T(Π_{r,t}(W))=rtΠ_{r,t}(W).                               (J2)

For these same sign parts, the selector values are

Sel(Π_{r,t}(W))=(r−t)Π_{r,t}(W),
Sel(Π_{++}(W))=Sel(Π_{--}(W))=v,
Sel(Π_{+-}(W))=hΠ_{+-}(W),
Sel(Π_{-+}(W))=−hΠ_{-+}(W).                            (J2s)

The four parts reconstruct W and recover themselves without mixing:

W=Π_{++}(W)+Π_{+-}(W)+Π_{-+}(W)+Π_{--}(W);                 (J3)

Π_{r,t}(Π_{p,q}(W))=Π_{p,q}(W) if r=p and t=q,
Π_{r,t}(Π_{p,q}(W))=v otherwise.                            (J4)

They are the unique four-part decomposition with these sign properties. Precisely: if supplied parts W_{r,t} have W=Σ_(r,t)W_{r,t}, W̃_{r,t}=rW_{r,t} and W*_{r,t}=tW_{r,t}, then W_{r,t}=Π_{r,t}(W) for each pair. No such sign identity is assumed for an arbitrarily selected part of W.

The two-sign decompositions already present in the graph are exactly the indicated regroupings:

P_u^~(W)=Π_{++}(W)+Π_{+-}(W),
P_{−u}^~(W)=Π_{-+}(W)+Π_{--}(W),

𝒫_{∅,*}(W)=Π_{++}(W)+Π_{-+}(W),
𝒫_{{0},*}(W)=Π_{+-}(W)+Π_{--}(W),

H_W=Π_{++}(W)+Π_{--}(W),
A_W=Π_{+-}(W)+Π_{-+}(W),
Sel(W)=h[Π_{+-}(W)−Π_{-+}(W)].                            (J5)

Each part keeps exactly the corresponding recovered coefficients:

Π_{r,t}(W)=Σ_{I∈𝓑_{r,t}}x_I(W)e_I.                          (J6)

The selector sublists are precisely the following inherited-order unions:

𝓑_H = 𝓑_{u,u} ∪ 𝓑_{−u,−u},
𝓑_A = 𝓑_{u,−u} ∪ 𝓑_{−u,u}.                           (J6s)

The four lists partition the complete blade listing. Their lengths are, in the displayed sign-pair order, 28, 36, 36 and 28. These count the explicitly listed blade positions, without requiring any coefficient at a position to be nonvoid. Their derivation is included below.

For each of the seven supplied derivatives, the part-taking expressions commute with that individual derivative:

∂_μ(Π_{r,t}(W))=Π_{r,t}(∂_μW).                             (J7)

No corresponding interchange with multiplication by a generator or with the entire contracted expression D is asserted.

The forcing

1. The available signs and operation identities

PB-0001 supplies the two-sided additive rules, sign products, product-negation laws and the difference criterion. Scalar centrality is 04 rule 8. The two signs satisfy

uu=u,    u(−u)=−u,    (−u)u=−u,    (−u)(−u)=u.

They are distinct: if u=−u, then h=u+u=u+(−u)=v, contrary to the input's guard. Thus the four sign-pair positions describe disjoint possibilities; no coefficient order or sign of a magnitude is involved.

PB-0007, including its exported M1 identities, gives

(W̃)̃=W,    (W*)*=W,    (W̃)*=(W*)̃,
(T(W))̃=W*,    (T(W))*=W̃,    T(T(W))=W.

It also derives additivity and scalar compatibility of these operations. In particular, both fix k, r and t. Those established identities, rather than a general spectral theorem, supply every operation used below.

1a. Export the complete additive and scalar operation laws

Use the four-term definition J1 on the full supplied sum. The parent operation laws and both distributive rules give

Π_{r,t}(W+Z)
  =kk[(W+Z)+r(W̃+Z̃)+t(W*+Z*)+rt(T(W)+T(Z))]
  =kk[W+Z+rW̃+rZ̃+tW*+tZ*+rtT(W)+rtT(Z)]
  =kk[W+rW̃+tW*+rtT(W)]
     +kk[Z+rZ̃+tZ*+rtT(Z)]
  =Π_{r,t}(W)+Π_{r,t}(Z).

Only the proved additive association and commutation were used to collect the two full groups. No product of two nonscalar expressions was reordered and no summand was omitted.

For the supplied scalar c, reversion reverses its two factors while star preserves their order; both fix c. Their composite T reverses the product and also fixes c. Consequently the full scalar calculation is

Π_{r,t}(cW)
  =kk[cW+rW̃c+tcW*+rtT(W)c]
  =kkc[W+rW̃+tW*+rtT(W)]
  =cΠ_{r,t}(W).

The middle step uses only the supplied scalar centrality of c, including its passage past k,r,t. No derivative or inverse of c occurs. All four constituent operations send v to v by PB-0007; therefore

Π_{r,t}(v)=kk[v+rv+tv+rtv]=v.

The scalar law at c=−u and PB-0001's product-negation rule give Π_{r,t}(−W)=−Π_{r,t}(W). Finally, write W−Z=W+(−Z), apply the already proved sum law, then the negative law:

Π_{r,t}(W−Z)
  =Π_{r,t}(W)+Π_{r,t}(−Z)
  =Π_{r,t}(W)−Π_{r,t}(Z).

This proves every part of J1a for each of the four actual sign pairs. The maps have not been asserted to preserve multiplication.

2. Each constructed part has its two stated signs

Reverse the actual four terms in J1, using scalar compatibility:

(Π_{r,t}(W))̃
  =kk[W̃+rW+tT(W)+rtW*].                              (J8)

Multiplying J1 by r and using rr=u gives

rΠ_{r,t}(W)
  =kk[rW+W̃+rtW*+tT(W)].

These are the same four summands under PB-0001's proved additive commutation and association. This proves the first equality of J2 without exchanging any nonscalar product factors.

Apply star to J1:

(Π_{r,t}(W))*
  =kk[W*+rT(W)+tW+rtW̃].                              (J9)

Multiplying J1 by t, with tt=u, gives the same terms. This proves the second equality. Apply star to the first and then use the second:

T(Π_{r,t}(W))=(rΠ_{r,t}(W))*=rtΠ_{r,t}(W).

No sign of W itself was supplied. The identities follow from the constructed expressions.

3. Recombination uses all four terms

The declared inversion gives hk=u, and 04 rule 8 gives kh=hk=u. For either specified r, adding the t=u and t=−u expressions cancels exactly their star and T terms:

Π_{r,u}(W)+Π_{r,−u}(W)
  =kk[(W+rW̃+W*+rT(W))
       +(W+rW̃−W*−rT(W))]
  =kkh(W+rW̃)
  =k(W+rW̃).                                         (J10)

The last equality follows from kkh=k(kh)=k. Add J10 for r=u and r=−u:

Σ_(r,t)Π_{r,t}(W)
  =k[(W+W̃)+(W−W̃)]
  =khW=W.

That is J3. None of its four components has been removed as irrelevant. A component may happen to equal v for a particular W; the formula does not require that event.

4. Recovery of parts and uniqueness

Take a supplied Z with the two displayed sign identities Z̃=pZ and Z*=qZ. Their composition gives T(Z)=pqZ. Substitute these actual identities into J1 and distribute scalar factors:

Π_{r,t}(Z)
  =kk[u+rp+tq+rtpq]Z
  =kk(u+rp)(u+tq)Z.                                  (J11)

When r=p, rp=u and the first parenthesis is h; when r and p are the two different written signs, rp=−u and that parenthesis is v. The second parenthesis has the same two cases for t,q. Consequently J11 is Z if both signs match, since kkhh=(kh)(kh)=u, and is v otherwise by absorption.

Apply this result to Z=Π_{p,q}(W), whose two sign identities are already J2. This proves every case of J4, including the repeated-part identity.

Now suppose the supplied four parts W_{p,q} have the stated recombination and signs. Apply the exported addition law J1a at the three actual prefixes of their four-term reconstruction:

Π_{r,t}(W)
  =Π_{r,t}(((W_{++}+W_{+-})+W_{-+})+W_{--})
  =Π_{r,t}((W_{++}+W_{+-})+W_{-+})+Π_{r,t}(W_{--})
  =Π_{r,t}(W_{++}+W_{+-})+Π_{r,t}(W_{-+})+Π_{r,t}(W_{--})
  =Π_{r,t}(W_{++})+Π_{r,t}(W_{+-})
      +Π_{r,t}(W_{-+})+Π_{r,t}(W_{--})
  =Σ_(p,q)Π_{r,t}(W_{p,q}).

Applying J11 to the four actually supplied parts leaves exactly W_{r,t}. This proves uniqueness. Conversely J2 and J3 construct such a decomposition. The claim is an equivalence for these specific sign conditions, not a uniqueness claim about arbitrary decompositions.

5. The old decompositions are exact coarse views

Equation J10 is already the reversion-positive or reversion-negative row sum. For either t, add the r=u and r=−u formulas instead. The reversion and T summands cancel, giving

Π_{u,t}(W)+Π_{−u,t}(W)=k(W+tW*).                       (J12)

The definition of θ gives θ(∅)=u and θ({0})=−u. Thus for t=u the right side is PB-0004's 𝒫_{∅,}(W); for t=−u it is 𝒫_{{0},}(W). Together with J10 these prove the first four regroupings in J5.

Add the two diagonal sign positions directly:

Π_{++}(W)+Π_{--}(W)
  =kk[(W+W̃+W*+T(W))
       +(W−W̃−W*+T(W))]
  =kkh(W+T(W))
  =k(W+T(W))=H_W.                                    (J13)

Adding the two remaining positions gives, by the same displayed cancellation of the reversion and star terms,

Π_{+-}(W)+Π_{-+}(W)=k(W−T(W))=A_W.                    (J14)

For each sign part, J2 gives

Sel(Π_{r,t}(W))=(r−t)Π_{r,t}(W).

At the two diagonal positions r=t, this coefficient is v. At (u,−u) it is h, and at (−u,u) it is −h. These are all the cases of J2s. Apply PB-0006 (1)'s selector-addition law at the three actual additions in J3, retaining these four exact values:

Sel(W)=Sel(Π_{++}(W))+Sel(Π_{+-}(W))
         +Sel(Π_{-+}(W))+Sel(Π_{--}(W))
      =hΠ_{+-}(W)−hΠ_{-+}(W).

This gives the last identity of J5. It also agrees with PB-0007's Sel(W)=hÃ_W, since the reversion signs of the two terms in J14 are opposite.

These are recombinations of one full W. They introduce neither competing choices of W nor separately asserted equations for its parts.

6. Finite coefficient membership and complete listing

PB-0008 supplies the recovered-coefficient identities for reversion, star and T. Apply them to the four-term definition J1 using PB-0004's additive/scalar coefficient rules:

𝓒_I(Π_{r,t}(W))
  =kk[u+rη(I)+tθ(I)+rtη(I)θ(I)]x_I(W)
  =kk(u+rη(I))(u+tθ(I))x_I(W).                           (J15)

This is precisely the scalar sign calculation of J11. It equals x_I(W) when η(I)=r and θ(I)=t, and equals v otherwise. The reconstruction part of PB-0004 therefore gives J6. It is coefficient recovery, not an assumed independence of blade names, that licenses this identification.

Every blade has the two derived signs η(I),θ(I), each one of u,−u. Each index consequently belongs to exactly one 𝓑_{r,t}. This exhausts the full finite listing before any count is made. It requires no decision about whether a supplied x_I(W) is nonvoid.

The stated list lengths are derived from the supplied six spatial labels as follows. Subsets of spatial size zero consist of the empty list; subsets of size one are the six single labels. The size-two and size-three listings are respectively

12 13 14 15 16   23 24 25 26   34 35 36   45 46   56;

123 124 125 126   134 135 136   145 146   156
234 235 236       245 246       256
345 346           356
456.

Here concatenated labels name index lists. There are 15 pair positions and 20 triple positions. Taking the complement in {1,2,3,4,5,6} pairs each size-four list with a pair, each size-five list with a singleton, and the size-six list with the empty list. This gives the seven spatial-list lengths 1,6,15,20,15,6,1.

The reversal construction of PB-0003/PB-0008 gives the blade signs at lengths zero through seven as

length:       0   1   2   3   4   5   6   7
reversal:     u   u  −u  −u   u   u  −u  −u.

For completeness, each next listed length appends its greatest generator. In the reversed word it passes all the preceding generators once. The actual successive lists of crossings have lengths zero, one, two, three, four, five and six. Their sign products are respectively u,−u,u,−u,u,−u,u, obtained by pairing adjacent factors (-u)(-u)=u; multiplying these into the preceding reversal sign gives exactly the displayed finite row. PB-0008(P0) identifies the sign of this permitted ordering route with its defined η sign, because both reorder the same reversed word to the same increasing word. No arbitrary-length exponent or induction assumption is used.

Without index 0, star has sign u. With index 0, star has sign −u and the blade length is one larger than its spatial list. The complete classification is therefore

spatial-list size    number of lists    η without 0    η with 0
        0                1                u             u
        1                6                u            −u
        2               15               −u            −u
        3               20               −u             u
        4               15                u             u
        5                6                u            −u
        6                1               −u            −u.

Reading the matching rows gives lengths 28 for (u,u), 36 for (u,−u), 36 for (−u,u) and 28 for (−u,−u). Because τ(I)=η(I)θ(I), the displayed sign products are u exactly at the two diagonal pairs and −u exactly at the two off-diagonal pairs. PB-0008 defines 𝓑_H and 𝓑_A by precisely these τ signs. Their membership therefore agrees in both directions with the inherited-order unions J6s. These counts merely report the actual lists used by J6.

7. Derivatives retain the same refinement

V02-PB-0011(D1,D3) supplies vanishing derivatives of u,v,h,k and the interchange of each individual supplied derivative with reversion, star and their composite T. The signs r,t are u or −u, so their derivatives are void by its additive-negation rule. For the actual products used in J1, 04's product rule then gives

∂_μ(kk)=(∂_μk)k+k(∂_μk)=vk+kv=v,
∂_μ(rt)=(∂_μr)t+r(∂_μt)=vt+rv=v.

Each last equality is derived void by absorption. These displayed product identities, not an unexported general derivative assertion, supply the constant factors in the next expansion.

Apply the product and addition rules to J1 with those actual derivative identities retained:

∂_μ(Π_{r,t}(W))
  =kk[∂_μW+r∂_μ(W̃)+t∂_μ(W*)+rt∂_μ(T(W))]
  =kk[∂_μW+r(∂_μW)̃+t(∂_μW)*+rtT(∂_μW)]
  =Π_{r,t}(∂_μW).

This is J7. The contraction D includes left multiplication by the specified generators. J7 alone does not move Π through those multiplications, and no such cancellation of coupled contributions has been made.

What is not forced

The four sublists classify every blade position under two supplied operations. They do not assert nonvoid coefficients at any position or select a support for Ψ. No component of Ψ was required to vanish.

The part-taking expressions preserve addition and scalar multiplication, but they are not asserted to preserve products. No one of the four parts, nor either coarse pair, is asserted to have products confined to itself or a separately imposed equation. Products and the source retain all contributions required by the actual equation. Individual derivative commutation in J7 does not assert commutation with the whole contracted derivative D.

Uniqueness refers to the displayed two sign conditions and the exact sum. The only inverse used is k under the input's guard on h; no coefficient inverse is used.

What it means

The two supplied involutions organize the whole expression into four uniquely recoverable parts under their stated reversion and star signs and exact sum. Reversion, star, and the split defined by T are then different regroupings of that same complete refinement. This gives the graph a common place for its algebraic descriptions to meet: a change of grouping does not choose a different input or discard one of its coefficients. The four parts commute with each individual supplied derivative, while their products and their roles in the full equation must still be carried explicitly.

Exact differentiation connects the source, mixed source and residual (V03-PB-0020)

LAYER: 6 FORCED BY: V01-PB-0001, V01-PB-0006, V02-PB-0011, V02-PB-0014, V03-PB-0019

Declares

u=unity, v=void, h=u+u and B=e₁e₂ retain 04's definitions; Sel,D,L,F,R retain V01-PB-0006's definitions. In particular,

Sel(W)=W̃−W*,       L(W)=hD(W)B,
F(W)=W e₀ Sel(W),  R(W)=L(W)−F(W).

Mix(X,Y)=Xe₀Sel(Y)+Ye₀Sel(X) is V02-PB-0014's mixed-source definition. Cmp_C(V):=L(V)−Mix(C,V) retains the exact finite expression defined in V03-PB-0019. W,X,Y are supplied full written elements. The labels μ,ν are any of the seven derivative labels already supplied in 04. W_μ:=∂_μW abbreviates a derived written expression; it declares neither an independent input nor a solution of R=v.

All derivatives below act on written identities under their retained hypotheses. They are the exact operations in 04. No parameter, perturbation size, evaluation point, limiting operation or discarded term is introduced. Where R(W)=v is used it is an additional established or retained identity for that same W. The axiom supplies that identity for Ψ, not for every named W.

What is forced

For every supplied full written input,

∂_μ Sel(W)=Sel(∂_μW),
∂_μ D(W)=D(∂_μW),
∂_μ L(W)=L(∂_μW).                                  (E1)

The source and mixed source obey the complete derivative identities

∂_μ F(W)=Mix(W,∂_μW),                              (E2)

∂_μ Mix(X,Y)
   =Mix(∂_μX,Y)+Mix(X,∂_μY).                       (E3)

Consequently the exact residual derivative is

∂_μ R(W)=L(W_μ)−Mix(W,W_μ)=Cmp_W(W_μ).               (E4)

Under the additional identity R(W)=v, every one of the seven derived expressions therefore satisfies

Cmp_W(W_μ)=v, or equivalently L(W_μ)=Mix(W,W_μ),       (E5)

R(W_μ)=Mix(W,W_μ)−F(W_μ),                          (E6)

and its own full equation holds exactly when

R(W_μ)=v iff Mix(W,W_μ)=F(W_μ).                    (E7)

No self-source of a new solution has been dropped: E5 is the exact derivative of the original identity, whereas E6 explicitly restores the self-source when the residual of the derived input W_μ is asked for. Neither equality sets W_μ or any of its coefficients to void.

The forcing

1. Differentiate the selector without changing its operations

V02-PB-0011(D1) gives derivative compatibility with negation and the void derivatives of u, v and h. Its D3 gives compatibility with reversion and star. Substitute those identities into the existing selector expression:

∂_μ Sel(W)
  =∂_μ(W̃−W*)
  =∂_μ(W̃)−∂_μ(W*)
  =(∂_μW)̃−(∂_μW)*
  =Sel(∂_μW).

This is the first identity of E1. Each involution still acts on the entire written element.

2. Differentiate all seven terms of D and the two fixed factors of L

For the e₀ term of D, 04's product rule and the supplied generator derivative give

∂_μ(e₀∂₀W)
  =(∂_μe₀)(∂₀W)+e₀∂_μ∂₀W
  =e₀∂_μ∂₀W.

The omitted product in the last line is derived void from ∂_μe₀=v and absorption. For every actual spatial label i, the same 04's product rule calculation and the proved derivative-negation law give

∂_μ(−eᵢ∂ᵢW)=−eᵢ∂_μ∂ᵢW.

Insert these seven displayed instances into derivative additivity:

∂_μD(W)
  =e₀∂_μ∂₀W−e₁∂_μ∂₁W−e₂∂_μ∂₂W
   −e₃∂_μ∂₃W−e₄∂_μ∂₄W−e₅∂_μ∂₅W−e₆∂_μ∂₆W.

The commuting-derivative rule in 04 changes each of these actual second derivatives to ∂_ν∂_μW. The resulting expression is exactly D(∂_μW), proving the second identity of E1. No generator has been moved past another written factor.

04's product rule applied to the actual product B gives

∂_μB=(∂_μe₁)e₂+e₁(∂_μe₂)=ve₂+e₁v=v.

Together with ∂_μh=v, this gives

∂_μL(W)
  =∂_μ[hD(W)B]
  =(∂_μh)D(W)B+h[∂_μD(W)]B+hD(W)(∂_μB)
  =hD(∂_μW)B
  =L(∂_μW).

Only the first and third displayed products became void, by their proved constant-factor derivatives and absorption. This completes E1.

3. Differentiate the full quadratic source

Apply 04's product rule twice to the three written factors of F, in their original order:

∂_μF(W)
  =∂_μ[W e₀ Sel(W)]
  =(∂_μW)e₀Sel(W)
    +W(∂_μe₀)Sel(W)
    +W e₀∂_μSel(W).

The middle term is void because ∂_μe₀=v. Substitution of E1's selector identity leaves

∂_μF(W)
  =(∂_μW)e₀Sel(W)+W e₀Sel(∂_μW)
  =Mix(W,∂_μW),

where the last line is the definition of Mix and additive commutation. This proves E2. Both differentiated source terms are present.

For two independently supplied inputs, use the same two product calculations on the complete definition of Mix:

∂_μMix(X,Y)
  =(∂_μX)e₀Sel(Y)+X e₀Sel(∂_μY)
    +(∂_μY)e₀Sel(X)+Y e₀Sel(∂_μX).

Regroup these four additive terms without moving their factors:

∂_μMix(X,Y)
  =[(∂_μX)e₀Sel(Y)+Y e₀Sel(∂_μX)]
    +[X e₀Sel(∂_μY)+(∂_μY)e₀Sel(X)]
  =Mix(∂_μX,Y)+Mix(X,∂_μY).

This proves E3 with both differentiated inputs retained.

4. Distinguish the derivative of a residual from a residual of a derivative

Differentiate the defining difference for R and use E1–E2:

∂_μR(W)
  =∂_μL(W)−∂_μF(W)
  =L(W_μ)−Mix(W,W_μ)
  =Cmp_W(W_μ).

The last equality uses exactly V03-PB-0019's definition. This proves E4 without a solution premise. Now additionally retain the actual identity R(W)=v. By substitution inside the supplied derivative and V02-PB-0011(D1),

∂_μR(W)=∂_μv=v.

Thus E4 is the void difference of the two sides of E5. V01-PB-0001's void-difference criterion gives E5 in both displayed forms. It does not say that the right side is F(W_μ).

To ask for the residual of that derived expression, write its own definition and substitute E5:

R(W_μ)=L(W_μ)−F(W_μ)
       =Mix(W,W_μ)−F(W_μ).

This is E6. The same void-difference criterion gives E7 in both directions. The condition in E7 is one equality between complete written elements; it is not split into void conditions on its terms.

What is not forced

E1–E4 are operation identities for supplied written inputs. E5–E7 additionally retain R(W)=v. They do not prove that an arbitrary input satisfies the equation, that any derivative of a solution is another solution, or that the derivative expression can be selected independently of its source W.

The exact derivative identity is not a finite-change approximation. V01-PB-0006 and V02-PB-0014 still give, for a separately supplied finite addition H,

R(W+H)=R(W)+L(H)−Mix(W,H)−F(H).

No identity in this record deletes F(H) from that equation. E5 does not give permission to infer a finite family of solutions W+aW_μ, nor does it introduce a derivative inverse.

The present claims are the displayed identities and implications. No converse from these derivative balances to the undifferentiated condition R(W)=v is asserted here.

What it means

Differentiating the equation produces a relation between a solution and its actual derivative. The quadratic source contributes both differentiated factors, collected exactly by the mixed source. The derivative’s own equation is precisely E7: its residual includes its self-source and vanishes exactly under the additional complete equality stated there. The same mixed expression therefore connects exact finite changes with exact derivative identities, without using an approximation or discarding a term from either calculation.

Exact source differences generate the complete allowed output and determine its scalar contractions (V04-PB-0023)

LAYER: 6 FORCED BY: V01-PB-0001, V01-PB-0004, V01-PB-0008, V01-PB-0009, V02-PB-0014

Declares

All inputs are supplied written expressions under 04 and the parents' equality and coefficient-recovery scope. The only scalar inverse used is k, under the existing nonvoid guard on h=u+u. No equation is imposed on an input by forming its source.

• e_I retains 04's blade definition; 𝓑,G,𝓒_I retain PB-0004's definitions; η,τ,w_I retain PB-0008's definitions; s(I,J) retains PB-0009's definition; Sel,F retain PB-0006's definitions; Mix retains V02-PB-0014's definition.

• I_s={1,2,3,4,5,6} and Ω_s=e_(I_s) retain V01-PB-0009's meanings. A coefficient at {0} is the e₀ coefficient; a coefficient at the empty index is the scalar coefficient.

• For each actual K≠{0},I_s, use the two summands p_K=e_I, q_K=e_J of V01-PB-0009 S6p, with their specified sign s_K^w. In its first case I=∅, J={0}△K, and p_K=u. In its second case its prescribed generator label gives I={m}, J={m}△({0}△K). These are the prescribed output witnesses, not selected supports for Ψ.

• Define the actual source combinations

  G_{ {0} } := k F(e₀),
  G_K := k s_K^w Mix(p_K,q_K)  for K≠{0},I_s.        (S0)
  

  The name G_K is indexed by a blade and does not replace the generator list G.

• A supplied scalar list 𝛇=(ζ_K) is indexed by the complete fixed list 𝓑. Define

  Λ_𝛇(Y) := Σ_(K∈𝓑) ζ_K 𝓒_K(Y).                    (S1)
  

  This same list is retained as Y changes. No derivative condition on its entries is part of S1. The coefficient entries ζ_K are distinct from the written blade undoings b_K and the fixed derivative weights b_{μ}.

• The fixed source-evaluation list 𝓣 consists of F(e₀) followed, at every actual K≠{0},I_s, by F(p_K), F(q_K), F(p_K+q_K) in that order. Repeated values need not be removed. This is one finite list determined by the already proved output witnesses.

• All sums start at v and use the inherited order. An actual finite scalar combination of sources means a supplied presentation Σ_a c_a F(X_a), including the empty presentation. It does not mean a single source preimage.

What is forced

For all supplied written Y,Z, every supplied scalar c and every J in 𝓑,

Λ_𝛇(v)=v,
Λ_𝛇(Y+Z)=Λ_𝛇(Y)+Λ_𝛇(Z),
Λ_𝛇(cY)=cΛ_𝛇(Y),
Λ_𝛇(−Y)=−Λ_𝛇(Y),
Λ_𝛇(Y−Z)=Λ_𝛇(Y)−Λ_𝛇(Z),
Λ_𝛇(e_J)=ζ_J.                                       (S2)

Every allowed output blade has the following exact scalar combination of complete source values:

Mix(p_K,q_K)
  =F(p_K+q_K)−F(p_K)−F(q_K)
  =h s_K^w e_K             (K≠{0},I_s),
G_K=e_K                    (K≠I_s).                  (S3)

Consequently the complete output decomposition is

Y=𝓒_(I_s)(Y)Ω_s+Σ_(K≠I_s)𝓒_K(Y)G_K.                 (S4)

In particular,

𝓒_(I_s)(Y)=v
  iff Y has an actual finite scalar presentation
      using complete source values F(X_a).           (S5)

The forward direction is furnished explicitly by S0, S3 and S4. The reverse direction applies to every such supplied presentation.

For the same supplied list 𝛇, the following five statements are equivalent:

Λ_𝛇(F(X))=v for every supplied X;                     (S6a)
Λ_𝛇 vanishes on every entry of 𝓣;                    (S6b)
ζ_K=v for every K≠I_s;                               (S6c)
Λ_𝛇(Y)=ζ_(I_s)𝓒_(I_s)(Y) for every supplied Y;       (S6d)
Λ_𝛇(Mix(X,Y))=v for every supplied pair X,Y.          (S6e)

Thus the complete fixed-list scalar-contraction class that annihilates the full source has one remaining coefficient. No coefficient ζ_K is cancelled or divided.

The forcing

1. Recover the fixed-list scalar laws before applying them to sources

V01-PB-0004 C1 and C3 give the actual reconstruction and the coefficient laws for v, addition, scalar multiplication and negation. V01-PB-0001 gives additive association/commutation, product signs, scalar commutation and void absorption.

For addition, let A, C and U be the actual preceding accumulations of ζ_K𝓒_K(Y), ζ_K𝓒_K(Z) and ζ_K𝓒_K(Y+Z). At their empty prefixes A=C=U=v, so U=A+C. At the next prescribed K, coefficient addition and distribution give

U'=U+ζ_K𝓒_K(Y+Z)
   =(A+C)+ζ_K[𝓒_K(Y)+𝓒_K(Z)]
   =(A+ζ_K𝓒_K(Y))+(C+ζ_K𝓒_K(Z))=A'+C'.

Applying this equality at each entry of the already fixed list reaches S2's addition identity. Only the additive order has changed.

For the scalar law the empty prefixes satisfy U=v=cv=cA. At the next actual K,

U'=U+ζ_K𝓒_K(cY)
   =cA+ζ_K c𝓒_K(Y)
   =c[A+ζ_K𝓒_K(Y)]=cA'.

Here only the supplied scalar c has moved. The final accumulations give the scalar law. With negation the corresponding exact step is

U'=−A+ζ_K[−𝓒_K(Y)]
   =−[A+ζ_K𝓒_K(Y)]=−A'.

Its initial equality is v=−v, giving the negation law. Every contribution for input v has the factor 𝓒_K(v)=v, so the accumulator remains v. The difference law follows by applying addition to Y+(−Z) and then negation.

For the supplied blade e_J, the actual singleton presentation has coefficient u at J and v at every different index, including the empty-index cases. Its contraction accumulator stays v before J, becomes v+ζ_Ju=ζ_J at J, and remains ζ_J after every later void contribution. This proves Λ_𝛇(e_J)=ζ_J. No source property or derivative condition entered any of these arguments.

2. Recover each whole source difference from its two ordered mixed terms

V02-PB-0014 M2 gives, for every supplied pair p,q,

F(p+q)−F(p)−F(q)=Mix(p,q)
                =p e₀Sel(q)+q e₀Sel(p).

This parent identity already retains all four ordered occurrences in F(p+q) before subtracting its two complete self-sources. For the prescribed p=e_I,q=e_J, V01-PB-0008 P0 and V01-PB-0009 S1 give directly

Mix(p,q)=[w_J s(I,J)+w_I s(J,I)]e_K,
I△{0}△J=K.                                           (S7)

In the first prescribed case, I=∅ and J={0}△K, so I△{0}△J=K. In the second, I={m} and J={m}△({0}△K), so the two {m} toggles and the two {0} toggles cancel and again I△{0}△J=K. Thus both mixed products have the same displayed output K. Their sign factors are the S1 signs already used in S7. The witness claims PB-0009(S6,S6p), applied to this actual p_K+q_K, give

𝓒_K(F(p_K+q_K))=h s_K^w.

For each individual blade p_K=e_I or q_K=e_J, PB-0008(P0) and PB-0009(S1) give respectively

F(e_I)=w_I s(I,I)e_{ {0} },
F(e_J)=w_J s(J,J)e_{ {0} },

because I△{0}△I={0} and J△{0}△J={0}. Since the present target K differs from {0}, coefficient recovery makes each of their K coefficients v. Apply PB-0004(C3) to the complete source difference from M2:

𝓒_K(Mix(p_K,q_K))
  =𝓒_K(F(p_K+q_K))−𝓒_K(F(p_K))−𝓒_K(F(q_K))
  =h s_K^w−v−v=h s_K^w.

S7 already gives the whole mixed output as a scalar multiple of e_K; PB-0004(C1) identifies that scalar with its K coefficient. Consequently

Mix(p_K,q_K)=h s_K^w e_K,
G_K=k s_K^w h s_K^w e_K=(kh)(s_K^w s_K^w)e_K=e_K.    (S8)

The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. The specified sign is u or −u, so its self-product is u by PB-0001's two sign products. These equalities justify the last line. This uses the witness's stated coefficient, without importing a formula from the witness proof.

At the remaining target {0}, the supplied generator images give

Sel(e₀)=e₀−(−e₀)=he₀,
F(e₀)=e₀e₀he₀=he₀,
G_{ {0} }=khe₀=e₀.                                   (S9)

These equalities establish S3 at every target except I_s. Every source difference is obtained from the full mixed identity, without an additional blade-product parent or a separate four-term re-derivation.

3. Reconstruct the whole source-generated class

V01-PB-0004 C1 gives the entire input presentation

Y=Σ_(K∈𝓑)𝓒_K(Y)e_K.

Move the one I_s summand to the front by finite additive regrouping. Replace each remaining blade with its exact G value from S3. This gives S4. If 𝓒_(I_s)(Y)=v, the first summand is v and the remaining sum is a literal finite scalar combination of source values: at {0} insert S9, and at each other target insert S3's three complete source values, carrying the scalar factors of S0. Negative weights use only the permitted additive negatives. This supplies the forward direction of S5, rather than an existence claim without a construction.

Conversely, take any actually supplied presentation Y=Σ_a c_aF(X_a). V01-PB-0009 G1 gives 𝓒_(I_s)(F(X_a))=v at each actual entry. Coefficient addition and scalar compatibility therefore give, at each prefix,

𝓒_(I_s)(Y_prefix+c_aF(X_a))
  =𝓒_(I_s)(Y_prefix)+c_av
  =𝓒_(I_s)(Y_prefix).

The empty prefix has recovered coefficient v. Each next equality preserves it, so the coefficient of the supplied final presentation is v. This covers the empty presentation and proves the reverse direction. This result is additive generation by full source values; it has not solved F(X)=Y for one X.

4. Determine every fixed-list contraction in the stated class

S6a immediately gives S6b by substitution of the explicitly supplied inputs in 𝓣. Under S6b, use the laws already established in S2 and the exact complete source combinations S3. At each nonexceptional target,

ζ_K=Λ_𝛇(e_K)=Λ_𝛇(G_K)
   =k s_K^w[Λ_𝛇(F(p_K+q_K))−Λ_𝛇(F(p_K))−Λ_𝛇(F(q_K))]
   =v.

At {0} the same use of S9 gives ζ_{ {0} }=kΛ_𝛇(F(e₀))=v. These cover all targets except I_s; hence S6c holds. The coefficients of 𝛇 have not been divided, and their nonvoidness was never assumed.

Under S6c, every term in S1 other than the I_s term is v. The accumulator remains v before that position, becomes ζ_(I_s)𝓒_(I_s)(Y) there, and is unchanged by the later void terms. This proves S6d for every supplied Y. Substituting Y=F(X) and using V01-PB-0009 G1 gives S6a again.

To include S6e, apply Λ to V02-PB-0014 M2's complete mixed-source identity, using the already proved difference law:

Λ_𝛇(Mix(X,Y))
  =Λ_𝛇(F(X+Y))−Λ_𝛇(F(X))−Λ_𝛇(F(Y)).

Under S6a all three values are v, giving S6e. Conversely, under S6e take the actual pair (X,X). The parent's diagonal identity is Mix(X,X)=hF(X), so

v=Λ_𝛇(Mix(X,X))=hΛ_𝛇(F(X)).

Multiply by the existing k to obtain S6a. This uses the one declared guard, not a new cancellation rule. All five statements are therefore equivalent.

What is not forced

S5 characterizes finite scalar combinations of complete source values. It does not assert that every element in that class equals one F(X), that the witnesses solve R=v, or that their coefficients can be specified simultaneously as one source output.

S6 concerns a single fixed scalar list retained while inputs change. It does not classify nonlinear, input-dependent or solution-only relations. Its coefficients may have nonvoid derivatives; S2 is an algebraic law and supplies no permission to commute a derivative through Λ without its additional product terms.

The derived missing output is an output relation, not an absent input coefficient. The arbitrary full Y is reconstructed before the coefficient condition in S5 is imposed. The reconstructed presentation includes every coefficient of that supplied Y.

What it means

The source's already derived individual witnesses can be combined through the exact mixed-source identity to isolate every permitted output blade. This gives a complete finite description of the output generated by source values and, by applying a fixed coefficient contraction on those exact generators, determines every scalar relation in that contraction class. The same relation annihilates the mixed source precisely when it annihilates the quadratic source. The nonlinear single-source image remains a distinct question.

The source-factor domain has an exact Γ-sign criterion and a reversible active-part recovery (V04-PB-0026)

LAYER: 6 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0007, V01-PB-0008, V02-PB-0011

Declares

The operations and scalar guards retain 04 and the cited interfaces, including h=u+u and k=h⁻¹. T(X)=(X̃)* and A_X=k(X−T(X)) retain their definitions in V01-PB-0007. The source factor K_A(X)=e₀Ã_X retains its definition in V01-PB-0007; write K_A as the abbreviation for K_A(A) when the displayed input is A. Define the operation

Γ(Y):=e₀T(Y)e₀.

K is an actual supplied written factor. U is an actual supplied written expression wherever KUK=K is retained. The scalar c_fac in F7 is a supplied scalar expression, distinct from any word-reordering sign. The finite blade list, λ₀(I), recovered coefficients and whole-blade signs retain V01-PB-0004 and V01-PB-0008 P0. The scalar signs ε_μ retain their definition in V01-PB-0002, carried by V01-PB-0004's C_μ.

What is forced

For all supplied Y,Z and scalars a, Γ is additive, carries scalar factors, fixes scalars, reverses products, undoes itself and commutes with each individual supplied derivative:

Γ(Y+Z)=Γ(Y)+Γ(Z), Γ(aY)=aΓ(Y), Γ(a)=a,
Γ(YZ)=Γ(Z)Γ(Y), Γ(Γ(Y))=Y,
Γ(e_μ)=−e_μ, ∂_μΓ(Y)=Γ(∂_μY).                    (F1)

Its native whole-blade action is

Γ(e_I)=τ(I)λ₀(I)e_I.                              (F2)

The eight written lengths 0 through 7 give respectively u,−u,−u,u,u,−u,−u,u. Consequently Γ(K)=K iff the recovered coefficients of K at lengths 1,2,5,6 are v, equivalently iff all coefficients at τ(I)λ₀(I)=−u are v. Every retained coefficient remains unrestricted.

For T(A)=−A, A_A=A and K_A(A)=e₀Ã. The two actual transformations A↦e₀Ã and K↦K̃e₀ undo one another and give

T(A)=−A iff Γ(e₀Ã)=e₀Ã,
Γ(K)=K ⇒ A=K̃e₀, T(A)=−A, K_A=K.                 (F3)

For this A and every supplied H with T(H)=H, A_(A+H)=A and K_A(A+H)=K. Thus every supplied Γ-fixed K is attained by precisely its recovered active part, together with an arbitrary supplied T-fixed part. This assertion concerns source factors, not the residual equation.

For K=c_fac u and any supplied U, with no further guard on c_fac,

KUK=K iff (c_fac c_fac)𝓒_∅(U)=c_fac
             and (c_fac c_fac)𝓒_I(U)=v for every nonempty I. (F7)

Such a U supplies c=𝓒_∅(U) with (c_fac c_fac)c=c_fac. Conversely that actual scalar identity supplies U=cu. More fully, all such U are exactly

U=cu+Σ_(I≠∅)d_Ie_I,
(c_fac c_fac)c=c_fac, (c_fac c_fac)d_I=v.

The forcing

1. Derive the operation before its domain

The two-sided e₀ product is e₀e₀=u. Directly from 04, T(e₀)=(ẽ₀)=e₀=−e₀. V01-PB-0007 gives T(T(Y))=Y and T(YZ)=T(Z)T(Y). Consequently

Γ(YZ)=e₀T(Z)T(Y)e₀
      =(e₀T(Z)e₀)(e₀T(Y)e₀)=Γ(Z)Γ(Y),
Γ(Γ(Y))=e₀T(e₀T(Y)e₀)e₀
        =e₀(−e₀)Y(−e₀)e₀=Y.

The first inserted e₀e₀ is precisely u; the two negatives in the second line multiply to u by V01-PB-0001. Distribution and T's scalar law give

Γ(Y+Z)=Γ(Y)+Γ(Z),   Γ(aY)=aΓ(Y),
Γ(a)=e₀ae₀=a,      Γ(v)=v, Γ(−Y)=−Γ(Y).

At μ=0, Γ(e₀)=e₀(−e₀)e₀=−e₀. At each of the six supplied spatial labels, T(eᵢ)=eᵢ and the one generator exchange gives Γ(eᵢ)=e₀eᵢe₀=−eᵢe₀e₀=−eᵢ. Finally the derivative compatibility of V02-PB-0011 D1,D3, together with ∂_μe₀=v, gives

∂_μΓ(Y)=vT(Y)e₀+e₀T(∂_μY)e₀+e₀T(Y)v=Γ(∂_μY).

2. Recover Γ's sign from the native whole-blade interface

V01-PB-0008 P0 gives T(e_I)=τ(I)e_I. V01-PB-0004 C0 at m=0 gives C₀(e_I)=e₀e_Ie₀=λ₀(I)e_I, because ε₀=u. Hence, on every blade,

Γ(e_I)=e₀T(e_I)e₀=τ(I)e₀e_Ie₀=τ(I)λ₀(I)e_I.

This is F2 without a count-parity formula. For the displayed length reading, write ρ_m for the actual product of −u factors along a word of that stated length. If 0 is absent, θ(I)=u and λ₀(I)=ρ_m. If it is present, θ(I)=−u and λ₀(I)=ρ_(m−1), so θ(I)λ₀(I)=ρ_m in that case too. The reversion sign at the next listed length is the previous sign times the actual exchanges moving that added largest generator across the previous word. These are the eight separate written sign calculations:

Written length	ρ_m	Reversion sign from its preceding written product	τ(I)λ₀(I)=η(I)ρ_m
0	u	u	u
1	−u	u·u=u	−u
2	(−u)(−u)=u	(−u)u=−u	−u
3	u(−u)=−u	u(−u)=−u	u
4	(−u)(−u)=u	(−u)(−u)=u	u
5	u(−u)=−u	uu=u	−u
6	(−u)(−u)=u	(−u)u=−u	−u
7	u(−u)=−u	u(−u)=−u	u

Each ρ entry is its preceding written value multiplied by the next actual −u; each reversion entry uses the corresponding preceding ρ. These seven specified applications use PB-0003's contextual exchange. V01-PB-0008 P0's route-independence identifies their actual reordering products with η(I), even though that interface's displayed route moves the smallest label first.

Reconstruct K=Σ_I 𝓒_I(K)e_I by V01-PB-0004 and distribute Γ over that full supplied list. Its I coefficient is τ(I)λ₀(I)𝓒_I(K). Where the sign is u it already agrees. Where it is −u, equality is equivalent to h𝓒_I(K)=v. The guarded inversion gives hk=u, and rule 8 gives kh=hk=u, so this is equivalent to 𝓒_I(K)=k(h𝓒_I(K))=v. Conversely the displayed coefficient conditions make every term fixed. This proves both directions at the full written input before its fixed-sign refinement.

3. The correspondence with the active part

For T(A)=−A, the definition gives A_A=k[A−(−A)]=khA=A, so K_A(A)=e₀Ã. Apply star to T(A)=−A and use V01-PB-0007 M1 to obtain Ã=−A*. Since T(Ã)=A* by that same claim, this is T(Ã)=−Ã. Product reversal then gives

T(K_A)=T(e₀Ã)=T(Ã)T(e₀)=(−Ã)(−e₀)=Ãe₀,
Γ(K_A)=e₀Ãe₀e₀=e₀Ã=K_A.

Conversely Γ(K)=K implies T(K)=e₀Ke₀ by multiplication on both sides by e₀. Set A=K̃e₀. Reversion and T commute by V01-PB-0007, so

T(A)=T(e₀)T(K̃)=−e₀(T(K))̃
     =−e₀(e₀Ke₀)̃=−e₀e₀K̃e₀=−A,
Ã=(K̃e₀)̃=e₀K,
e₀Ã=e₀e₀K=K.

Starting instead at A, the reverse construction is (e₀Ã)̃e₀=Ae₀e₀=A. Thus both directions undo their actual inputs. V01-PB-0007's sign-characterized uniqueness gives A_(A+H)=A for each supplied T-fixed H. Conversely, if an actual X has K_A(X)=K, the reverse construction gives

A_X=(K_A(X))̃e₀=K̃e₀=A,
X=A_X+H_X=A+H_X, T(H_X)=H_X.

The last two statements are V01-PB-0007's complete decomposition. Thus these and only these actual X attain K. This establishes F3's precise factor domain, with the residual condition still separate.

4. Scalar factors expose their exact witness condition

For scalar c_fac, T(c_fac e₀)=−c_fac e₀ and e₀(c_fac e₀)̃=c_fac u. Scalar centrality therefore gives KUK=(c_fac c_fac)U. Reconstruct U through V01-PB-0004. Comparing every coefficient to c_fac u proves F7, including all nonempty positions. Its empty coefficient is the scalar identity (c_fac c_fac)c=c_fac. Conversely U=cu gives KUK=(c_fac c_fac)cu=c_fac u from precisely that identity. More generally all actual witnesses have the recovered form

U=cu+Σ_(I≠∅)d_Ie_I,   (c_fac c_fac)c=c_fac, (c_fac c_fac)d_I=v.

No coefficient has been divided by c_fac or its product. An already guarded scalar inverse, when actually supplied, can discharge this obligation at its own scope; no such guard is added here.

What is not forced

F1–F3 describe Γ and the complete source-factor domain. They make no witness-availability statement for an arbitrary K. F7 retains its exact scalar products and contains no cancellation of c_fac. Attainment of a source factor by A+H is algebraic; the same A+H must still satisfy its whole residual equation if it is to be a solution.

What it means

The source factor can be recovered from its active part and returned to it. A whole-blade sign gives its complete algebraic domain. This domain statement and the separate supplied-witness product condition have distinct premises, which remain visible to later completion constructions.

The direct generator calculation T(e₀)=−e₀ in §1 is also the {0} instance of V01-PB-0008(P0); that general blade interface is used in §2 for the full list.

Six exact bivector pairs resolve their complete product information and derivative corners (V04-PB-0031)

LAYER: 6 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V01-PB-0008, V02-PB-0011, V07-PB-0042

Declares

Retain u,v,h,k,B from 04 with its guard. T,A_X,H_X,K_A retain V01-PB-0007’s definitions; Sel,D,L,F,R retain V01-PB-0006’s; Mix retains V02-PB-0014’s definition; coefficient recovery and λ_μ retain V01-PB-0004’s. ε_μ and ρ retain V01-PB-0002’s and V01-PB-0003’s definitions carried by V01-PB-0004. All inputs are supplied written expressions.

Begin with the unrestricted product

Prod_A(Δ):=Δe₀Ã.                                     (BC1)

This definition is valid for every supplied A,Δ, without a T sign or a relation between A and an active part.

The pair index is the full finite list satisfying i<j with i,j spatial and disjoint from {1,2}. Its increasing enumeration is (3,4),(3,5),(3,6),(4,5),(4,6),(5,6). For each listed (i,j), define

C_{ij}:=eᵢeⱼ,       I_{ij}:=BC_{ij},
P_{ij,σ}:=k(u+σI_{ij})       for σ in {u,−u},
A^{pair}_{ij}(f,g):=fB+gC_{ij},
A^{pair}_{ij,σ}:=B+σC_{ij},       K_{ij,σ}:=e₀(A^{pair}_{ij,σ})̃.

The scalar operands f,g need not be derivative-constant or invertible. The two signs are retained together. In a calculation with one actual pair, abbreviate C,I,P_σ,A(f,g),A_σ,K_σ as local references to precisely those displayed pair expressions. Put L^{in}{ij}={1,2,i,j} and L^{out}{ij}={0,…,6}\L^{in}_{ij}. These names are finite lists, not new spaces.

For any supplied Δ define the two collected derivative expressions

D_out^{ij}(Δ):=Σ_{μ in L^{out}_{ij}} b_{{μ}}∂_μΔ,
D_in^{ij}(Δ):=Σ_{μ in L^{in}_{ij}} b_{{μ}}∂_μΔ,
b_{{0}}=e₀, b_{{i}}=−eᵢ as in V01-PB-0008 P5u.       (BC2)

Each sum uses its actual increasing label order. Right-null at K has the ordered meaning ΔK=v of V07-PB-0042. At the actual source factor this reads ΔK_A(X)=v, with K_A(X)=e₀Ã_X as defined in V01-PB-0007. The named blade I_{ij} is distinct from a coefficient index; coefficient indices in this record are denoted J.

What is forced

At T(Δ)=Δ and A=A_X, the exact recovery is

F(X+Δ)−F(X)=Mix(X,Δ)=hProd_A(Δ),
R(X+Δ)−R(X)=h[D(Δ)B−Prod_A(Δ)].                    (BC3)

Thus source equality is exactly Prod_A(Δ)=v; equal residuals are exactly D(Δ)B=Prod_A(Δ). Under R(X)=v the latter tests the new solution. Preserving both sides separately is exactly Prod_A(Δ)=v and D(Δ)=v. For any supplied A,Δ with Prod_A(Δ)=v,

(∂_μΔ)e₀Ã+Δe₀(∂_μA)̃=v                          (BC4)

for each actual μ. These are the derivatives of the same retained undifferentiated product identity.

Each of the six pairs supplies

BB=CC=−u,    BC=CB,    e₀B=Be₀,    e₀C=Ce₀,
B̃=−B, C̃=−C, B*=B, C*=C,
II=u,  Ĩ=I*=T(I)=I,   BI=IB=−C,
P_++P_−=u,   P_σP_σ=P_σ,   P_σP_{−σ}=v,
T(P_σ)=P_σ,    ∂_μP_σ=v,    P_σ≠v.                 (BC5)

Both P_σ commute with B and e₀. The complete label routing and whole-input recovery are

e_μI=I e_μ, e_μP_σ=P_σe_μ                 for μ in L^{out}_{ij},
e_μI=−I e_μ, e_μP_σ=P_{−σ}e_μ            for μ in L^{in}_{ij},
P_σI=IP_σ=σP_σ,
W=P_+WP_++P_+WP_−+P_−WP_++P_−WP_−.                (BC5a)

The four displayed products recover by multiplying W by their own left and right factors. They are all void exactly when W=v. In particular e₁P_σ=P_{−σ}e₁ and P_σe₁=e₁P_{−σ}, for each pair and each sign.

The complete product identity, with no T condition on Δ, is

Prod_{A(f,g)}(Δ)Be₀
  =(f−g)ΔP_++(f+g)ΔP_−.                            (BC6)

Consequently

Prod_{A(f,g)}(Δ)=v
  iff (f−g)ΔP_+=v and (f+g)ΔP_−=v.                (BC7)

The factors f−g and f+g are not cancelled. Also T(A(f,g))=Ã(f,g)=−A(f,g), so A_{A(f,g)}=A(f,g). For the two complete outputs,

kProd_{A_σ}(Δ)Be₀=ΔP_{−σ},
Δ=k[Prod_{A_+}(Δ)+Prod_{A_−}(Δ)]Be₀.               (BC8)

Under T(Δ)=Δ, their individual null descriptions are exactly

Prod_{A_σ}(Δ)=v
  iff Δ=P_σΔP_σ
  iff Δ=P_σH_YP_σ for a supplied Y.                (BC9)

The final existence statement is constructive: a supplied null Δ uses Y=Δ; any supplied Y in the displayed expression gives a T-fixed null Δ. In particular P_σ itself is a nonvoid derivative-constant member. For every Δ the two products recover that same whole Δ by BC8; both products are void exactly when Δ=v.

As a non-use identification, BC9 is V05-PB-0035 S4b at ι=T and positive part sign u, with K=K_σ,U=kBe₀ for either retained pair sign σ. BC10 below supplies KUK=K and E_L=P_σ, while BC5 gives T(E_L)=E_L. The proof of BC9 here uses its own products.

The factor-witness application is the explicit pair of ordered products

K_σ(kBe₀)=(kBe₀)K_σ=P_{−σ},
K_σP_{−σ}=P_{−σ}K_σ=K_σ,
K_σ(kBe₀)K_σ=K_σ.                                 (BC10)

These products introduce no nonscalar inverse notation.

For every supplied Δ already satisfying Δ=P_σΔP_σ, the exact derivative structure is

∂_μΔ=P_σ(∂_μΔ)P_σ,
D_out^{ij}(Δ)=P_σD_out^{ij}(Δ)P_σ,
D_in^{ij}(Δ)=P_{−σ}D_in^{ij}(Δ)P_σ,
D(Δ)=D_out^{ij}(Δ)+D_in^{ij}(Δ).                   (BC11)

It follows in both directions that

D(Δ)=v iff D_out^{ij}(Δ)=v and D_in^{ij}(Δ)=v.      (BC12)

At (i,j)=(3,4) these are exactly

e₀∂₀Δ−e₅∂₅Δ−e₆∂₆Δ=v,
−e₁∂₁Δ−e₂∂₂Δ−e₃∂₃Δ−e₄∂₄Δ=v.

For a supplied base solution X whose actual constructed active part is A_σ, and Δ=P_σH_YP_σ, the replacement X+Δ is a solution iff both complete balances BC12 hold. The existence of such a base solution is not supplied by this conditional statement.

The forcing

1. Use the unrestricted comparison and retain the differentiated product

At A=A_X, Prod_A(Δ)=ΔK_A(X) by association. T(Δ)=Δ gives Sel(Δ)=v by V01-PB-0007's stated selector criterion. Apply V01-PB-0006(3)–(5) with its same selector-null increment H=Δ: the source difference in (3) is Δe₀Sel(X)=hΔe₀Ã_X=hProd_A(Δ), using V01-PB-0007's stated identity Sel(X)=hÃ_X. Its residual difference is therefore h[D(Δ)B−Prod_A(Δ)]. Under the same Sel(Δ)=v, V02-PB-0014’s definition gives directly

Mix(X,Δ)=Xe₀Sel(Δ)+Δe₀Sel(X)=Δe₀Sel(X).

This identifies that source difference and proves every equality of BC3.

The guarded inversion gives hk=u, and rule 8 gives kh=hk=u. Multiply each difference by k. The void-difference criterion gives source equality iff Prod_A(Δ)=v, and residual equality iff D(Δ)B=Prod_A(Δ). Under the separate R(X)=v premise, residual equality is exactly R(X+Δ)=v. Under the already retained Sel(Δ)=v, V01-PB-0006(5) states the separate-side condition as Δe₀Sel(X)=v and D(Δ)=v. Its first condition is hProd_A(Δ)=v iff Prod_A(Δ)=v. Equivalently B0's actual undoing −B turns L(Δ)=hD(Δ)B=v into D(Δ)=v. This proves the separate-side criterion without assuming the right-null condition first.

For BC4 apply V02-PB-0011 D7z to the actual product Δ(e₀Ã)=v. D3 and ∂_μe₀=v give

∂_μ(e₀Ã)=e₀(∂_μA)̃,
(∂_μΔ)e₀Ã+Δe₀(∂_μA)̃=v.

The undifferentiated product remains the premise of that application.

2. Derive the six finite word constructions

The membership rule leaves the labels 3,4,5,6. At first label 3 the larger permitted labels are 4,5,6; at first label 4 they are 5,6; at first label 5 it is 6; at first label 6 none is larger. These are exactly the six displayed pairs. In each, the four labels 1,2,i,j are distinct. For C=eᵢeⱼ V01-PB-0003’s adjacent exchange and 04’s squares give

CC=eᵢ(eⱼeᵢ)eⱼ=−(eᵢeᵢ)(eⱼeⱼ)=−u.

The same calculation gives BB=−u. The complete contextual exchange chain for the two pairs is

e₁e₂eᵢeⱼ=−e₁eᵢe₂eⱼ
           =eᵢe₁e₂eⱼ
           =−eᵢe₁eⱼe₂
           =eᵢeⱼe₁e₂.

Thus BC=CB. The two other actual chains are

e₀e₁e₂=−e₁e₀e₂=e₁e₂e₀,
e₀eᵢeⱼ=−eᵢe₀eⱼ=eᵢeⱼe₀.

They give e₀B=Be₀ and e₀C=Ce₀. Reversion gives B̃=e₂e₁=−B and C̃=eⱼeᵢ=−C; star fixes each supplied spatial letter and hence each pair. These computations use only the actual distinctness of the four labels, so apply to each of the six supplied entries without sampling one.

Now association and that proved commutation give

II=BCBC=(BB)(CC)=u,
BI=BBC=−C=BCB=IB,
Ĩ=C̃B̃=CB=I,        I*=I,        T(I)=I.

The finite distributions in the two actual signs are

(u+I)(u+I)=h(u+I),
(u−I)(u−I)=h(u−I),
(u+I)(u−I)=(u−I)(u+I)=v.

Multiply by kk and use kkh=k to obtain the three product rules for P. Addition gives P_++P_−=kh=u. T fixes k,u,I, so fixes P. The previously derived commutations give the two claimed commutations of P. Each generator derivative is void, and V02-PB-0011 D1 gives ∂k=∂u=v; applying the complete product rule of 04 to these displayed finite words gives ∂P_σ=v.

Their nonvoidness uses our recovered coefficients rather than a supposition about signs or positivity. I is the already increasing nonempty four-letter blade e_{12ij}. Consequently V01-PB-0004 C1 recovers 𝓒_∅(P_σ)=k. If P_σ=v, its C3 coefficient law forces k=v, hence u=hk=v. Then h=u+u=v, contradicting its guard. Thus P_σ is nonvoid. The same recovery at the nonempty index shows I is distinct from both u and −u if that distinction is needed; no ordering of scalars has been introduced.

Use V01-PB-0004(C0) at this actual blade I=e_{12ij}:

ε_μe_μI e_μ=λ_μ({1,2,i,j})I.

Right-multiply by e_μ. Since e_μe_μ=ε_μ and ε_με_μ=u, the left side reduces to e_μI, proving e_μI=λ_μ({1,2,i,j})I e_μ. The explicit sign accumulation on these four letters is

ρ({1,2,i,j})=((−u)(−u))((−u)(−u))=u.

For a label outside the list, the definition λ_μ(L)=ρ(L∖{μ}) gives λ_μ(L)=ρ(L)=u. For a label inside the list, scalar commutation of the actual four factors gives ρ(L)=(−u)ρ(L∖{μ}). Multiplication by −u, with (−u)(−u)=u, gives λ_μ(L)=(−u)ρ(L)=−u. These are the two exhaustive membership cases for the supplied seven labels, now obtained from the exported whole-blade conjugation and its explicit scalar products. Distributing the definition of P gives

e_μP_σ=P_σe_μ       if μ is outside L^{in}_{ij},
e_μP_σ=P_{−σ}e_μ    if μ is inside L^{in}_{ij}.

The same II=u gives, in both orders,

P_σI=k(I+σu)=σk(u+σI)=σP_σ=IP_σ.

For the whole-input formula insert the actual unity P_++P_− on both sides:

W=(P_++P_−)W(P_++P_−)
  =P_+WP_++P_+WP_−+P_−WP_++P_−WP_−.

Multiplication on either side by one of the two P factors keeps exactly its matching term and makes each other term void using the mixed products in BC5. Thus the four products recover, all are void when W=v, and their reconstruction gives W=v when all four are void. This proves all of BC5a before its derivative use.

3. Resolve the entire scalar-pair product without dividing its coefficients

Since T and reversion fix f,g and negate B,C, they negate A(f,g). Its constructed active part is therefore k[A−(−A)]=khA=A. Using BI=−C gives

A(f,g)=B(f−gI).

From their actual expressions and II=u,

IP_+=P_+,       IP_−=−P_−,
f−gI=(f−g)P_++(f+g)P_−.

The factor f−gI commutes with B,e₀ because I does. Therefore, keeping Δ in its original left position,

Prod_{A(f,g)}(Δ)Be₀
  =−Δe₀B(f−gI)Be₀
  =−Δ(f−gI)e₀(BB)e₀
  =Δ(f−gI)
  =(f−g)ΔP_++(f+g)ΔP_−.

This proves BC6 with both scalar factors present. The right multiplier Be₀ is undone on both sides by −e₀B:

(Be₀)(−e₀B)=−BB=u,
(−e₀B)(Be₀)=−e₀(BB)e₀=u.

Thus the left side of BC6 is void iff Prod is void. Right multiplication of the right side by P_+ recovers exactly its first summand, and multiplication by P_− recovers exactly its second. If the complete sum is void both recoveries are void; conversely two void summands give a void sum. This is BC7. It uses the proved complementary products, not a general inference from a sum to its terms.

At f=u,g=σ the two instances of BC6 give Prod_{A_σ}(Δ)Be₀=hΔP_{−σ}. Apply k and add the two identities. Since P_++P_−=u this proves BC8. At f=u,g=v, the same formula gives Prod_B(Δ)Be₀=Δ; this particular single factor retains the whole operand.

For BC10, reversion of A_σ gives K_σ=−e₀B(u−σI). Every factor here commutes with I, and e₀ commutes with B. Directly in both orders,

K_σ(kBe₀)=−ke₀B(u−σI)Be₀=k(u−σI)=P_{−σ},
(kBe₀)K_σ=−kBe₀e₀B(u−σI)=k(u−σI)=P_{−σ}.

Also (u−σI)P_{−σ}=u−σI and its reversed order has the same value. Multiply the displayed expression for K_σ by P_{−σ} on either side to get K_σ itself. Association now proves the last BC10 equality.

4. Recover all null operands at either retained sign

BC8 and the written undoings say Prod_{A_σ}(Δ)=v iff ΔP_{−σ}=v. Add ΔP_σ+ΔP_{−σ}=Δ to obtain Δ=ΔP_σ. With T(Δ)=Δ, apply T to this equality, reverse the factors and use T(P_σ)=P_σ. It gives Δ=P_σΔ. Combining yields Δ=P_σΔP_σ. Conversely this sandwich multiplied on the right by P_{−σ} is void. These implications prove the first equivalence of BC9.

For arbitrary supplied Y, V01-PB-0007’s part-sign claim gives T(H_Y)=H_Y. Hence T(P_σH_YP_σ)=P_σH_YP_σ. Its right product by P_{−σ} is void, so it has the required right-null relation at K_σ. Conversely for a given T-fixed null Δ, its own H part is

H_Δ=k[Δ+T(Δ)]=khΔ=Δ.

Take the actual Y=Δ in the sandwich already proved. This proves the construction and completeness in BC9. At Y=u the sandwich is P_σuP_σ=P_σ, nonvoid and derivative-constant by §2. No arbitrary family is selected by success in a trial: BC9 describes all operands meeting its explicit sign and product conditions.

5. Retain and separate the two actual derivative collections

Differentiate the actual identity Δ=P_σΔP_σ. The full product-rule expansion of 04 has three summands; the first and last vanish because ∂_μP_σ=v. Therefore ∂_μΔ=P_σ(∂_μΔ)P_σ.

Use the exported BC5a label routing. The scalar sign in b_{{μ}} does not alter these relations. Substitute the derivative sandwich in every actual b_{{μ}}∂_μΔ and use the appropriate displayed relation. Summing the outside and inside lists gives their two corner identities in BC11. Their combined seven terms are exactly D(Δ), proving its final identity.

Left multiplication of D(Δ) by P_σ returns D_out; left multiplication by P_{−σ} returns D_in. This follows by using P_σP_σ=P_σ and P_σP_{−σ}=v in their actual corner expressions. Thus D=v implies both recovered parts void. Conversely both parts void imply D=v by their sum. This proves BC12 without setting an individual derivative to void.

Finally retain R(X)=v and A_X=A_σ, and supply Δ=P_σH_YP_σ. Sections 1 and 4 give Sel(Δ)=v and Prod_{A_X}(Δ)=v. Consequently R(X+Δ)=hD(Δ)B. Undo h and B by k and −B, and use BC12, to obtain the claimed equivalence for this same supplied base and difference. Each condition remains attached to that use.

What is not forced

The full mixed-source comparison precedes this conditional family. The pair index is exactly the increasing pairs from {3,4,5,6}, as derived by its membership rule. The result classifies these pair products at their stated premises. No member fB+gC is assumed to be Ψ or its active part. A proposed base with that actual part retains its own equation as an additional premise.

BC7 does not undo f±g and does not assume a scalar domain without zero divisors. BC12 separates two particular recoverable corners; it does not separate their constituent derivative terms. The products and derivative collections have their displayed algebraic scope.

What it means

For each of the six declared pairs, the two complete product outputs in BC8 reconstruct every supplied difference Δ. Under T(Δ)=Δ, each individual null class is exactly Δ=P_σH_YP_σ as in BC9.

For a supplied base X with R(X)=v and A_X=A_σ, and a supplied difference Δ=P_σH_YP_σ, the two complete balances BC12 are necessary and sufficient for R(X+Δ)=v. These are the premises of the conditional application: the existence of that base is not supplied here, and its active part cannot be changed independently of the difference’s corner. BC11 explains the two recoverable derivative corners without making the individual derivatives void.

The all-generator sign operation and its derivative compatibility (V07-PB-0043)

LAYER: 6 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0006, V01-PB-0007, V02-PB-0011

Declares

For each supplied full written X, put

Gr(X):=e₀X*e₀.

This is one expression built from the supplied star and products. The names even and odd, when used, mean respectively Gr(X)=X and Gr(X)=−X. The sign ρ(I) is V01-PB-0003's actual written product of −u, once per generator of e_I in its listed order, with ρ(∅)=u. Define ε₀:=u and εᵢ:=−u for i=1,2,3,4,5,6. D retains V01-PB-0006's definition and T retains V01-PB-0007's. The complete words B=e₁e₂ and Ω=e₀e₁e₂e₃e₄e₅e₆ are written products.

What is forced

The actual seven-term derivative can be written

D(X)=Σ_(μ=0,1,2,3,4,5,6) ε_μe_μ∂_μX.                (G0d)

For supplied X,Y and scalar c,

Gr(X+Y)=Gr(X)+Gr(Y),    Gr(cX)=cGr(X),
Gr(XY)=Gr(X)Gr(Y),      Gr(Gr(X))=X,
Gr(c)=c,    Gr(v)=v,    Gr(−X)=−Gr(X).              (G1)

At every supplied generator and every actual blade,

Gr(e_μ)=−e_μ,        Gr(e_I)=ρ(I)e_I,
Gr(B)=B,             Gr(Ω)=−Ω.                       (G2)

The operation has the following compatibility identities at every supplied X:

Gr(X̃)=(Gr(X))̃=e₀T(X)e₀,    Gr(X*)=(Gr(X))*,
Gr(T(X))=T(Gr(X)),
∂_μGr(X)=Gr(∂_μX),
Gr(D(X))=−D(Gr(X)).                                (G3)

For either actual sign σ=u or −u these yield the two separate implications

T(X)=σX ⇒ T(Gr(X))=σGr(X),
Gr(X)=σX ⇒ Gr(D(X))=−σD(X).                        (G3a)

Each statement includes X=v and includes a void output.

The forcing

1. Product, scalar and involution identities

Insert the supplied e₀e₀=u and use star's product rule:

Gr(X)Gr(Y)=e₀X*e₀e₀Y*e₀
          =e₀X*Y*e₀=e₀(XY)*e₀=Gr(XY).

Distribution through the two exterior factors gives additivity. Star fixes c and c commutes with the generators, so Gr(cX)=cGr(X) and Gr(c)=ce₀e₀=c. Absorption gives Gr(v)=v; the product-sign laws of V01-PB-0001 give Gr(−X)=−Gr(X). V01-PB-0007's double-star identity gives

Gr(Gr(X))=e₀(e₀X*e₀)*e₀
          =e₀(−e₀)X(−e₀)e₀=X.

The two negative signs combine to u and the two adjacent e₀ pairs give u. These calculations prove G1.

2. Apply the operation to each actual word

At e₀, Gr(e₀)=e₀(−e₀)e₀=−e₀. For each spatial i, V01-PB-0003's adjacent exchange gives

Gr(eᵢ)=e₀eᵢe₀=−eᵢe₀e₀=−eᵢ.

The empty word is u and Gr(u)=u. If an actual prefix w has acquired the sign c, the next generator g gives, using G1,

Gr(wg)=Gr(w)Gr(g)=(cw)(−g)=c(−u)wg.

Apply this displayed equality to the actual generator listing of e_I. Its sign is exactly the same ρ(I) supplied by V01-PB-0003. The B word has two negative factors which combine to u. The Ω word has seven negative factors: pair the first with the second, the third with the fourth, and the fifth with the sixth. Those three products give u and the last factor remains −u. This proves G2 without any unprovided word or length rule.

3. The operation identities under reversion and differentiation

V01-PB-0007 gives commutation of reversion and star. Reverse the defining expression, retaining order:

(Gr(X))̃=e₀(X*)̃e₀=e₀(X̃)*e₀=Gr(X̃)=e₀T(X)e₀.

Apply star instead:

(Gr(X))*=(−e₀)X(−e₀)=e₀Xe₀=Gr(X*).

Composing these two equalities gives commutation with T. For each actual derivative label, V02-PB-0011(D3) and 04's product rule give

∂_μ(e₀X*e₀)
  =(∂_μe₀)X*e₀+e₀∂_μ(X*)e₀+e₀X*(∂_μe₀)
  =e₀(∂_μX)*e₀=Gr(∂_μX).

The first and third contributions are void by the supplied generator derivatives and absorption. Substituting ε₀=u and the six εᵢ=−u into V01-PB-0006's definition of D gives G0d. G1 and G2 give at each of those seven labels

Gr(ε_μe_μ∂_μX)=−ε_μe_μGr(∂_μX)
               =−ε_μe_μ∂_μGr(X).

Add those seven equalities to obtain G3's final identity. If Gr(X)=σX, the derivative of the scalar sign is void by V02-PB-0011(D1), so D(σX)=σD(X); substitution gives the stated sign consequence. Commutation with T likewise gives T(Gr(X))=σGr(X) whenever T(X)=σX.

What is not forced

Gr is the specific displayed operation e₀X*e₀; each use retains that definition. Commutation with each individual derivative produces the stated sign reversal for D because D also contains its generator factors. No sign hypothesis on the distinguished Ψ is supplied here.

What it means

One operation negates every generator while preserving product order. Its written identities separate signs of complete expressions and track how those signs behave under the supplied derivatives. Consumers can use these identities directly without constructing the operation again inside another decomposition.

The expression e₀T(X)e₀ in G3 is also V04-PB-0026's Γ(X). This is a non-use identification of the written expressions; the proof above uses only the stated star, reversion and e₀ products.

The full equation has one coupled index-0 view, and its two residual partitions meet in the same four parts (V02-PB-0013)

LAYER: 7 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0005, V01-PB-0006, V01-PB-0007, V01-PB-0008, V01-PB-0009, V01-PB-0010, V02-PB-0011, V02-PB-0012

Declares

The scope is the full supplied written element, with no support or derivative restriction. W denotes an arbitrary supplied written element; its specialization to Ψ uses the asserted axiom only where stated.

• u,v,h,k retain 04's scalar meanings; Sel,D,L,F,R retain PB-0006's definitions; the two involutions are 04's operations; 𝓒_I retains PB-0004's coefficient recovery. The blade-product sign χ is PB-0005's finite construction. In particular L(W)=h(D(W))B, F(W)=We₀Sel(W), and R(W)=L(W)−F(W).

• The two part expressions are

  X:=k(W+W*),    Y:=k e₀(W−W*).                         (V1)
  

  Their properties and reconstruction are proved below. X and Y are constructed from W; they are not selected values of Ψ. X' and Y' name supplied alternative parts only in the uniqueness proof, where their star-fixed properties and displayed recombination are retained hypotheses.

• α(Z):=e₀Ze₀ abbreviates this written product for any supplied Z. It does not denote the star operation. A written element is called spatial here when its recovered presentation has only indices contained in {1,2,3,4,5,6}. The equivalent star-fixed criterion is proved and stated in V3a; §6 retains it for its coefficient routing.

• D_s(Z):=e₁(∂₁Z)+e₂(∂₂Z)+e₃(∂₃Z)+e₄(∂₄Z)+e₅(∂₅Z)+e₆(∂₆Z). A dot abbreviates ∂₀, retaining exactly that supplied derivative label.

• S_X:=X̃−X and V_Y:=α(Ỹ)+Y abbreviate the two displayed source expressions.

• The source-side parts and residual-side parts are

  F_0:=X V_Y+α(Y)S_X,
  F_1:=α(X)S_X+Y V_Y,
  
  E_0:=h(Ẏ−D_sX)B−F_0,
  E_1:=h(Ẋ+D_sY)B−F_1.                                 (V2)
  

  The subscripts distinguish the star-positive and e₀-multiplied placements below. They do not label additional inputs.

• Π_{r,t} retains V02-PB-0012's four sign-part expressions. I_s={1,2,3,4,5,6}, I_perp={3,4,5,6} and Ω_s=e_{I_s} retain PB-0009's meanings.

• When discussing coefficients, x_I(W):=𝓒_I(W). For a spatial index list J, 0J means {0}∪J in increasing order. Concatenated subscripts such as 03456 are index lists, not scalar numbers.

What is forced

For every supplied Z, the e₀ products defining α give

α(α(Z))=Z,    Ze₀=e₀α(Z),    α(Z)e₀=e₀Z.             (V0)

Under the graph's retained equality and nonvoid reading, every supplied W has the unique representation

W=X+e₀Y,    X*=X,    Y*=Y,                              (V3)

with X,Y given by V1. Both X and Y may contain every spatial blade, including the empty blade. Their coefficient presentations are

X=Σ_{J⊆{1,…,6}}x_J(W) e_J,
Y=Σ_{J⊆{1,…,6}}x_{0J}(W) e_J.                             (V4)

These are the full two complementary 64-position lists; they require no coefficient to be nonvoid and omit none of W's 128 positions. For any supplied Z, the exact spatial criterion is

Z*=Z iff 𝓒_I(Z)=v at every index I containing 0
     iff Z has its full presentation on the spatial index list. (V3a)

The complete derivative, source and residual are exactly

D(W)=(Ẏ−D_sX)+e₀(Ẋ+D_sY),
F(W)=F_0+e₀F_1,
R(W)=E_0+e₀E_1,                                       (V5)

where all four expressions F_0,F_1,E_0,E_1 are star-fixed. Therefore the conjunction

h(Ẋ+D_sY)B=α(X)S_X+Y V_Y,
h(Ẏ−D_sX)B=X V_Y+α(Y)S_X                              (V6)

is equivalent to the full equation R(W)=v. Both equations hold for W=Ψ. Conversely, the two equations for a supplied W reconstruct precisely L(W)=F(W); no second-order implication or omitted condition is used.

The two equations in V6 and the two reversion-resolved equations of PB-0010 are coarse views of the same four residual parts:

E_0=Π_{++}(R(W))+Π_{-+}(R(W)),
e₀E_1=Π_{+-}(R(W))+Π_{--}(R(W)).                       (V7)

The full equation is equivalent to all four Π_{r,t}(R(W))=v together. PB-0010 R8 enforces precisely the two parts with r=−u; its companion R9 enforces precisely the two parts with r=u. V6 enforces the two columns with t=u and t=−u. These regroupings change no equation's content and assert no separate equation for a part of W.

PB-0009's universally void source-output coefficient is retained in the original output convention:

𝓒_{I_s}(F_0)=v.                                       (V8)

For every supplied W, its complete residual coefficient is exactly

𝓒_{I_s}(R(W))=h𝓒_{I_perp}(Ẏ−D_sX).                    (V8a)

Consequently the one collected derivative condition

𝓒_{I_perp}(Ẏ−D_sX)=v,                                 (V9)

is equivalent to 𝓒_{I_s}(R(W))=v, without requiring the whole residual to vanish. It expands, with every term retained, to

∂₀x_{03456}(W)+∂₁x_{13456}(W)+∂₂x_{23456}(W)
  −∂₃x_{456}(W)+∂₄x_{356}(W)−∂₅x_{346}(W)+∂₆x_{345}(W)=v.         (V10)

The seven-term expansion is obtained by applying PB-0009(G0r), its unrestricted residual-coefficient identity, to this same W and combining it with V8a. The full equation R(W)=v supplies V9–V10; the actual input W=e₀ below has V9–V10 while R(e₀)=−he₀≠v. V10 is one collected equation on the displayed seven derivative contributions.

For every supplied spatial Z (equivalently Z*=Z by V3a),

𝓒_{I_s}(ZB)=𝓒_{I_perp}(Z).                            (V25)

This coefficient route retains the spatial hypothesis on Z.

The forcing

1. Construct and recover both complementary parts

PB-0007's additive/scalar compatibility and double-star identity give

X*=k(W*+(W*)*)=k(W*+W)=X.

For Y, star fixes k and sends e₀ to −e₀, so

Y*=k(−e₀)(W*−W)
   =k e₀(W−W*)=Y.                                     (V11)

The two product signs cancel using PB-0001; no order of nonscalar factors is changed. The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. Since e₀e₀=u,

X+e₀Y
  =k(W+W*)+k(e₀e₀)(W−W*)
  =k[(W+W*)+(W−W*)]
  =khW=W.                                              (V12)

To prove uniqueness, suppose an actual second expression satisfies W=X'+e₀Y' with X'* =X' and Y'* =Y'. Applying star gives W*=X'−e₀Y'. Adding yields W+W*=hX', so multiplying by k gives X'=X. Subtracting yields W−W*=h e₀Y', so multiplying on the left by k e₀ gives Y'=Y. This establishes both directions of V3.

For its coefficient content, PB-0008 gives 𝓒_I(W*)=θ(I)x_I(W). Here θ(I)=u if 0 is absent and θ(I)=−u if 0 is present. V1 and coefficient recovery consequently retain exactly the no-0 coefficients in X and exactly the with-0 terms in e₀Y. A with-0 blade is already written e₀e_J, because the input's order puts 0 first. Left multiplication by e₀ removes just that adjacent pair by e₀e₀=u, leaving coefficient x_{0J}(W) and spatial blade e_J. This proves V4 without any additional sign.

Equivalently, if a supplied Z is star-fixed, its recovered with-0 coefficient z_I satisfies z_I=−z_I; adding z_I gives hz_I=v, and multiplying by k gives z_I=v. Conversely a presentation using only spatial blades is star-fixed by the declared star images. This proves both directions of V3a and why the star-fixed X and Y have complete spatial presentations. No inverse of a coefficient has entered.

The full sign fan makes the same construction explicit:

X=Π_{++}(W)+Π_{-+}(W),
e₀Y=Π_{+-}(W)+Π_{--}(W),                               (V13)

by J5 of V02-PB-0012. Every original component remains in exactly one of these two sums.

2. Expand every derivative contribution

V02-PB-0011 supplies the individual derivative identities used here; the input already gives ∂_μe₀=v. Thus 04's product rule gives

∂_μ(e₀Y)=e₀∂_μY.

For each spatial label i, PB-0003's contextual exchange gives eᵢe₀=−e₀eᵢ. Applying it separately to each of the six terms gives

D_s(e₀Y)=−e₀D_sY.                                     (V14)

Use the complete D=e₀∂₀−D_s in the identity W=X+e₀Y:

D(W)
  =e₀Ẋ+e₀(e₀Ẏ)−D_sX−D_s(e₀Y)
  =e₀Ẋ+Ẏ−D_sX+e₀D_sY
  =(Ẏ−D_sX)+e₀(Ẋ+D_sY).                               (V15)

Right multiplication by the declared B and multiplication by h yield

L(W)=h(Ẏ−D_sX)B+e₀h(Ẋ+D_sY)B.                         (V16)

Only h, a scalar, has moved. B remains on the right of both complete expressions; it has not been commuted through X,Y or their derivatives. Both derivative combinations remain present.

3. Expand the source without losing either factor

The product rules for reversion and star give

W̃=X̃+Ỹe₀,
W*=X−e₀Y.

For any supplied Z, the elementary products

e₀α(Z)=e₀e₀Ze₀=Ze₀,
α(Z)e₀=e₀Ze₀e₀=e₀Z,
α(α(Z))=e₀e₀Ze₀e₀=Z                                 (V17)

use only e₀e₀=u and association. This proves V0. Therefore Ỹe₀=e₀α(Ỹ), and subtracting the preceding two equalities gives

Sel(W)=(X̃−X)+e₀[α(Ỹ)+Y]=S_X+e₀V_Y.                 (V18)

Also We₀=Xe₀+α(Y). Multiply these complete two-term expressions, retaining all four ordered products:

F(W)
  =[Xe₀+α(Y)][S_X+e₀V_Y]
  =Xe₀S_X+Xe₀e₀V_Y+α(Y)S_X+α(Y)e₀V_Y
  =e₀α(X)S_X+X V_Y+α(Y)S_X+e₀Y V_Y
  =[X V_Y+α(Y)S_X]+e₀[α(X)S_X+Y V_Y]
  =F_0+e₀F_1.                                         (V19)

Every equality uses distribution, association, V17, or a scalar/sign law already proved. Neither α(Y)S_X nor Y V_Y has been set to v, and no product order was exchanged to combine them.

For the promised star properties, X and Y are star-fixed by V11. Commutation of the two involutions makes their reversions star-fixed as well. Direct star application gives

(α(Z))*=(e₀Ze₀)*=(−e₀)Z*(−e₀)=α(Z*),

so α preserves star-fixedness. The product and addition rules then show that S_X,V_Y,F_0,F_1 are star-fixed. Each individual derivative preserves star-fixedness by V02-PB-0011, each eᵢ is star-fixed, and B*=B. Thus the two inner expressions h(Ẏ−D_sX)B and h(Ẋ+D_sY)B in V16 are star-fixed. Multiplication of the second by e₀ makes that whole second term star-negative. The constructed E_0 and E_1 themselves are star-fixed.

4. Both directions of the whole-equation equivalence

Subtract V19 from V16 with PB-0001's additive rules. The result is exactly

R(W)=E_0+e₀E_1.                                       (V20)

Since E_0 and E_1 are star-fixed, apply the unique decomposition V1–V3 to this residual itself:

k[R(W)+R(W)*]=E_0,
k e₀[R(W)−R(W)*]=E_1.                                 (V21)

If R(W)=v, the established involution and void rules make both expressions on the left void, hence E_0=E_1=v. The difference criterion then gives the two equations V6. Conversely, if both equations V6 hold, their differences E_0,E_1 are void, and V20 reconstructs R(W)=v. This is a written equivalence, not an inference from only one of the equations.

For W=Ψ, the axiom supplies R(Ψ)=v, so the pair follows. For any other supplied W, retaining or establishing R(W)=v is a separate premise; V6 is its exact test, not a proof that every W satisfies it.

5. The reversion and index-0 routes group the same residual

Apply J5 of V02-PB-0012 to the same residual R(W). The two star coarse sums are

k[R(W)+R(W)*]=Π_{++}(R(W))+Π_{-+}(R(W)),
k[R(W)−R(W)*]=Π_{+-}(R(W))+Π_{--}(R(W)).                 (V22)

Compare the first with V21. Multiplying the second equality of V21 on the left by e₀, using e₀e₀=u and scalar centrality, identifies the second with e₀E_1. This proves V7.

For any supplied written Z, the four reconstruction and recovery identities J3–J4 give

Z=v iff Π_{r,t}(Z)=v for all four actual sign pairs.       (V23)

For necessity, each part-taking expression maps v to v by the exported J1a of V02-PB-0012. For sufficiency, the sum J3 of four void expressions is v. Taking Z=R(W) yields the four-part equation stated above.

PB-0010 establishes that its R8 relation is equivalent to R(W)=R(W)̃. This equality is equivalent to k[R(W)−R(W)̃]=v: subtraction and k give one direction, while multiplying by h and the difference criterion gives the converse. By the reversion regrouping J5, that expression is

Π_{-+}(R(W))+Π_{--}(R(W)).

If this sum is void, apply either of its two sign-part expressions. J1a carries the map through this actual two-term sum and maps its void output to v; J4 then recovers exactly the corresponding summand. Thus each summand is void. Conversely two void summands give a void sum. R8 therefore enforces exactly those two residual parts. In the same manner, PB-0010 R9 is equivalent to R(W)=−R(W)̃, which makes the complementary reversion sum Π_{++}(R(W))+Π_{+-}(R(W)) void and hence both its parts void.

The first star column E_0=v in V7 enforces Π_{++}R and Π_{-+}R; the other column e₀E_1=v enforces Π_{+-}R and Π_{--}R. Here e₀E_1=v is equivalent to E_1=v by left multiplication by e₀, a displayed written undoing, not nonscalar inverse notation.

Accordingly both pairs of whole-expression equations group the same four residual parts. All groups must be retained to recover the whole equation. These are not extra independent restrictions and do not assert autonomous equations for the four parts of W.

6. Carry the source-output boundary into the coupled view

PB-0009 G1 gives 𝓒_{I_s}(F(W))=v for every supplied W, independently of whether R(W)=v. In V19, F_1 is star-fixed. Its reconstructed blade presentation is therefore spatial, so e₀F_1 contains only blades whose indices include 0. The spatial index I_s cannot occur there. PB-0004/PB-0008 recovery applied to the actual full sum yields

𝓒_{I_s}(F(W))=𝓒_{I_s}(F_0)=v,

which is V8. The universally void source coefficient remains in the original F convention; no input coefficient is identified with that output or removed.

For any spatial written Z, PB-0008's product coefficient rule routes a term e_J in ZB to index J△{1,2}. It reaches I_s precisely when J=I_s△{1,2}=I_perp. The sign for this one contribution is u, because

e_{3456}B=e₃e₄e₅e₆e₁e₂=e₁e₂e₃e₄e₅e₆=Ω_s.          (V24)

PB-0003's contextual exchange moves each of e₁,e₂ across the four displayed distinct factors; the four signs in each passage pair to u. The sign in PB-0008 P1 is χ(I_perp,{1,2}). PB-0005 (4) identifies it with the sign u in this permitted reduction of the same original product to the same output blade Ω_s. There is no other contributing J. Hence

𝓒_{I_s}(ZB)=𝓒_{I_perp}(Z).                            (V25)

Retain arbitrary W and apply this to the actual star-fixed expression Z=Ẏ−D_sX. In the unrestricted residual R(W)=E_0+e₀E_1, E_1 is star-fixed, so its full recovered presentation is spatial. Consequently e₀E_1 has no I_s coefficient by the same actual index reasoning just used for e₀F_1. Coefficient additivity, the entire definition of E_0 and the proved source identity give

𝓒_{I_s}(R(W))
  =𝓒_{I_s}(E_0)
  =h𝓒_{I_s}((Ẏ−D_sX)B)−𝓒_{I_s}(F_0)
  =h𝓒_{I_perp}(Ẏ−D_sX).

This proves V8a for every supplied W, before any equation premise. If its left side is v, multiplication by k gives V9. If V9 holds, multiplication by h and absorption give its left side v. Thus V9 is equivalent exactly to this one residual-coefficient condition. The stronger R(W)=v implies it by recovery of a void input, but has not been used in either direction of the equivalence.

Apply PB-0009(G0r) to this same arbitrary W. Its asserted unrestricted identity has left side 𝓒_{I_s}(R(W)) and right side h times the seven-term expression in V10. Equating it with V8a and multiplying the entire equality by k gives

𝓒_{I_perp}(Ẏ−D_sX)
  =∂₀x_{03456}(W)+∂₁x_{13456}(W)+∂₂x_{23456}(W)
     −∂₃x_{456}(W)+∂₄x_{356}(W)−∂₅x_{346}(W)+∂₆x_{345}(W),

because kh=u on both complete sides. This applies the parent's claimed coefficient identity; no proof-local row or separate sign calculation is needed. The displayed expression being v is exactly V10 and exactly V9. No individual summand is inferred void from its void sum. Under R(W)=v this recovers PB-0009(G2); without that premise it states precisely the one condition 𝓒_{I_s}(R(W))=v.

The stated distinction is strict at the already supplied generator e₀. Its two constructed index-0 operands are

X=k[e₀+(−e₀)]=v,
Y=ke₀[e₀−(−e₀)]=kh(e₀e₀)=u.

V02-PB-0011 gives the void derivatives of v and u. Thus Ẏ−D_sX=v, proving V9 and hence V10 at this actual W. PB-0010(R10) supplies its full derivative and source and the nonvoid residual R(e₀)=−he₀. No additional witness is introduced. Its I_s coefficient is v because {0} and I_s are distinct recovered positions. This explicit written example proves that the full equation is not a necessary hypothesis for the collected balance; it is not a solution of R=v or a selected support for Ψ.

What is not forced

The decomposition is a reversible writing of the complete W. Choosing Y=v, restricting X or Y to selected blades, deleting a spatial derivative, or declaring one equation autonomous would be an additional premise. None was used here. The terms called Ẋ and Ẏ refer only to the supplied derivative ∂₀.

The reversion pair and the index-0 pair are equivalent complete presentations of one residual equation. Their conjunctions agree through the complete four-part residual criterion; an individual row or column is only a part of that equation. The displayed decompositions assert no separately selectable parts satisfying their own equations.

The source-output coefficient identity holds for every supplied W. The collected derivative balance is equivalent exactly to 𝓒_{I_s}(R(W))=v by the unrestricted residual-coefficient identity. The full equation supplies that coefficient condition but is not necessary for it, as the displayed e₀ calculation proves. The balance is a condition on the collected derivative expression; Ω_s retains its input definition. The parent sharp source and mixed-output witnesses retain their original scope and are not promoted to solutions of R=v by this rewriting.

Only k under h's existing guard is used as a scalar inverse. The displayed e₀ products use e₀e₀=u and introduce no nonscalar inversion operation. Every original source and derivative contribution remains in V5–V6.

What it means

The occurrence of index 0 gives a reversible way to write the complete input as two spatial expressions. Their equations retain the crossed derivative terms and all four ordered source products. This view and the reversion-resolved view meet in the same four parts of one residual: one groups them by star sign, the other by reversion sign. Their complete conditions therefore agree. The missing source output exposes an exact residual-coefficient identity for every input. Its void derivative balance is one component condition, supplied by the full equation but also meaningful without it; it is neither a discarded coefficient of W nor a separately imposed equation for a part of W.

The complete center determines the source and residual laws under central multiplication (V03-PB-0017)

LAYER: 7 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V01-PB-0008, V01-PB-0009, V02-PB-0011, V02-PB-0012, V02-PB-0014

Declares

All claims concern supplied written expressions under the equality and actual-hypothesis scope of the parents. No support, coefficient condition, or new equation is imposed on the distinguished field Ψ.

• u,v,h,k retain 04's scalar meanings; 𝓑,G,λ_m(I),𝓒_I retain PB-0004's definitions; ρ(I) retains PB-0003's sign accumulation; ε_m is PB-0002's generator-square sign; Ω_all retains PB-0009's word. In this record Ω := Ω_all=e₀e₁e₂e₃e₄e₅e₆. This is a specified written element, not a new scalar primitive.

• Sel,D,L,F,R retain V01-PB-0006’s definitions; T,A_W,H_W,K_A(W) are V01-PB-0007’s expressions and Mix is V02-PB-0014’s two-input source expression. In particular T(W)=(W̃)*, F(W)=We₀Sel(W), and L(W)=hD(W)B in the original source convention. Π_{r,t} is exactly V02-PB-0012's four-part expression, for the four supplied sign pairs r,t∈{u,−u}. A prefixed minus on a sign denotes its already proved additive negative.

• A supplied written Z is called central here when ZX=XZ for every supplied written X. The proof below gives a finite generator criterion and its complete coefficient characterization; this word adds no hypothesis by itself.

• a,b,c,d are supplied scalar expressions, with no inverse, square root, nonvoid guard, or derivative condition supplied merely by naming them. Put

  Z(a,b):=a+bΩ,
  Zᶜ(a,b):=a−bΩ,
  n(a,b):=aa+bb.
  

  The superscript c abbreviates the displayed expression. Its identification with star and reversion, and its product with Z, are proved below.

• D(Z) applies the existing seven-derivative contraction to the written Z. The expressions Da,Db similarly use its supplied scalar arguments. W,X,Y below are arbitrary supplied written inputs unless an actual equation is stated. z_I:=𝓒_I(Z) in the center proof refers to recovered coefficients of its arbitrary input Z, before it has the two-term form.

• The multiplication law for two central expressions uses the actual supplied pair (a,b) and actual supplied pair (c,d).

• I denotes an entry of 𝓑; μ,m,p,q denote supplied generator labels in G at their actual uses. In the center proof p is a retained member of a specified nonempty proper I and q is a retained absent label. These are finite index labels, not scalar coefficients.

What is forced

The complete blade has the exact products and derivative identities

ΩΩ=−u,        Ω̃=Ω*=−Ω,        T(Ω)=Ω,
∂_μΩ=v for each μ in G.                                  (C1)

For every supplied Z, the following are equivalent:

Z is central;
Ze_m=e_mZ for each of the seven supplied generators;
Z=a+bΩ, with a=𝓒_∅(Z) and b=𝓒_G(Z).                     (C2)

The last representation is unique in those two scalar coefficients. For the displayed expressions, with no extra condition on their coefficients,

Z(a,b)Z(c,d)=Z(ac−bd,ad+bc),
Z̃=Z*=Zᶜ,         T(Z)=Z,
ZZᶜ=ZᶜZ=n(a,b)u.                                      (C3)

For every supplied W, the full selector, constructed parts, and joint-sign parts obey

Sel(ZW)=ZᶜSel(W),
A_(ZW)=Z A_W,          H_(ZW)=Z H_W,
Π_{r,t}(ZW)=aΠ_{r,t}(W)+bΩΠ_{−r,−t}(W).                  (C4)

The source and mixed-source laws are

F(ZW)=n(a,b)F(W),
Mix(ZX,ZY)=n(a,b)Mix(X,Y).                               (C5)

These are identities for the complete inputs, including arbitrary supplied scalars a,b. Their derivative terms occur in the full residual identity

D(Z)=Da+(Db)Ω,
D(ZW)=(D(Z))W+ZD(W),
L(ZW)=ZL(W)+h(D(Z))WB,
R(ZW)=Z R(W)+(Z−n(a,b)u)F(W)+h(D(Z))WB.                 (C6)

The scalar member Z(a,v)=a has n(a,v)=aa and gives, for every supplied scalar a and written W,

D(aW)=(Da)W+aD(W),
L(aW)=aL(W)+h(Da)WB,    F(aW)=aaF(W),
R(aW)=aR(W)+a(u−a)F(W)+h(Da)WB.                    (C6a)

Under R(W)=v, R(aW)=v is equivalent to a(u−a)F(W)+h(Da)WB=v. Under R(W)=v and all seven ∂_μa=v, it is equivalent to a(u−a)F(W)=v. Both implications in each equivalence keep those hypotheses.

Accordingly, if the actual scalar identity n(a,b)=u holds, then F(ZW)=F(W). The solution criterion retains both actual hypotheses explicitly:

Under n(a,b)=u and R(W)=v:
  R(ZW)=v iff (Z−u)F(W)+h(D(Z))WB=v
          iff (D(Z))WB=WK_A(W)(u−Z).                  (C7)

This retains the entire collected condition. If also every ∂_μa=∂_μb=v, it reduces exactly to (Z−u)F(W)=v, without arbitrary-factor cancellation.

There is an unconditional same-source instance supplied by the complete blade itself:

F(ΩW)=F(W),
R(ΩW)=ΩR(W)+(Ω−u)F(W).                                  (C8)

ΩW=W iff W=v.                                          (C8a)

Thus an actually nonvoid supplied W has a distinct explicitly constructed input ΩW with the same entire source. This concerns equality of source expressions, not solutions of R=v.

The written expression −k(Ω+u) undoes Ω−u on both sides. Therefore

R(W)=v and R(ΩW)=v
  iff L(W)=v and F(W)=v
  iff D(W)=v and F(W)=v.                                (C9)

No inverse operation on a nonscalar is introduced by this displayed product. C9 describes two supplied full expressions and their exact equations; it is not an existence assertion about a nonvoid solution of R=v.

Repeated multiplication by the same supplied Ω closes the actual finite cycle

W → ΩW → −W → −ΩW → W,
F(W)=F(ΩW)=F(−W)=F(−ΩW).                               (C10)

Under an actual retained nonvoid condition on W, the four displayed inputs are pairwise distinct. (C11)

This completes that specified finite cycle. It does not assert that these four inputs exhaust the equal-source expressions of W.

The forcing

1. The complete blade and its two-sided product

PB-0009 S5 already proves centrality of Ω, Ω̃=Ω*=−Ω, and T(Ω)=Ω. Those are whole-word results from the fixed seven-generator listing, not assumptions based on the name of the word.

For its square, remove the actual adjacent equal pairs using the supplied generator squares, with all intermediate words retained:

ΩΩ̃=e₀e₁e₂e₃e₄e₅(e₆e₆)e₅e₄e₃e₂e₁e₀
   =−e₀e₁e₂e₃e₄(e₅e₅)e₄e₃e₂e₁e₀
   = e₀e₁e₂e₃(e₄e₄)e₃e₂e₁e₀
   =−e₀e₁e₂(e₃e₃)e₂e₁e₀
   = e₀e₁(e₂e₂)e₁e₀
   =−e₀(e₁e₁)e₀
   = e₀e₀=u.

Only the scalar −u supplied by each spatial square is moved; PB-0001's sign rules combine adjacent negatives.

Substitute Ω̃=−Ω and use PB-0001's product sign rule:

Ω(−Ω)=−(ΩΩ)=u,
ΩΩ=−u.

For each actual μ, V02-PB-0011(D6w) gives ∂_μΩ=v. Its finite word-prefix product-rule step starts at ∂_μu=v and appends each of the seven supplied generators, whose derivatives are v. This proves C1 without any scalar square root.

2. The finite generator criterion determines the entire center

If Z commutes with every supplied written X, it commutes with each supplied e_m by substitution. Conversely, suppose the seven generator equalities hold. Z commutes with the empty word u by the two-sided unity law. At an actual word prefix W for which ZW=WZ, and its actual next generator e_m, association and the retained generator equality give

Z(We_m)=(ZW)e_m=(WZ)e_m
        =W(Ze_m)=W(e_mZ)=(We_m)Z.

Write this step at each actual position of any supplied blade. It proves commutation with that blade. Z commutes with each scalar coefficient by 04's scalar centrality. Distributing over the actual finite blade presentation of an arbitrary supplied X therefore gives ZX=XZ. This proves equivalence of centrality with the seven displayed equalities.

PB-0004 defines C_m(Z)=ε_m e_m Z e_m, proves C_m(e_I)=λ_m(I)e_I, and recovers all coefficients. The generator equality is equivalent to C_m(Z)=Z: its forward direction multiplies e_mZ=Ze_m on the right by e_m and on the left by the scalar ε_m, using ε_mε_m=u. For the reverse, start with ε_m e_m Z e_m=Z, multiply on the left by e_m, and associate:

e_m(ε_m e_m Z e_m)
  =ε_m(e_m e_m)Z e_m
  =ε_mε_m Z e_m=Ze_m=e_mZ.

The final equality follows because the other side of the multiplied equality was e_mZ. Thus it is exactly the required generator commutation.

Take the full recovered presentation Z=Σ_I z_I e_I. PB-0004(C0a) gives its entire conjugation presentation and 𝓒_I(C_m(Z))=λ_m(I)z_I; PB-0004(C4) then gives

C_m(Z)=Z
  iff λ_m(I)z_I=z_I for each I in 𝓑.                   (P1)

The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. When λ_m(I)=−u, this scalar equality gives z_I=−z_I. Adding z_I, using the existing k, and associating yields

hz_I=v,
z_I=(kh)z_I=k(hz_I)=kv=v.                              (P2)

There is no cancellation of z_I in this argument. Only the existing guarded h is undone.

Now take any actual index I that is neither empty nor the whole G. Its finite membership listing supplies a label p in I and a label q absent from I. PB-0004 defines λ_m(I)=ρ(I∖{m}). The finite product ρ(I) has one −u factor for each actual member. Move the p factor through the other scalar factors to write

ρ(I)=(−u)ρ(I∖{p}),
(−u)ρ(I)=ρ(I∖{p})=λ_p(I),
λ_q(I)=ρ(I∖{q})=ρ(I).

The middle equality uses (−u)(−u)=u. Starting at u, each actual −u factor changes u to −u or −u to u by PB-0001's sign products, so ρ(I) is u or −u. If it is u then λ_p(I)=−u; if it is −u then λ_q(I)=−u. In either case one of the seven retained generator conditions and P2 forces z_I=v. This accounts for every nonempty proper blade index; it is not a case split on arbitrary coefficient values.

At the empty index, every λ_m is u. At G, each λ_m is the product of the six actual negative signs contributed by the other six labels, hence u. Thus the center conditions remove no coefficient at those two positions. Reconstructing the whole input with PB-0004 C1 gives

Z=z_∅u+z_GΩ=z_∅+z_GΩ.

Conversely, every displayed a+bΩ commutes with each supplied X: its scalar part does by 04, and its Ω part does by PB-0009 S5, with distribution retaining both terms. Coefficient recovery at ∅ and G yields a and b respectively, since these are distinct blade indices. Thus the two coefficients are unique. This proves all directions of C2, without assuming a particular scalar ring or that central elements must be scalar expressions.

3. The finite multiplication and conjugation laws

All scalar factors in this paragraph commute by 04. Expand the four actual products and use C1:

(a+bΩ)(c+dΩ)
  =ac+adΩ+bcΩ+bdΩΩ
  =(ac−bd)+(ad+bc)Ω.

The reversion and star rules, their scalar fixing, and Ω̃=Ω*=−Ω give

(a+bΩ)̃=a−bΩ,
(a+bΩ)*=a−bΩ.

Their composite sends a+bΩ back to itself, since T fixes scalars and Ω. Specialize the multiplication formula to c=a,d=−b. Scalar commutation cancels the two mixed summands:

(a+bΩ)(a−bΩ)=aa+bb+(−ab+ba)Ω=aa+bb.

The same calculation with the factors exchanged gives the identical result. A scalar n equals nu by the right-unity law. This proves C3. It supplies the actual undoing Zᶜ when the additional identity n=u is retained, but does not assert that n is nonvoid or invertible for arbitrary a,b.

4. Retain both factors in the selector and all parts of the input

The just-proved Zᶜ is central as another instance of C2. The reversion and star product rules, their scalar laws, and centrality therefore give

(ZW)̃=W̃Z̃=ZᶜW̃,
(ZW)*=Z*W*=ZᶜW*,
Sel(ZW)=Zᶜ(W̃−W*)=ZᶜSel(W).                           (P3)

The composite T reverses product order. C3 and centrality, not an order-preserving assumption about T, give

T(ZW)=T(W)T(Z)=T(W)Z=ZT(W).

Substitute this identity in the actual constructed part expressions:

A_(ZW)=k[ZW−ZT(W)]=Zk[W−T(W)]=ZA_W,
H_(ZW)=k[ZW+ZT(W)]=Zk[W+T(W)]=ZH_W.                    (P4)

For the joint refinement, first use the special Ω identities in its full formula:

Π_{r,t}(ΩW)
  =kk[ΩW+r(−ΩW̃)+t(−ΩW*)+rtΩT(W)]
  =Ωkk[W−rW̃−tW*+rtT(W)]
  =ΩΠ_{−r,−t}(W).

Here (−r)(−t)=rt is an already proved scalar sign law. The complete addition and scalar laws exported as V02-PB-0012 J1a give

Π_{r,t}(aW+bΩW)
  =aΠ_{r,t}(W)+bΠ_{r,t}(ΩW)
  =aΠ_{r,t}(W)+bΩΠ_{−r,−t}(W).

Every actual sign pair is retained. This proves C4. The coarse selector parts are preserved in the stated multiplicative sense, while the finer two-sign parts mix in their paired positions; no part is set to v.

5. Central multiplication scales the complete source by n

Using P3, centrality of Z and Zᶜ, and C3,

F(ZW)
  =(ZW)e₀Sel(ZW)
  =ZW e₀ ZᶜSel(W)
  =ZZᶜ W e₀Sel(W)
  =n(a,b)F(W).

No derivative occurs, so a and b need not have void derivatives. For the two-input expression, both terms must be retained:

Mix(ZX,ZY)
  =(ZX)e₀Sel(ZY)+(ZY)e₀Sel(ZX)
  =ZX e₀ZᶜSel(Y)+ZY e₀ZᶜSel(X)
  =ZZᶜ[Xe₀Sel(Y)+Ye₀Sel(X)]
  =n(a,b)Mix(X,Y).

This proves C5 on the complete inputs. If the supplied identity n=u holds, the first equality is exactly equal source output for W and ZW. No assertion that this construction exhausts all equal-source inputs is used.

6. The derivative product rule keeps both central coefficient contributions

For every actual μ, the 04 product rule and C1 give

∂_μZ=∂_μa+(∂_μb)Ω+b∂_μΩ
     =∂_μa+(∂_μb)Ω.

PB-0008 P5u writes D as the seven-term sum with b_{{μ}}=ε_μe_μ. Substitute in that complete contraction and distribute it:

D(Z)=Σ_μ ε_μe_μ[∂_μa+(∂_μb)Ω]
     =Da+(Db)Ω.                                         (P5)

This is the displayed seven-generator expression for D(Z). Apply V02-PB-0011(D7p) to the actual ordered pair Z,W and replace each b_{{μ}} by ε_μe_μ using PB-0008 P5u:

D(ZW)=(D(Z))W+Σ_μ ε_μe_μZ(∂_μW).

C2 permits moving this Z past each e_μ. Let S_pre be the preceding accumulation of ε_μe_μ∂_μW. At the empty accumulated sum, v=Zv; at a next term, left distribution gives ZS_pre+Z(ε_μe_μ∂_μW)=Z(S_pre+ε_μe_μ∂_μW). Applying that equality at the seven supplied labels and using PB-0008 P5u to identify their sum with D(W) gives

D(ZW)=(D(Z))W+ZD(W).                                  (P6)

No analogous movement of D(Z) through W has occurred.

Multiply P6 on the right by B and on the left by h, and use scalar centrality:

L(ZW)=h(D(Z))WB+ZL(W).

Subtract C5's full source:

R(ZW)=ZL(W)+h(D(Z))WB−nF(W)
      =Z[L(W)−F(W)]+(Z−nu)F(W)+h(D(Z))WB.

This proves C6, including its displayed D identity. Set b=v. V02-PB-0011(D1) gives Db=v, and the defining expressions give Z=a and n=aa. C5 and C6 therefore give C6a, with a−aa=a(u−a) by distribution. If R(W)=v its term is void; if every ∂_μa=v, all seven terms of Da are void. The remaining whole residuals and the void-difference criterion give both stated equivalences. No individual factor is cancelled.

Under exactly n=u and R(W)=v, C6 becomes the first complete condition C7 by the void-difference criterion. V01-PB-0007's source-factor identity writes F(W)=hWK_A(W). Thus the condition is h[(Z−u)WK_A(W)+(D(Z))WB]=v. Multiply by k, use the void-difference criterion, and commute only the central Z−u. This gives (D(Z))WB=WK_A(W)(u−Z), and reversing those written steps gives the other direction. If every actual derivative of a and b is v, P5 gives D(Z)=v. The reduced form stated here retains those seven pairs of derivative hypotheses and is the undivided product (Z−u)F(W)=v.

7. The supplied complete blade has exact source and residual identities

Take the actual pair a=v,b=u. Their derivatives are v by V02-PB-0011. The definitions and C1–C3 give

Z(v,u)=Ω,
n(v,u)=vv+uu=u,
D(Ω)=v.

C5 and C6 therefore give C8 with no new coefficient condition. To undo Ω−u, expand the finite product in both orders, using Ω's centrality and ΩΩ=−u:

(Ω−u)(Ω+u)=ΩΩ+Ω−Ω−u=−u−u=−h,
(Ω+u)(Ω−u)=ΩΩ−Ω+Ω−u=−h.

Multiplying either product by the scalar −k gives

(Ω−u)[−k(Ω+u)]=u,
[−k(Ω+u)](Ω−u)=u.                                     (P7)

If ΩW=W, the void-difference criterion gives (Ω−u)W=v. Left-multiplication by −k(Ω+u) and P7 give W=v. Conversely, W=v makes ΩW=v=W by absorption. This proves C8a. Under an actual retained nonvoid condition on W, equality ΩW=W is therefore excluded, while C8 still gives equal source outputs.

Suppose both R(W)=v and R(ΩW)=v. C8 then gives (Ω−u)F(W)=v. Left-multiply this actual equality by the displayed second product factor in P7 and associate:

F(W)=uF(W)=[−k(Ω+u)](Ω−u)F(W)=v.

Since R(W)=L(W)−F(W)=v, the established difference criterion gives L(W)=F(W)=v. Conversely, if L(W)=v and F(W)=v, their difference R(W) is v, and C8 gives R(ΩW)=v. This proves its first equivalence. Finally L(W)=hD(W)B and PB-0006(B0) give

kL(W)(−B)=k[hD(W)B](−B)=D(W).

Thus L(W)=v implies D(W)=v; conversely D(W)=v gives L(W)=hvB=v. Keeping the same F(W)=v on both sides gives the second equivalence in C9.

The only scalar inverse in every displayed chain is k, whose h guard is inherited from 04. P7 is an explicit product identity made of already permitted terms. No new nonscalar inversion operation, positivity, or conclusion about unprovided inputs has been used.

8. Close the actual cycle and check every distinct pair

Association, C1 and the product sign law give the four successive products

Ω(W)=ΩW,
Ω(ΩW)=(ΩΩ)W=−W,
Ω(−W)=−ΩW,
Ω(−ΩW)=−(ΩΩ)W=W.

Apply C8's source equality at each of those actual inputs in order. The resulting equalities are exactly C10. The last value returns to the first input; no infinite iteration is needed.

In addition to P7, the other sign of the unit difference has the explicit two-sided product

(Ω+u)(u−Ω)=Ω−ΩΩ+u−Ω=h,
(u−Ω)(Ω+u)=Ω+u−ΩΩ−Ω=h,

so k(u−Ω) undoes Ω+u on both sides. Also Ω is undone by −Ω because Ω(−Ω)=(−Ω)Ω=u by C1.

There are six pairs in the four displayed positions. Equality of W and ΩW gives W=v by C8a. Equality of W and −W gives hW=v, hence W=v after multiplication by k. Equality of W and −ΩW gives (Ω+u)W=v, hence W=v after multiplication by the displayed undoing of Ω+u. Equality of ΩW and −W gives the same product equality and the same conclusion. Equality of ΩW and −ΩW gives hΩW=v; multiply by k and then −Ω to obtain W=v. Finally equality of −W and −ΩW negates to W=ΩW, already handled. Each equality therefore contradicts an actual retained nonvoid condition on W. This proves C11 on every pair, without cancelling W or treating nonvoidness itself as an inversion permission.

What is not forced

The entire-center characterization is a characterization of commutation under the existing operations. Its two coefficients are the recovered scalar coefficients of the displayed written element; Ω retains its seven-generator definition.

C5 gives the same scalar factor n(a,b) for the complete source and mixed source. Under n=u it gives one explicitly constructed family of equal-source inputs. It does not classify the complete nonlinear image of F or every pair of inputs with the same source. Under n=u and R(W)=v, the full equation for ZW has exactly the additional collected condition C7.

C10–C11 close one actual four-position multiplication cycle and prove its distinctness under their stated nonvoid condition. This finite cycle does not classify all inputs with the same source. Its members satisfy the full equation only under the applicable residual criteria, including C9 for the first two members.

C7 is one collected condition on the displayed sum. The extra derivative hypotheses used for its reduced form remain additional hypotheses. Even then, only the specific factor Ω−u in C9 is undone by its stated two-sided product with −k(Ω+u); no arbitrary central or scalar factor is cancelled.

The four-part identity in C4 does not assign separate equations to the parts or remove any coefficient. Every supplied W remains complete. C9 is an exact statement about equations for W and ΩW; it asserts no nonvoid solution.

What it means

The supplied operations determine exactly which written expressions commute with the whole algebra: each has the form a+bΩ, with its two coefficients recoverable. Multiplication of a complete input by such an expression changes every source coefficient through the same scalar aa+bb. Repeated multiplication by the particular factor Ω closes the four-position cycle W, ΩW, −W, −ΩW. All four have the same entire source, and the written undoings prove that they are pairwise distinct whenever the supplied W is actually nonvoid. This completes that specified cycle without classifying all inputs with that source. The full equation retains more information: its derivative side transforms by the central expression itself and also keeps the derivatives of its coefficients. Under n=u and R(W)=v, the complete solution criterion is exactly C7; for the particular factor Ω it has the two-input form C9.

Every actual finite derivative list retains its complete complementary source pairs (V03-PB-0021)

LAYER: 7 FORCED BY: V01-PB-0001, V01-PB-0006, V02-PB-0011, V02-PB-0014, V03-PB-0019, V03-PB-0020

Declares

The scalar expressions u,v,h retain 04's meanings; Sel,D,L,F,R retain PB-0006's definitions; Mix retains V02-PB-0014's definition. Cmp_C(V)=L(V)−Mix(C,V) retains the same exact expression used in V03-PB-0019(C1). W is a supplied full written element. No support, coefficient or derivative condition is imposed on W by this declaration.

An actual finite ordered list of supplied derivative labels is given. Its ordered positions are denoted P and its label at position p is μ_p. P denotes this supplied finite list of positions only. It is not an assumption about an infinite index set. A selection S from P means an actual choice of which of those positions to retain, with their inherited order. P\S denotes the complementary selection. Equal derivative labels at distinct positions remain distinct occurrences.

For each selection needed in the displayed finite calculation, W_S is the written expression formed from W by applying the selected derivatives in list order: begin at W and apply each selected ∂_μ to the entire preceding expression. In particular W_empty=W and W_P is the complete listed derivative. Likewise ∂_P Z abbreviates that actual list of operations applied to any supplied written Z. The notation ∂_empty Z means Z, not a new derivative operation.

The finite list of all selections is constructed explicitly along P. At the empty prefix its only selection is empty. At the next actual position p, replace each previous selection S by the two selections S and S with p appended. Thus each final selection records one left/right assignment for every supplied position. No infinite iteration or assumed integer parameter enters this construction.

If P is nonempty, let p_first denote its first actual position. Pairs(P) denotes the finite selections S that contain p_first and omit at least one position of P. Each pairs S with its nonempty complement. It names one representative of each proper complementary pair; it does not select a component of Ψ or solution alternative.

Every sum below is the finite accumulation of its specified actual terms starting at v. If its list has no terms its value is v. No factorial, binomial coefficient, scalar count inverse, convergence rule or extra scalar guard is introduced.

For the displayed two- and three-position instances, the positions are called a,b,c, their supplied labels are μ,ν,ρ, and subscripts on W denote their actual derivative selections. These position names are not scalar values.

What is forced

For the supplied finite list, including its empty case, the full ordered-product expansion is

∂_P F(W)
   =Σ_(S selected from P) W_S e₀ Sel(W_(P\S)).          (N1)

For a nonempty actual list, pair those terms by complementary selections. The exact result is

∂_P F(W)
   =Mix(W,W_P)
     +Σ_(S in Pairs(P)) Mix(W_S,W_(P\S)).               (N2)

For that same nonempty actual list, the full residual derivative is

∂_P R(W)
   =Cmp_W(W_P)−Σ_(S in Pairs(P)) Mix(W_S,W_(P\S)).       (N3)

For that nonempty list, under the additional identity R(W)=v, N3 gives the forced complete balance

Cmp_W(W_P)=Σ_(S in Pairs(P)) Mix(W_S,W_(P\S)).           (N4)

For one position, the right side is the empty accumulation v, giving V03-PB-0020(E5). Under R(W)=v, the two- and three-position specializations are respectively

Cmp_W(W_ab)=Mix(W_a,W_b),                               (N5)

Cmp_W(W_abc)
   =Mix(W_a,W_bc)+Mix(W_b,W_ac)+Mix(W_c,W_ab).          (N6)

All these derivative expressions are derived from the same W. In particular, retaining R(W)=v, if the two labels in N5 are the same supplied μ, the complete repeated-derivative result is

Cmp_W(∂_μ∂_μW)=hF(∂_μW).                              (N7)

The self-source on the right of N7 is retained exactly. Repeated labels do not justify merging occurrences by deleting their terms.

The forcing

1. Establish the empty prefix and the local assignment step

At the empty prefix the single selection and its complement are both empty. By the declarations,

W_empty e₀ Sel(W_empty)=W e₀ Sel(W)=F(W).

Since ∂_empty F(W) means F(W), this is exactly N1 at that prefix.

Consider a prefix P_old already written, with its N1 identity, and its actual next position p. Fix one of that prefix's actual selections S. Differentiate its one product in the original order:

∂_(μ_p)[W_S e₀ Sel(W_(P_old\S))]
  =(∂_(μ_p)W_S)e₀Sel(W_(P_old\S))
    +W_S(∂_(μ_p)e₀)Sel(W_(P_old\S))
    +W_S e₀∂_(μ_p)Sel(W_(P_old\S)).

The middle term is void from the supplied generator derivative and absorption. V03-PB-0020(E1) moves the remaining last derivative through Sel. Because p is the next actual position, the definitions of the selected derivative expressions give

∂_(μ_p)W_S=W_(S with p appended),
∂_(μ_p)W_(P_old\S)=W_((P_old\S) with p appended).

Thus the whole differentiated term is

W_(S with p appended) e₀ Sel(W_(P_old\S))
   +W_S e₀ Sel(W_((P_old\S) with p appended)).         (N8)

The first term assigns the new derivative occurrence to the left factor; the second assigns it to the right factor. Both are kept. The earlier assignments in S and its complement are unchanged.

2. Carry that exact calculation through the supplied list

Differentiate the whole previous N1 identity by additivity and substitute N8 for every one of its actual summands. The new first terms are exactly the selections containing p; the new second terms are exactly those omitting p. Each previous selection contributes one of each. No final selection belongs to both lists, because its entry at p is fixed by which term produced it. No final selection is missing, because removing its entry at p gives its unique previous selection in the preceding constructed list.

For each original selection, additive commutation puts its two produced terms in the declared order: first the selection omitting p, then the selection with p appended. This is precisely the declaration's update of the complete selection list. Hence the resulting expression is N1 for the new prefix. Starting from the displayed empty-prefix equality, attach this local calculation at every position actually present in P. Its last equality is N1 for the supplied full list.

The proof is the stated finite chain of product-rule equalities. It asserts neither an infinite family of independently existing derivative data nor an infinite representation of W. Only the actual operations needed for this finite calculation have been written.

3. Pair complementary assignments without discarding occurrences

Assume now that P is nonempty. The term with S=empty and the term with S=P in N1 are

W e₀Sel(W_P),           W_P e₀Sel(W).

Their sum is Mix(W,W_P) by its definition.

Every remaining S and its complement are both nonempty proper selections. Exactly one of them contains p_first, because that actual position was assigned to exactly one of the two factors. Thus exactly one belongs to Pairs(P). At that representative S, the two original N1 summands are

W_S e₀Sel(W_(P\S)),
W_(P\S) e₀Sel(W_S).

Their sum is exactly Mix(W_S,W_(P\S)). Since the representative is determined by the same first position for every pair, no proper assignment is missed or counted in two pairs. Finite additive regrouping therefore gives N2.

This occurrence argument is unchanged when derivative labels repeat. Two different assignments may then produce equal written expressions. Their summands still occur at their distinct positions in N1 and are collected by the same complementary-pair rule. No distinctness of the expression values was used.

4. Carry L through the actual derivative list and retain the complete residual

V03-PB-0020(E1) gives the actual one-step equality

∂_(μ_p)L(Z)=L(∂_(μ_p)Z)

for every supplied written Z. Beginning at L(W), apply this identity at each actual position of P, with Z the preceding written derivative expression. Its final equality is

∂_P L(W)=L(W_P).                                      (N9)

Derivative additivity and the proved derivative-negation law likewise give, at every actual step, the derivative of a difference as the difference of derivatives. Applying those local equalities to R(W)=L(W)−F(W) and using N9 and N2 gives

∂_P R(W)=L(W_P)−Mix(W,W_P)
             −Σ_(S in Pairs(P)) Mix(W_S,W_(P\S)).

The first two terms are precisely Cmp_W(W_P), proving N3.

If additionally R(W)=v, substitute that retained identity into the actual derivative expression. The first derivative is void by V02-PB-0011(D1); every further derivative is another instance of that same void-derivative identity. Hence ∂_P R(W)=v for this nonempty supplied list. In N3, move the entire final sum to the other side using the additive laws. This gives N4. No constituent mixed term is set to void.

5. Write the first finite instances in full

For the one-position list a, there is no proper selection containing a and omitting another actual position. Thus Pairs(P) is empty and N4 reduces to the previously proved first derivative equation.

For the two-position list a,b the entire ordered expansion N1 is

∂_ν∂_μF(W)
  =W e₀Sel(W_ab)
    +W_a e₀Sel(W_b)
    +W_b e₀Sel(W_a)
    +W_ab e₀Sel(W).

Its endpoint terms give Mix(W,W_ab), and its remaining two terms give Mix(W_a,W_b). Substitution into N4 proves N5.

For the three-position list a,b,c, N1 has the eight actual terms

W e₀Sel(W_abc),
W_a e₀Sel(W_bc),
W_b e₀Sel(W_ac),
W_c e₀Sel(W_ab),
W_ab e₀Sel(W_c),
W_ac e₀Sel(W_b),
W_bc e₀Sel(W_a),
W_abc e₀Sel(W).

Group its first and last terms, then its second and seventh, third and sixth, and fourth and fifth terms. These four paired sums are exactly

Mix(W,W_abc),
Mix(W_a,W_bc),
Mix(W_b,W_ac),
Mix(W_c,W_ab).

They give N6 from N4. Choosing the representative by p_first in §3 may write the last two mixed expressions with reversed arguments; V02-PB-0014(MixOps)'s symmetry makes their displayed values identical.

Finally take the two actual labels in N5 to be the same μ. Both W_a and W_b are then the same written expression ∂_μW, while W_ab is ∂_μ∂_μW. V02-PB-0014(M2) gives

Mix(∂_μW,∂_μW)=hF(∂_μW).

Substitution into N5 proves N7. The factor h records the two original product-rule assignments; its terms have not been deleted.

What is not forced

This record constructs exact consequences for each supplied finite list. It is not an infinite derivative expansion, a convergence theorem, a regularity assumption, or a reconstruction of W from derivatives.

N1–N3 need no solution premise. N4–N7 additionally use the retained identity R(W)=v. Neither the source terms in N4 nor its left-side difference are declared separately void. No inverse of their operators or inputs is introduced, and no termwise splitting of a collected void expression is used.

N4 concerns the exact derivatives of one full solution identity. It does not assert that W_P separately solves its own nonlinear equation. Its own residual would additionally subtract F(W_P), just as V03-PB-0020(E6) explicitly does at the first step. The derivative-list construction assigns no support to W or to its derivatives.

The displayed implications run from R(W)=v to the stated derivative balances. No converse to the undifferentiated equation is asserted here.

What it means

For every supplied finite derivative list, the complete differentiated source is obtained by assigning each actual derivative occurrence to one of its two factors. Complementary assignments pair into the existing mixed source. Repeated labels preserve their distinct occurrences, so the resulting multiplicities and self-source contributions remain explicit. This is a finite written construction for the chosen list; it neither assumes an infinite derivative expansion nor turns the resulting derivative expressions into independent solutions.

The retained-core scalar family has explicit products and an exact derivative criterion (V04-PB-0029)

LAYER: 7 FORCED BY: V01-PB-0001, V01-PB-0004, V01-PB-0006, V01-PB-0007, V02-PB-0011, V04-PB-0026, V04-PB-0031

Declares

Scalar guards retain 04; D retains V01-PB-0006’s definition, T retains V01-PB-0007’s and Γ retains V04-PB-0026’s. I_{34} and P_{34,±} are the (3,4) instances defined in V04-PB-0031. For a supplied scalar t, define

W₄=e₁P_{34,+}, Z₄=e₁I_{34},
M_t=u+tW₄, N_t=u−tW₄,
K_t=M_tP_{34,+}Γ(M_t), Δ_t=T(N_t)P_{34,−}N_t.

There is no added guard or derivative rule on t. All formulas below concern these actual written products. Right-null at K means ΔK=v.

What is forced

The fixed words have the complete ordered products

Γ(P_{34,±})=P_{34,±}, P_{34,∓}e₁=e₁P_{34,±},
T(W₄)=e₁P_{34,−}, Γ(W₄)=−e₁P_{34,−},
W₄W₄=Γ(W₄)Γ(W₄)=v,
W₄Γ(W₄)=P_{34,−}, Γ(W₄)W₄=P_{34,+},
W₄+Γ(W₄)=Z₄, Γ(W₄)−W₄=−e₁, Z₄Z₄=u,
P_{34,−}W₄P_{34,+}=W₄,
P_{34,+}W₄=v, Γ(W₄)P_{34,+}=v,
W₄P_{34,+}=W₄, P_{34,+}Γ(W₄)=Γ(W₄).             (E1a)

For every actual supplied scalar t, the full expressions are

M_tN_t=N_tM_t=u,
T(M_t)T(N_t)=T(N_t)T(M_t)=u,
Γ(M_t)Γ(N_t)=Γ(N_t)Γ(M_t)=u,
K_t=P_{34,+}+tZ₄+ttP_{34,−},
Γ(K_t)=K_t, P_{34,+}K_tP_{34,+}=P_{34,+}, K_tP_{34,+}K_t=K_t,
Δ_t=P_{34,−}−te₁−ttP_{34,+},
T(Δ_t)=Δ_t, Δ_tK_t=v, Δ_t≠v.                      (E1)

At t=u, K_t=u+e₁I_{34} and Δ_t=−I_{34}−e₁.

The full derivative and its exact criterion are

D(Δ_t)=−D(t)(e₁+htP_{34,+}),
D(Δ_t)=v iff ∂_μt=v for every μ∈{0,1,2,3,4,5,6}.    (E2)

No guard, inverse or positivity condition on t or u+tt is used.

The forcing

1. Cite the fixed-pair products and recover Γ on the fixed words

V04-PB-0031 BC5 at (3,4) supplies I_{34}I_{34}=u,T(I_{34})=I_{34}, both complementary P products, both T signs and all their void derivatives; BC5a supplies the e₁ exchange in E1a. The guarded inversion gives hk=u and rule 8 gives kh=hk=u. The scalar difference is

P_{34,+}−P_{34,−}=k[(u+I_{34})−(u−I_{34})]=I_{34}.

BC5a also gives e₀I_{34}=I_{34}e₀. The operation in V04-PB-0026 F1 therefore gives Γ(I_{34})=e₀T(I_{34})e₀=I_{34}, and additivity gives Γ(P_{34,±})=P_{34,±}. These are the only additional fixed-word operation statements needed for §2.

2. Obtain the complementary mixed products

With W₄=e₁P_{34,+}, apply V04-PB-0026 F1 and V01-PB-0007’s product reversal; 04 gives T(e₁)=(ẽ₁)*=e₁. The preceding equalities give

P_{34,−}W₄P_{34,+}=P_{34,−}e₁P_{34,+}P_{34,+}=e₁P_{34,+}=W₄,
Γ(W₄)=Γ(P_{34,+})Γ(e₁)=−P_{34,+}e₁=−e₁P_{34,−},
T(W₄)=T(P_{34,+})T(e₁)=P_{34,+}e₁=e₁P_{34,−}=−Γ(W₄).

The four ordered quadratic products are

W₄Γ(W₄)=−e₁P_{34,+}e₁P_{34,−}
            =−e₁e₁P_{34,−}P_{34,−}=P_{34,−},
Γ(W₄)W₄=−e₁P_{34,−}e₁P_{34,+}
            =−e₁e₁P_{34,+}P_{34,+}=P_{34,+},
W₄W₄=e₁P_{34,+}e₁P_{34,+}=e₁e₁P_{34,−}P_{34,+}=v,
Γ(W₄)Γ(W₄)=e₁P_{34,−}e₁P_{34,−}=e₁e₁P_{34,+}P_{34,−}=v.

Their additive combinations, using P_{34,+}±P_{34,−} above, are

W₄+Γ(W₄)=e₁(P_{34,+}−P_{34,−})=e₁I_{34}=Z₄,
Γ(W₄)−W₄=−e₁(P_{34,−}+P_{34,+})=−e₁,
Z₄Z₄=W₄W₄+W₄Γ(W₄)+Γ(W₄)W₄+Γ(W₄)Γ(W₄)=u.

Also P_{34,+}W₄=P_{34,+}e₁P_{34,+}=e₁P_{34,−}P_{34,+}=v, and Γ(W₄)P_{34,+}=−e₁P_{34,−}P_{34,+}=v. The other two ordered products are

W₄P_{34,+}=e₁P_{34,+}P_{34,+}=e₁P_{34,+}=W₄,
P_{34,+}Γ(W₄)=−P_{34,+}e₁P_{34,−}
              =−e₁P_{34,−}P_{34,−}=−e₁P_{34,−}=Γ(W₄).

These explicit products supply the complete construction that follows.

3. Expand the full scalar family without discarding its quadratic term

The complete products of M_t and N_t use W₄W₄=v:

M_tN_t=u−tW₄+tW₄−ttW₄W₄=u,
N_tM_t=u+tW₄−tW₄−ttW₄W₄=u.

Apply V01-PB-0007’s product reversal for T and V04-PB-0026 F1 for Γ, both fixing u, to obtain both corresponding ordered undoings in E1. Distribution in the defining K_t keeps all four terms:

K_t=P_{34,+}+tW₄P_{34,+}+tP_{34,+}Γ(W₄)
                  +ttW₄P_{34,+}Γ(W₄)
   =P_{34,+}+tZ₄+ttP_{34,−}.

Γ reverses the defining triple product, fixes P_{34,+}, and undoes itself, so Γ(K_t)=K_t. By E1a, P_{34,+}M_t=P_{34,+} and Γ(M_t)P_{34,+}=P_{34,+}. Hence

P_{34,+}K_tP_{34,+}=P_{34,+},
K_tP_{34,+}K_t
  =M_tP_{34,+}Γ(M_t)P_{34,+}M_tP_{34,+}Γ(M_t)
  =M_tP_{34,+}Γ(M_t)=K_t.

T reverses Δ_t=T(N_t)P_{34,−}N_t and fixes P_{34,−}, so T(Δ_t)=Δ_t. Its right product uses the actual adjacent undoing:

Δ_tK_t=T(N_t)P_{34,−}N_tM_tP_{34,+}Γ(M_t)
      =T(N_t)P_{34,−}P_{34,+}Γ(M_t)=v.

Since P_{34,−}=u−P_{34,+}, E1a gives Γ(W₄)P_{34,−}=Γ(W₄)−Γ(W₄)P_{34,+}=Γ(W₄) and P_{34,−}W₄=W₄−P_{34,+}W₄=W₄. With T(W₄)=−Γ(W₄), the full four-term expansion is

Δ_t=(u+tΓ(W₄))P_{34,−}(u−tW₄)
     =P_{34,−}+tΓ(W₄)P_{34,−}−tP_{34,−}W₄−ttΓ(W₄)P_{34,−}W₄
     =P_{34,−}+tΓ(W₄)−tW₄−ttΓ(W₄)W₄
     =P_{34,−}−te₁−ttP_{34,+}.

At t=u, the two expansions give K_t=P_{34,+}+Z₄+P_{34,−}=u+e₁I_{34} and Δ_t=P_{34,−}−e₁−P_{34,+}=−I_{34}−e₁. No additional condition on t or its derivatives has entered.

For nonvoidness, V01-PB-0004 recovers the singleton-{1} coefficient of Δ_t as −t, because P_{34,+},P_{34,−} have only empty and {1,2,3,4} coefficients. If Δ_t=v this forces t=v and the same expansion gives Δ_t=P_{34,−}. V04-PB-0031 BC5 already proves P_{34,−}≠v, so the supposed Δ_t=v is impossible. This establishes Δ_t≠v for every actual t without deciding whether t itself is void or nonvoid.

4. Keep the entire derivative and give its exact undoing

V02-PB-0011 D1 supplies ∂_μu=∂_μk=v; 04 supplies each actual scalar’s product derivative. V04-PB-0031 BC5 gives ∂μP{34,±}=v. Thus the expanded Δ_t gives

∂_μΔ_t=−(∂_μt)e₁−[(∂_μt)t+t(∂_μt)]P_{34,+}
        =−(∂_μt)(e₁+htP_{34,+}).

The derivative of t is scalar under 04, so its two products with t are equal. Adding all seven original signed generator products gives D(Δ_t)=−D(t)(e₁+htP_{34,+}). The right factor has these two written undoings:

(e₁+htP_{34,+})(−e₁+htP_{34,−})
  =−e₁e₁+ht e₁P_{34,−}−ht P_{34,+}e₁+(ht)(ht)P_{34,+}P_{34,−}=u,
(−e₁+htP_{34,−})(e₁+htP_{34,+})
  =−e₁e₁−ht e₁P_{34,+}+ht P_{34,−}e₁+(ht)(ht)P_{34,−}P_{34,+}=u.

The two middle terms cancel by the actual exchange products of §1; the last product is void in each order. Thus D(Δ_t)=v iff D(t)=v. Now D(t)=e₀∂₀t−e₁∂₁t−e₂∂₂t−e₃∂₃t−e₄∂₄t−e₅∂₅t−e₆∂₆t is already on seven distinct singleton blades. Coefficient recovery says it is v exactly when ∂₀t=v and −∂ᵢt=v for each actual spatial label. Negation gives all seven conditions in E2; substitution proves the reverse. There is no analogous claim about D of an unrestricted multivector.

What is not forced

The expressions are finite for every supplied t; they are not a claim that every factor has this form. The right-null and derivative results at K_t apply to the same constructed Δ_t. E2 separates its seven actual scalar derivatives by an explicit right undoing and distinct recovered blades; that argument has this displayed scope.

An actual active base solution with factor K_t is not constructed by these algebraic products. Their use as a source-preserving difference requires the full comparison hypotheses, including the base equation wherever a new solution is asserted.

What it means

The finite scalar family retains its quadratic term, and its nonvoid right-null expression Δ_t at K_t has an exact derivative criterion because a written product undoes the remaining right factor. These words and undoings are available in describing the full class ZK_t=v at this same K_t. Using one such Z as a source-preserving difference still requires its T-fixed condition and an actual base with factor K_t; asserting a new solution also requires that base’s equation and the complete residual comparison.

The following are identifications of this direct construction, not proof parents. In V04-PB-0028 R1 take K=V=P_{34,+},W_c=tW₄,S_c=v. BC5 and E1a give its hypotheses, E_L=u−P_{34,+}P_{34,+}=P_{34,−} and r=P_{34,+}; its M,M_undo are M_t,N_t, and its full K′ is K_t. In V05-PB-0035 S7b take ι=T,ϑ=Γ,K_c=P_{34,+},P_out=v and Θ=P_{34,−}. E1 supplies the two undoings, while BC5 gives T(Θ)=Θ and ΘK_c=v. Its returned expression is exactly Δ_t. At t=u this same K_t is V04-PB-0028 R8's K_complete.

Complete joint-part product routing retains every ordered interaction (V04-PB-0034)

LAYER: 7 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V02-PB-0011, V02-PB-0012

Declares

All objects and the equality/nonvoid reading are inherited from 04 and the listed parents. W and Z range over supplied written elements without a support restriction or a solution premise.

• u, v, h=u+u and k=h⁻¹ retain their meanings from 04, with h≠v.
• W,Z are arbitrary supplied written elements. Reversion and star are 04's operations. Sel,D,L,F,R retain V01-PB-0006's definitions; T retains V01-PB-0007's, and B=e₁e₂ is the ordered product in 04.
• r,t,p,q,s,z are the scalar signs u or −u. The fixed sign-pair listing is (++),(+−),(−+),(−−), where + means u and − means −u.
• W_{r,t}:=Π_{r,t}(W), Z_{p,q}:=Π_{p,q}(Z), with the same four part expressions as V02-PB-0012. These names do not select supports; all four constructed parts are retained.
• ε_μ is the generator-square sign defined in V01-PB-0002 and carried by V01-PB-0004; θ_μ abbreviates V01-PB-0004's θ({μ}). μ follows 0,1,2,3,4,5,6. Put W_{r,t;μ}:=∂μW{r,t}; the semicolon names this supplied derivative, not a new coordinate.
• Every sign-pair sum is the explicit four-position listing above, or its paired sixteen-position listing with the first pair visited first. A restriction on that listing retains exactly the entries satisfying the displayed sign equality. Sums over μ use its seven-position list. All accumulations start at v; no arbitrary-arity operation is assumed.

What is forced

The generator signs and their roles in the supplied operations are

ε₀=u,  ε_i=−u,  θ₀=−u,  θ_i=u  (i=1,2,3,4,5,6),
e_μ*=θ_μe_μ,  D(W)=Σ_μ ε_μe_μ∂_μW.                   (Signs)

For supplied U,V having the retained signs Ũ=rU, U*=tU, Ṽ=pV and V*=qV, the complete product satisfies

Π_{s,z}(UV)=kk(u+ztq)(UV+srpVU).                         (P1)

Consequently it equals k(UV+srpVU) when z=tq, and v when z=−tq. The two reversion signs at the allowed star sign add to UV. This conclusion concerns the whole ordered product; UV and VU have not been identified.

For arbitrary full W,Z this gives the complete recovered output:

Π_{s,z}(WZ)
  =k Σ_{r,t,p,q : tq=z}
      [W_{r,t}Z_{p,q}+srp Z_{p,q}W_{r,t}].             (P2)

All sixteen input pairs are classified before each restricted output sum is formed. Each target star sign receives eight of those pairs, and each such pair retains both displayed ordered products.

With the supplied e₀ between the same operands, the corresponding law is

Π_{s,z}(Ue₀V)=kk(u−ztq)(Ue₀V+srpVe₀U).                (P3)

It gives the following complete source and derivative routing:

Π_{s,z}(F(W))
  =k Σ_{r,t,p,q : −tq=z}(p−q)
      [W_{r,t}e₀W_{p,q}+srpW_{p,q}e₀W_{r,t}],         (P4)

Π_{s,z}(L(W))
  =Σ_{μ,r,t : θ_μt=z} ε_μ
      [e_μW_{r,t;μ}B−sr B W_{r,t;μ}e_μ].             (P5)

Subtracting (P4) from (P5) is Π_{s,z}(R(W)), with no term discarded. The four simultaneous equalities of (P5) and (P4), one for each (s,z), hold exactly when R(W)=v. They therefore hold for the asserted Ψ. For arbitrary W they remain a simultaneous equivalent writing of its whole equation, not four autonomous equations.

The forcing

1. The operations on an actual ordered product

V01-PB-0007 proves double reversion, double star, their commutation and T(X)=(X̃)*. The two supplied product rules and the retained signs give

(UV)*=U*V*=(tU)(qV)=tqUV,
(UV)̃=ṼŨ=(pV)(rU)=rpVU,
T(UV)=((UV)̃)*=rp(VU)*=rptqVU.

Only scalar signs have been commuted. Substituting all four images in the Π formula defined in V02-PB-0012(J1) gives

Π_{s,z}(UV)
  =kk[UV+srpVU+ztqUV+szrptqVU]
  =kk[(u+ztq)UV+srp(u+ztq)VU]
  =kk(u+ztq)(UV+srpVU).

The second line collects the two UV occurrences and the two VU occurrences by additive commutation and distribution. It does not cancel or divide by either product. This proves (P1).

The guarded inverse in 04 gives hk=u; scalar commutation then gives kh=hk=u. If z=tq, then ztq=(tq)(tq)=u by the four sign products, so kk(u+ztq)=kkh=k. If z=−tq, then ztq=−u and that scalar factor is v. These two cases exhaust the signs, since u≠−u follows from h≠v. For the allowed z, the two actual reversion outputs give

k(UV+rpVU)+k(UV−rpVU)=khUV=UV.

Thus the apparently absent star outputs arise from the complete sign calculation and the surviving outputs recover the product.

2. Transport the full sixteen-pair object

V02-PB-0012(J3) writes W as all four W parts and Z as all four Z parts. For each actual prefix U_old of the W sum and next part U_next,

(U_old+U_next)Z=U_old Z+U_next Z.

For each one of those parts and each actual prefix V_old of the Z sum with next part V_next,

U_next(V_old+V_next)=U_next V_old+U_next V_next.

Both prefixes start at v, whose products are v by absorption. Applying these two equalities down the two four-position listings produces the sixteen ordered products, each once. No input was restricted before multiplication.

V02-PB-0012(J1a) supplies Π's additive and scalar laws. Applying those exported laws at each actual prefix of this sixteen-position product sum gives

Π_{s,z}(WZ)=Σ_{r,t,p,q}Π_{s,z}(W_{r,t}Z_{p,q}).

Each actual part has its two signs by J2. Apply (P1) to every occurrence. For z≠tq its output is v; for z=tq its output is the displayed bracket times k. Removing only the proved v summands and factoring k through the actual finite accumulator gives (P2).

The star routing can also be read as the exhaustive two-sign table

t     q       tq       −tq
u     u        u        −u
u    −u       −u         u

−u u −u u −u −u u −u.

For either fixed z, exactly two rows have tq=z. Each row has the four actual pairs of r,p. These are the eight entries of (P2). This finite count reports the listing; it supplies no scalar premise. Adding the four Π outputs recovers WZ by J3, also term by term by the recovery at the end of §1.

3. The e₀ insertion and the entire nonlinear source

For the same retained operand signs, the supplied e₀ images give

(Ue₀V)*=tU(−e₀)qV=−tqUe₀V,
(Ue₀V)̃=pVe₀rU=rpVe₀U,
T(Ue₀V)=−rptqVe₀U.

Insert these images into Π and collect both orders:

Π_{s,z}(Ue₀V)
  =kk[Ue₀V+srpVe₀U−ztqUe₀V−szrptqVe₀U]
  =kk(u−ztq)(Ue₀V+srpVe₀U).

This proves (P3). When z=−tq its scalar factor is kkh=k; when z=tq it is v. These are exactly the last column of the table.

V02-PB-0012(J2s), applied to every actual part of full W, gives

Sel(W_{p,q})=(p−q)W_{p,q}.

Apply Sel's additivity from V01-PB-0006(1) to the four-part reconstruction and retain the full first factor in F(W)=We₀Sel(W). The same prefix distributions used in §2, with the fixed e₀ kept in place, give

F(W)=Σ_{r,t,p,q}(p−q)W_{r,t}e₀W_{p,q}.

For the four (p,q) positions the actual factors p−q are v,h,−h,v, respectively. No condition on any W coefficient has been imposed; the two void scalar multipliers follow from the selector itself. Carry each scalar factor through Π and apply (P3). Its complete star-sign restriction is −tq=z, so the result is (P4). Every source occurrence that survives has retained both orders created by its reversion projection. No product of two nonvoid objects has been assumed nonvoid, and no cancelling sum has been split into termwise conditions.

4. The derivative term in the same refinement

V02-PB-0012(J3) reconstructs ∂μW, J7 identifies its parts as W{r,t;μ}, and J2 at that derivative input gives, for each actual μ,

∂_μW=Σ_{r,t}W_{r,t;μ},
(W_{r,t;μ})̃=rW_{r,t;μ},
(W_{r,t;μ})*=tW_{r,t;μ}.

The sign identities use V02-PB-0011(D3)'s derivative compatibility and (D1)'s void derivatives of u and −u. The prefix derivative and addition rules retain all four parts for each of the seven μ labels.

The squares and star images of the seven supplied generators in 04, with the definitions of ε_μ and θ_μ, give the values and star formula in (Signs). Substituting those seven values into the definition of D in V01-PB-0006 gives its displayed finite sum.

The original ordered B has B*=B and B̃=e₂e₁=−e₁e₂=−B, using V01-PB-0003's exchange at that actual pair. For V=W_{r,t;μ}, therefore,

(e_μVB)*=θ_μt e_μVB,
(e_μVB)̃=B̃Ṽẽ_μ=−rBV e_μ,
T(e_μVB)=−rθ_μt BV e_μ.

Substitution into Π and distribution yield

Π_{s,z}(e_μVB)
  =kk(u+zθ_μt)[e_μVB−srBV e_μ].

This equals k times its bracket when θ_μt=z, and v otherwise. Now expand the original whole derivative term, retaining its ordered B:

L(W)=h Σ_{μ,r,t} ε_μ e_μW_{r,t;μ}B.

This is the seven finite derivative sums, each with its four parts; it is obtained by distributing the already written definition of D. Applying the last projection formula at every occurrence and using hk=u gives precisely (P5). In particular the B on the left of the reversed summand was not moved back to the right.

5. Recombination supplies both directions of the equation

V02-PB-0012(J1a) supplies the additive, negative and void-image laws of Π. Hence

Π_{s,z}(R(W))=Π_{s,z}(L(W))−Π_{s,z}(F(W)).

If R(W)=v, every projected residual is v by its formula. V01-PB-0001's void-difference criterion makes each (P5) equal its corresponding (P4). Conversely, suppose those four equalities hold. Their four projected residuals are v. J3 reconstructs R(W) as their actual four-term sum, so R(W)=v. All four equations use the same W and the same derivatives and source factors. Substitution of the asserted Ψ uses its supplied equation and adds no further premise.

What is not forced

The completeness here is over the derived four-part listings and their ordered products. It does not state that the parts are ideals, that only one product order occurs, or that projected equations are independent. The derivative projection explicitly retains generator and B placement; commuting a part-taking operator with the whole D or L would require an additional identity, which has not been assumed.

The only scalar inverse is k under the original guard on h. All apparent output absences follow from the complete sign table and the associated v multiplier. The nonlinear source and every surviving quadratic term remain in (P4) and in the equivalent whole equation.

What it means

The four-part refinement now has an exact multiplication interface. Star fixes which output pair can receive an ordered input pair; reversion divides that pair into two explicit combinations of the original and reversed products. The same mechanism handles the e₀ insertion in the source and the ordered generator/B placement in the derivative term. These are complete views of one object, with a written recovery back to the original equation.

An actual witness gives complete criteria for both signed products and their derivatives (V05-PB-0035)

LAYER: 7 FORCED BY: V01-PB-0001, V01-PB-0007, V01-PB-0008, V02-PB-0011, V04-PB-0026

Declares

The operations, written elements and scalar guards retain 04 and the cited interfaces. In particular h=u+u and k=h⁻¹ retain 04’s nonvoid guard. Let ι be an actual supplied operation that is additive, carries scalar factors, fixes scalars, reverses products, undoes itself and commutes with each of the seven supplied derivatives. These are the interface hypotheses at every use. T retains V01-PB-0007’s definition and Γ retains V04-PB-0026’s definition. A second label ϑ names an actual operation with the same properties when it occurs; no commutation of ι and ϑ is needed. This conditional interface does not assert the existence of any further operation.

The two actual scalar signs are σ=u and σ=−u. Define the common part expression

Part_(ι,σ)(Y):=k[Y+σ ι(Y)].

H_Y and A_Y retain V01-PB-0007’s definitions. The parameterized Part expression applies to the complete supplied Y.

For the product results, K, U and P_out are arbitrary supplied written expressions. Claims using the actual witness identity KUK=K state it at their use; the algebraic recovery S4a itself needs no witness condition. Define the two ordered complement expressions on their supplied arguments,

E_L(K,U):=u−KU,    E_R(K,U):=u−UK.

Write E_L=E_L(K,U) only within a statement using that same K,U. Set

C_part:=P_out U+σ ι(P_out U)E_L,
N_Y^(ι,σ):=ι(E_L)Part_(ι,σ)(Y)E_L.                 (S0)

The dependence on the actual K,U is retained in every use. At ι=T, σ=u, write H₀=C_part and N_Y=N_Y^(T,u). Y is an actual written parameter and is not claimed unique. In a witness comparison the labels a,b distinguish supplied witnesses U_[a],U_[b]. Thus E_(L,a)=E_L(K,U_[a]), E_(R,a)=E_R(K,U_[a]), and likewise for the particular and null expressions bearing those labels. No label denotes a derivative.

A separate sign σ_fac, also one of u and −u, is used only for a factor and its refined witness. The products p=KV and r=VK then use the refined witness V; their complements are E_L(K,V) and E_R(K,V), the same parameterized expressions just defined, evaluated at that actual V.

In the change of factor and unknown, M,M_undo are actual expressions with M M_undo=M_undo M=u. The expression K_c is the factor before that transformation. Subscripts on U_c and E_(L,c)=E_L(K_c,U_c) indicate that same factor. The prime on E′L denotes E_L(K′,U′), not a new complement expression. The seven contraction factors are b{{μ}}=ε_μe_μ from V01-PB-0008 P5u; D retains V01-PB-0006’s definition. Right-null at a supplied K means the ordered relation ΔK=v. In the derivative statements K_{,μ}=∂μK, U{,μ}=∂μU and W(out,μ):=(∂μP_out)U+P_out U{,μ}. For supplied Δ define

Comp_μ(Δ):=−ΔK_{,μ}U−ι(U)ι(K_{,μ})ΔE_L,
Rem_μ(Δ):=ι(E_L)(∂_μΔ)E_L.                        (S8b)

What is forced

The operations T and Γ satisfy the declared operation interface. For each of the two actual signs σ and every supplied operation ι with that interface,

σσ=u,    ι(σ)=σ,    ∂_μσ=v.                        (S0a)

The two T instances of the defined part are

Part_(T,u)(Y)=H_Y,    Part_(T,−u)(Y)=A_Y.           (S0b)

For every actual K,U with KUK=K, the unrestricted product equation is completely described by

XK=P_out has a supplied completion
  iff P_out U K=P_out,
X=P_out U+Y E_L.                                  (S1)

Every supplied X with that product is recovered by the actual choice Y=X. This statement precedes any involution-sign condition.

Every written X has the unique complete two-sign decomposition

X=Part_(ι,u)(X)+Part_(ι,−u)(X),
ι(Part_(ι,σ)(X))=σ Part_(ι,σ)(X).                 (S2)

For actual K,U with KUK=K and either sign, the complete signed-product criterion is

ι(X)=σX, XK=P_out has a supplied completion
  iff P_out U K=P_out
      and ι(K)P_out=σ ι(P_out)K.                  (S3a)

For actual K,U with KUK=K and exactly the two S3a output conditions, all and only the supplied completions are

X=C_part+N_Y^(ι,σ).                                  (S3b)

For every supplied X of sign σ and arbitrary K,U, with no KUK=K hypothesis and irrespective of its product output,

X=XKU+σ ι(XKU)E_L+ι(E_L)XE_L.                    (S4a)

For actual K,U with KUK=K and every supplied Δ with ι(Δ)=σΔ,

ΔK=v iff Δ=ι(E_L)ΔE_L.                            (S4b)

Consequently all such signed null differences are exactly N_Y^(ι,σ). S4b retains both the displayed sign condition on Δ and its sandwich equation. For arbitrary X the products of its two recovered parts are

P_σ=Part_(ι,σ)(X)K=k[XK+σ ι(X)K],
P_u+P_(−u)=XK.                                   (S4c)

They are outputs of those actual parts, and are not declared the ι-parts of XK. Conversely, for actual K,U with KUK=K, two supplied outputs satisfying their respective S3a tests construct both signed parts; their sum has output P_u+P_(−u). With that same actual KUK=K, every supplied unrestricted completion is included by its actual P_σ.

If the two actual witnesses U_[a],U_[b] satisfy KU_[i]K=K for i=a,b, both the product tests and the complete signed family are witness-independent. Explicitly,

E_(L,i)E_(L,j)=E_(L,i),
E_(R,i)E_(R,j)=E_(R,j),
N_(j,N_(i,Y))=N_(i,Y).                            (S5a)

Under the same two actual witness identities and the common S3a output tests, C_(part,a)−C_(part,b) has sign σ and right product v. Under those same tests the actual parameter change

Y_[b]=C_(part,a)−C_(part,b)+N_(a,Y_[a])                     (S5b)

recovers precisely the same completion with witness U_[b]. Exchanging the two witness labels gives the reverse recovery. This is an exact equality of completed expressions, not uniqueness or a bijection of free parameters.

If the additional factor condition ι(K)=σ_fac K holds, every actual U with KUK=K supplies the refined witness

U_part=Part_(ι,σ_fac)(U),    V=U_part K U_part,
ι(U_part)=σ_fac U_part, K U_part K=K,
ι(V)=σ_fac V, KVK=K, VKV=V.                            (S6a)

For any actual K,V with ι(K)=σ_fac K, ι(V)=σ_fac V, KVK=K and VKV=V, not only the constructed V, the following ordered products hold:

pp=p, rr=r,
E_L(K,V)E_L(K,V)=E_L(K,V), E_R(K,V)E_R(K,V)=E_R(K,V),
pE_L(K,V)=E_L(K,V)p=v, rE_R(K,V)=E_R(K,V)r=v,
pK=K=Kr, rV=V=Vp,
E_L(K,V)K=KE_R(K,V)=VE_L(K,V)=E_R(K,V)V=v,
ι(p)=r, ι(r)=p, ι(E_L(K,V))=E_R(K,V), ι(E_R(K,V))=E_L(K,V).
                                                           (S6b)

At ι=Γ, σ_fac=u these same hypotheses additionally give

T(E_L(K,V))=e₀E_R(K,V)e₀,
T(E_R(K,V))=e₀E_L(K,V)e₀.                         (S6d)

The construction applies to either actual factor sign. It does not assert that an arbitrary factor has one sign, or that either sign furnishes an unsupplied U.

On the full written input, the relation between the two named operations is also exact before either sign is specified:

Γ(e₀Ỹ)=−e₀(T(Y))̃,
T(Y)=σY iff Γ(e₀Ỹ)=−σ(e₀Ỹ).                    (S6c)

The transformation Y↦e₀Ỹ is reversed by K↦K̃e₀. Its positive-Γ instance is precisely 0026's active-source-factor correspondence; its negative-Γ instance corresponds to the positive-T part. The latter is an algebraic factor correspondence, not an additional source factor in F.

For actual M M_undo=M_undo M=u and the exact factor relation

K′=M K_c ϑ(M),
P_M=ι(M)P_out ϑ(M_undo),
Θ=ι(M)XM,    X=ι(M_undo)ΘM_undo,                           (S7a)

the two changes of unknown undo one another, preserve either ι sign, and give

ι(X)=σX, XK′=P_out
  iff ι(Θ)=σΘ, ΘK_c=P_M.                         (S7b)

For arbitrary supplied U_c,U′ related by the displayed maps,

U′=ϑ(M_undo)U_cM_undo,    U_c=ϑ(M)U′M,
E′_L=M E_(L,c)M_undo.                                 (S7c)

The maps undo one another, and

K′U′K′=K′ iff K_cU_cK_c=K_c.

The S3a output tests are equivalent before and after the transformation. Define C_(part,c) by S0 at (K_c,U_c,P_M), and C′_part by S0 at (K′,U′,P_out), with the same ι and σ. Their expressions obey

C′_part=ι(M_undo)C_(part,c)M_undo.                                (S7d)

Their S0 sandwich expressions obey the same transformation with the actual parameter change Y_c=ι(M)Y′M, reversed by Y′=ι(M_undo)Y_cM_undo. When the equivalent witness identities hold, these are the complete signed right-null families; when the two output tests also hold, the particular expressions and their sums are the completions of S3.

For the actual factor change S7a, suppose also that actual K,V satisfy KVK=K, with r=VK,E_R=E_R(K,V), and

K_c=K+S_c, S_c r=v.

Then Kr=K,KE_R=v,r+E_R=u and S_c E_R=S_c, so for either sign the single output equation has the exact coupled form

ΘK_c=P_M iff ΘK=P_Mr and ΘS_c=P_ME_R.             (S7e)

Both equations concern that same Θ. At P_out=v its complete signed null class is

Θ=ι(E_L(K,V))Part_(ι,σ)(Y)E_L(K,V), ΘS_c=v,
X=ι(M_undo)ΘM_undo.                                        (S7f)

Under those same hypotheses and the additional S_c=v, all signed completions exist exactly when P_Mr=P_M and ι(K)P_M=σ ι(P_M)K; every such completion is

X=ι(M_undo)[P_MV+σ ι(P_MV)E_L(K,V)
              +ι(E_L(K,V))Part_(ι,σ)(Y)E_L(K,V)]M_undo. (S7g)

For arbitrary actual K,U, put K_{,μ}=∂μK and U{,μ}=∂_μU for their actual derivatives. The complement derivative is

∂_μE_L=−K_{,μ}U−KU_{,μ}.                           (S8a)

With the additional actual KUK=K, its differentiated identity is

K_{,μ}UK+KU_{,μ}K+KUK_{,μ}=K_{,μ}.

For arbitrary actual K,U and supplied ι(Δ)=σΔ, ΔK=v,

(∂_μΔ)K=−ΔK_{,μ},
∂_μΔ=Comp_μ(Δ)+Rem_μ(Δ),
ι(Rem_μ(Δ))=σRem_μ(Δ), ι(Comp_μ(Δ))=σComp_μ(Δ).  (S8c)

With the additional actual KUK=K, the two ordered products are

Rem_μ(Δ)K=v,    Comp_μ(Δ)K=−ΔK_{,μ}.

The expression Comp_μ has the same written formula for both signs. This is a derived cancellation of the two sign factors, not an omitted term. For arbitrary actual K,U and that same signed right-null Δ, with b_{{μ}}=ε_μe_μ, the complete seven-term contraction is

D(Δ)=Σ_μ b_{{μ}}Comp_μ(Δ)+Σ_μ b_{{μ}}Rem_μ(Δ),
D(Δ)K=−Σ_μ b_{{μ}}ΔK_{,μ},
D(Δ)=v iff Σ_μ b_{{μ}}Rem_μ(Δ)=−Σ_μ b_{{μ}}Comp_μ(Δ).     (S8d)

There is no assertion that D commutes with ι or preserves a sign.

For arbitrary actual K,U, the following complete derivatives of the S0 expressions hold:

∂_μPart_(ι,σ)(Y)=Part_(ι,σ)(∂_μY),
∂_μN_Y^(ι,σ)=ι(∂_μE_L)Part_(ι,σ)(Y)E_L
             +ι(E_L)Part_(ι,σ)(∂_μY)E_L
             +ι(E_L)Part_(ι,σ)(Y)(∂_μE_L),
∂_μC_part=W_(out,μ)+σ ι(W_(out,μ))E_L
                      +σ ι(P_out U)(∂_μE_L).     (S8e)

The part derivative needs only the stated operation interface; the two other identities retain all factors of the actual S0 expressions.

The forcing

1. Establish the unrestricted product before using either sign

Retain only the supplied KUK=K. Distribution and association give

E_LK=K−KUK=v,
E_LE_L=u−KU−KU+KUKU=u−KU=E_L,
KUE_L=KU−KUKU=v.

If XK=P_out, then P_out U K=XKUK=XK=P_out. Conversely if P_out U K=P_out, the supplied expression X₀=P_out U has product X₀K=P_out, and every written Y gives

(P_out U+YE_L)K=P_out U K+Y(E_LK)=P_out.

For an actual completion X, its own product gives the exact recovery

P_out U+XE_L=XKU+X(u−KU)=X.

This proves S1 in both directions, including all right-null freedom (the actual ordered relation ΔK=v). No involution condition or factor-sign hypothesis has entered.

2. Recover both signed parts of each actual expression

V01-PB-0007 gives T’s additive, scalar, reversed-product and involution laws; V02-PB-0011 D3 gives its individual derivative compatibility. V04-PB-0026 F1 gives these same properties for Γ. At σ=u, rule 7 gives σσ=u; at σ=−u, V01-PB-0001’s product-sign laws give (−u)(−u)=u. The operation interface fixes each scalar σ, and V02-PB-0011 D1 gives ∂_μu=∂_μ(−u)=v. These prove S0a. The definitions and the product-sign laws give, without a witness hypothesis,

Part_(T,u)(Y)=k[Y+T(Y)]=H_Y,
Part_(T,−u)(Y)=k[Y−T(Y)]=A_Y.

This proves S0b. The guarded inversion in 04 gives hk=u, and rule 8 gives kh=hk=u. For each supplied Y and either written sign, distribution and involution now give

ι(Part_(ι,σ)(Y))=k[ι(Y)+σY]
                 =σk[Y+σ ι(Y)]
                 =σ Part_(ι,σ)(Y).

The two expressions sum to

k[Y+ι(Y)]+k[Y−ι(Y)]=k(hY)=Y.

For any actual decomposition Y=Y_++Y_− with ι(Y_+)=Y_+ and ι(Y_−)=−Y_−, applying ι gives ι(Y)=Y_+−Y_−. Adding these two equalities gives hY_+=Y+ι(Y); subtracting gives hY_−=Y−ι(Y). Multiplication by the already supplied k recovers the two Part expressions. This proves existence and uniqueness at exactly the two sign conditions.

If ι(S)=σS, its own matching part is

Part_(ι,σ)(S)=k(S+σσS)=k(hS)=S,

and its opposite part is k(S−S)=v. These are actual calculations rather than assumptions suggested by the word “part”. Both signs were obtained from the original Y; neither requires a coordinate restriction.

3. Recover every signed null difference and the complete signed input

For the signed right-null classification retain actual KUK=K. Suppose ι(Δ)=σΔ and ΔK=v. Right multiplication by its complement gives

ΔE_L=Δ−ΔKU=Δ.

Apply ι to that whole equality:

ι(E_L)ι(Δ)=ι(Δ),
σ ι(E_L)Δ=σΔ,
ι(E_L)Δ=Δ.

The final step multiplies by the actual sign σ and uses σσ=u. Combining the left and right equalities gives Δ=ι(E_L)ΔE_L. Conversely that sandwich annihilates K on the right because E_LK=v. Its sign remains the stated antecedent; the sandwich equality alone did not prove it.

For each actual Y the constructed N_Y^(ι,σ) has the two required properties, by the complete reversed product and the displayed annihilation:

ι(N_Y^(ι,σ))
  =ι(E_L)ι(Part_(ι,σ)(Y))E_L
  =σ ι(E_L)Part_(ι,σ)(Y)E_L
  =σN_Y^(ι,σ),
N_Y^(ι,σ)K=ι(E_L)Part_(ι,σ)(Y)(E_LK)=v.

Every actual signed null Δ returns through Y=Δ, because §2 gives Part_(ι,σ)(Δ)=Δ and the proved sandwich equality returns Δ itself. This proves S4b and the full null family.

For the separate S4a identity let K,U again be arbitrary; it needs no witness condition. For supplied ι(X)=σX, the reversed product is

σ ι(XKU)E_L
  =σ ι(U)ι(K)ι(X)E_L
  =ι(U)ι(K)XE_L
  =[u−ι(E_L)]XE_L.

Add ι(E_L)XE_L to obtain XE_L. Adding XKU then gives X(KU+E_L)=X, proving S4a. In particular this recovery does not first guess a form for X; it is an identity on its complete supplied value.

4. Derive both output tests and every completion

Retain the actual KUK=K. Let ι(X)=σX and XK=P_out. The unrestricted calculation proves the first test P_out U K=P_out. Reversing the product gives the second:

σ ι(P_out)K=σ ι(XK)K
               =σ ι(K)ι(X)K
               =ι(K)XK=ι(K)P_out.

Conversely retain both displayed tests and put W_out=P_out U. The defined C_part=W_out+σ ι(W_out)E_L satisfies

C_part K=P_out U K+σ ι(W_out)E_LK=P_out.

Its complete difference from its required signed image is

C_part−σ ι(C_part)
  =W_out+σ ι(W_out)E_L−σ ι(W_out)−ι(E_L)W_out
  =[u−ι(E_L)]W_out−σ ι(W_out)[u−E_L]
  =ι(U)ι(K)P_out U−σ ι(U)ι(P_out)KU
  =ι(U)[ι(K)P_out−σ ι(P_out)K]U=v.

The second test removes precisely this bracket. The void-difference criterion gives C_part=σ ι(C_part); multiplication by σ yields ι(C_part)=σC_part. This constructs a signed completion from the exact tests.

Every other signed completion X has Δ=X−C_part of the same sign and ΔK=v. Section 3 recovers it through the actual Y=X−C_part. Conversely C_part+N_Y^(ι,σ) has sign σ and product P_out for every supplied Y. This proves all of S3, including necessity, sufficiency and reconstruction.

For the two actual parts of a general X, distribution gives

Part_(ι,σ)(X)K=k[XK+σ ι(X)K].

The two terms sum to XK by §2. Each has its actual signed preimage, so the two necessity calculations just proved apply to its own P_σ. Conversely apply sufficiency once at σ=u to its supplied P_u, and once at σ=−u to its supplied P_(−u). The recovered sum has those same parts by uniqueness in §2, and product P_u+P_(−u). This establishes S4c's two directions without replacing a coupled full equation by independently selected outputs.

5. Prove witness independence by ordered products and actual recovery

For actual U_[a],U_[b] with KU_[i]K=K, the first product test transfers directly:

P_out U_[a]K=P_out
  ⇒ P_out U_[b]K=P_out U_[a]KU_[b]K
               =P_out U_[a](KU_[b]K)=P_out U_[a]K=P_out.

The same calculation with labels exchanged proves the reverse. The second test contains neither witness. Thus the two-test criterion is witness-independent before selecting any particular completion.

For i,j each one of the actual labels a,b, E_(L,i)K=v gives

E_(L,i)E_(L,j)
  =E_(L,i)[u−KU_[j]]
  =E_(L,i)−[E_(L,i)K]U_[j]=E_(L,i).

Likewise KE_(R,j)=K−KU_[j]K=v gives

E_(R,i)E_(R,j)=[u−U_[i]K]E_(R,j)=E_(R,j).

Section 3 gives ι(N_(i,Y))=σN_(i,Y), hence §2 gives Part_(ι,σ)(N_(i,Y))=N_(i,Y). Applying the second sandwich now gives

N_(j,N_(i,Y))
  =ι(E_(L,j))ι(E_(L,i))Part_(ι,σ)(Y)E_(L,i)E_(L,j)
  =ι(E_(L,i)E_(L,j))Part_(ι,σ)(Y)
                                 [E_(L,i)E_(L,j)]
  =N_(i,Y).

Now retain the common S3a output tests. Both particular completions then have sign σ and the same product, so their difference Diff_ab=C_(part,a)−C_(part,b) has sign σ and Diff_ab K=v. Section 3 recovers that difference using either witness. The same is true of N_(a,Y_[a]), and consequently of their actual sum Y_[b] in S5b. Thus

C_(part,b)+N_(b,Y_[b])
  =C_(part,b)+Diff_ab+N_(a,Y_[a])
  =C_(part,a)+N_(a,Y_[a]).

Exchanging labels recovers every completion in the other direction. Only completed expressions have been identified; no uniqueness of the free Y representation has been asserted.

6. Refine an actual witness at either actual factor sign

Now additionally retain ι(K)=σ_fac K, where σ_fac is one of the two written signs. Apply ι to the actual KUK=K:

ι(K)ι(U)ι(K)=ι(K),
Kι(U)K=σ_fac K.

The second line used both actual scalar factors σ_fac and σ_fac σ_fac=u. For U_part=k[U+σ_fac ι(U)], §2 gives ι(U_part)=σ_fac U_part. Its complete product is

K U_part K=k[KUK+σ_fac Kι(U)K]
    =k[K+σ_fac σ_fac K]=k(hK)=K.

At V=U_part K U_part the reversed product has three sign factors:

ι(V)=ι(U_part)ι(K)ι(U_part)=σ_fac σ_fac σ_fac U_part K U_part=σ_fac V.

Repeated use of that one written K U_part K identity gives

KVK=K U_part K U_part K=(K U_part K)U_part K=K U_part K=K,
VKV=U_part K U_part K U_part K U_part=U_part(K U_part K)U_part K U_part
     =U_part K U_part K U_part=U_part(K U_part K)U_part=U_part K U_part=V.

No additional product identity for U was assumed. Set p=KV,r=VK and use U=V in the completion interface. The actual products are

pp=(KVK)V=KV=p,       rr=(VKV)K=VK=r,
pK=KVK=K=Kr,         rV=VKV=V=Vp.

Distribution now gives

E_L(K,V)E_L(K,V)=u−p−p+pp=u−p=E_L(K,V),
E_R(K,V)E_R(K,V)=u−r−r+rr=u−r=E_R(K,V),
pE_L(K,V)=E_L(K,V)p=p−pp=v,
rE_R(K,V)=E_R(K,V)r=r−rr=v,
E_L(K,V)K=K−KVK=v,        KE_R(K,V)=K−KVK=v,
VE_L(K,V)=V−VKV=v,        E_R(K,V)V=V−VKV=v.

Finally both factor and witness have sign σ_fac, so

ι(p)=ι(V)ι(K)=σ_fac σ_fac VK=r,
ι(r)=p,    ι(E_L(K,V))=u−r=E_R(K,V),    ι(E_R(K,V))=E_L(K,V).

The same calculation uses only S6b’s four stated identities, so it proves all those ordered products for every actual refined K,V, not only the V constructed above. At ι=Γ, σ_fac=u the definition Γ(Y)=e₀T(Y)e₀ gives T(E_L(K,V))=e₀E_R(K,V)e₀ and T(E_R(K,V))=e₀E_L(K,V)e₀, proving S6d under its stated sign and witness conditions.

For S6c, V01-PB-0007 gives the commutation of T with reversion, and 04 gives T(e₀)=(ẽ₀)=e₀=−e₀. On every supplied Y,

Γ(e₀Ỹ)=e₀T(e₀Ỹ)e₀
        =e₀T(Ỹ)(−e₀)e₀
        =−e₀(T(Y))̃.

The two written transformations undo their inputs:

(e₀Ỹ)̃e₀=Ye₀e₀=Y,
e₀(K̃e₀)̃=e₀e₀K=K.

If T(Y)=σY, substitution in the first equality proves Γ(e₀Ỹ)=−σe₀Ỹ. Conversely the latter equality and the first imply e₀(T(Y))̃=σe₀Ỹ. Multiplication on the left by e₀ gives (T(Y))̃=σỸ; reversion gives T(Y)=σY. More explicitly, for an arbitrary supplied Γ(K)=σ_fac K, multiplication of its defining equality by e₀ on both sides gives T(K)=σ_fac e₀Ke₀. Set Y=K̃e₀ and compute

T(Y)=T(e₀)T(K̃)=−e₀(T(K))̃
     =−σ_fac e₀(e₀Ke₀)̃=−σ_fac K̃e₀=−σ_fac Y.

Thus both signs are carried by the same full reversible transformation, with opposite sign labels. Its active-source use remains the case σ_fac=u and T(Y)=−Y already established in 0026.

7. Transfer the complete signed problem, its tests and its families

Retain actual M M_undo=M_undo M=u and K′=MK_cϑ(M). Reversing the two undoing products gives

ι(M)ι(M_undo)=ι(M_undo)ι(M)=u,
ϑ(M)ϑ(M_undo)=ϑ(M_undo)ϑ(M)=u.

Therefore the proposed changes of unknown undo one another:

ι(M_undo)[ι(M)XM]M_undo=X,
ι(M)[ι(M_undo)ΘM_undo]M=Θ.

If ι(X)=σX, reversing its transformed expression gives

ι(ι(M)XM)=ι(M)ι(X)M=σ ι(M)XM.

Applying the same calculation to the inverse proves the reverse sign implication. The full transformed product is

XK′=ι(M_undo)ΘM_undo MK_cϑ(M)=ι(M_undo)ΘK_cϑ(M).

Multiplying by ι(M) on the left and ϑ(M_undo) on the right recovers ΘK_c=P_M. The displayed undoing products also prove the reverse, establishing S7b without discarding any part of X or K_c.

For arbitrary U_c set U′=ϑ(M_undo)U_cM_undo. Substitution gives

K′U′K′
  =MK_cϑ(M)ϑ(M_undo)U_cM_undo MK_cϑ(M)
  =MK_cU_cK_cϑ(M).

Thus K_cU_cK_c=K_c implies K′U′K′=K′. Conversely, from K′U′K′=K′ multiply on the left by M_undo and on the right by ϑ(M_undo); substitution of K′ yields

K_c[ϑ(M)U′M]K_c=K_c.

The two witness maps undo one another by the same adjacent products. Also K′U′=MK_cU_cM_undo, and writing u=M M_undo gives

E′_L=u−K′U′=M[u−K_cU_c]M_undo=M E_(L,c)M_undo.

The output relation can be reversed as P_out=ι(M_undo)P_Mϑ(M). Direct ordered substitution in the first product test gives

P_out U′K′=ι(M_undo)[P_MU_cK_c]ϑ(M).

This equals P_out precisely when the bracket equals P_M, by its two displayed undoing factors. For the second test, reverse K′ and P_out in their complete written order:

ι(K′)P_out=ι(ϑ(M))ι(K_c)ι(M)ι(M_undo)P_Mϑ(M)
                =ι(ϑ(M))[ι(K_c)P_M]ϑ(M),
ι(P_out)K′=ι(ϑ(M))ι(P_M)M_undo MK_cϑ(M)
                =ι(ϑ(M))[ι(P_M)K_c]ϑ(M).

Thus their signed difference is

ι(K′)P_out−σ ι(P_out)K′
  =ι(ϑ(M))[ι(K_c)P_M−σ ι(P_M)K_c]ϑ(M).

Multiplication by ι(ϑ(M_undo)) on the left and ϑ(M_undo) on the right removes exactly these exterior factors. Its vanishing is therefore equivalent to the untransformed test. No interchange of ι and ϑ was used.

For the particular completions,

P_out U′=ι(M_undo)[P_MU_c]M_undo,
ι(P_out U′)E′_L
  =ι(M_undo)ι(P_MU_c)M_undo M E_(L,c)M_undo
  =ι(M_undo)[ι(P_MU_c)E_(L,c)]M_undo.

Adding with the actual factor σ proves C′part=ι(M_undo)C(part,c)M_undo. For the full null parameter, let Y_c=ι(M)Y′M. Direct distribution gives

Part_(ι,σ)(Y_c)
  =k[ι(M)Y′M+σ ι(M)ι(Y′)M]
  =ι(M)Part_(ι,σ)(Y′)M.

Use E′L=M E(L,c)M_undo and its reversed form in the complete sandwich:

ι(E′_L)Part_(ι,σ)(Y′)E′_L
  =ι(M_undo)ι(E_(L,c))ι(M)Part_(ι,σ)(Y′)M E_(L,c)M_undo
  =ι(M_undo)[ι(E_(L,c))Part_(ι,σ)(Y_c)E_(L,c)]M_undo.

The parameter change is reversed by Y′=ι(M_undo)Y_cM_undo, using the same two undoing identities. This proves exact transport of the entire family.

For S7e, KVK=K gives Kr=K and KE_R=K−KVK=v. The definition E_R=u−r gives r+E_R=u, and the retained S_c r=v gives S_c E_R=S_c−S_c r=S_c. These actual products give

[Θ(K+S_c)]r=ΘK, [Θ(K+S_c)]E_R=ΘS_c.

Thus the whole equation implies both displayed equations. Conversely their sum is Θ(K+S_c)=P_M(r+E_R)=P_M. For S7f take P_out=v in S7b and apply S4b at the same K,V to ΘK=v; the second equation is retained as ΘS_c=v. If S_c=v it is automatic. S3a–S3b at U=V then give S7g, because VK=r. Every recovery uses the same Θ and the same supplied V.

8. Differentiate the actual identities with all varying factors retained

The supplied ι commutes with every individual ∂_μ by its retained interface hypothesis; the cited parents give this for T and Γ. V02-PB-0011 D1 and S0a give the void derivatives of k and each actual sign. Therefore

∂_μPart_(ι,σ)(Y)=Part_(ι,σ)(∂_μY).

04’s product rule on E_L and KUK=K gives S8a with its two and three actual contributions respectively. In particular, a derivative of U has not been omitted. The constructed null family has the full derivative

∂_μN_Y^(ι,σ)
  =ι(∂_μE_L)Part_(ι,σ)(Y)E_L
   +ι(E_L)Part_(ι,σ)(∂_μY)E_L
   +ι(E_L)Part_(ι,σ)(Y)(∂_μE_L).

For the particular completion write W_(out,μ)=(∂μP_out)U+P_out U{,μ}. Its full derivative is

∂_μC_part=W_(out,μ)+σ ι(W_(out,μ))E_L
                     +σ ι(P_out U)(∂_μE_L).

These are exact differentiated expressions, not finite changes with higher-order terms deleted.

For the actual signed null Δ, differentiation gives both

ι(∂_μΔ)=σ∂_μΔ,
(∂_μΔ)K+ΔK_{,μ}=v.

Apply the already proved recovery S4a to the same actual ∂μΔ and its actual product −ΔK{,μ}:

∂_μΔ=−ΔK_{,μ}U+σ ι(−ΔK_{,μ}U)E_L
                         +ι(E_L)(∂_μΔ)E_L.

The middle term has the ordered reduction

σ ι(−ΔK_{,μ}U)E_L
  =−σ ι(U)ι(K_{,μ})ι(Δ)E_L
  =−σσ ι(U)ι(K_{,μ})ΔE_L
  =−ι(U)ι(K_{,μ})ΔE_L.

This proves S8c’s reconstruction and explains the identical formula for its forced term at both signs. The remainder is signed because

ι(Rem_μ)=ι(E_L)ι(∂_μΔ)E_L=σRem_μ,

and Comp_μ inherits the same sign from ∂_μΔ−Rem_μ. No witness identity has been used in these derivative and sign recoveries. Now retain KUK=K. It gives E_LK=v, hence Rem_μK=v. To recover the other right product, multiply the actual derivative product by UK:

(∂_μΔ)KUK=(∂_μΔ)K=−ΔK_{,μ},
−ΔK_{,μ}UK=−ΔK_{,μ}.

Consequently Comp_μK=−ΔK_{,μ}UK=−ΔK_{,μ}, since its second term contains E_LK=v. This establishes the complete ordered claim.

Finally multiply the seven recovered derivative equalities on the left by their actual b_{{μ}} and add them in the supplied order. Distribution gives the first S8d equality. Independently, the differentiated null identity gives, for each of the seven labels,

[b_{{μ}}(∂_μΔ)]K=b_{{μ}}[(∂_μΔ)K]=−b_{{μ}}ΔK_{,μ}.

Their complete sum gives the second equality without KUK=K. The final iff is additive rearrangement of the first equality, not cancellation of K. No individual derivative is declared void, and no sign property is transferred through the left generator multiplication without a separate calculation.

What is not forced

The unrestricted product statement S1 precedes the signed statements. S3–S5's completion and full right-null classifications retain actual KUK=K. The claims concern the resulting expressions and make no parameter-uniqueness assertion. S4c records the actual two signed outputs of one supplied X. It does not say that a general right-null X has two separately right-null parts, that the parts of XK are the products of the parts of X, or that the two signed outputs of an unrestricted completion are independently determined by P_out alone.

Witness refinement starts with a supplied witness. S6b is stated for any refined pair meeting all four displayed identities. S7 retains the actual two-sided products M M_undo=M_undo M=u. S7e retains KVK=K and S_c r=v and derives the four ordered products it uses. Neither statement supplies a retained core or an unsupplied witness.

The derivative statements describe derivatives of actual retained identities, and keep their complete seven-term sum. They do not identify an independently supplied algebraic slot with a realizable derivative. S8a's differentiated witness identity and S8c's two right-product identities retain KUK=K. S8c's reconstruction and signs and S8d need arbitrary actual K,U and the displayed signed right-null Δ. S8e needs only the actual S0 expressions and the operation/derivative interface. The algebraic product constructions alone contain no residual-equation condition.

What it means

For actual K,U with KUK=K, the product recovery gives the compatibility tests, the complete signed families and their changes under replacement of the supplied witness. S6’s refinement additionally retains the factor sign ι(K)=σ_fac K; its four identities keep both products and both signs visible. The transformations in S7 retain their actual two-sided undoing products, and S7e also retains KVK=K and S_c r=v.

The algebraic recovery S4a itself needs no KUK=K. For actual signed right-null Δ, the reconstruction and signs in S8c and the complete seven-term identities in S8d hold for arbitrary actual K,U. KUK=K is needed for S8a’s differentiated witness identity and S8c’s two separate right-product identities. These distinct scopes let the same written interfaces serve T-fixed completion, Γ-fixed witness refinement and retained-core changes without transferring an unused hypothesis.

The direct T(e₀)=−e₀ calculation in S6c is the singleton instance of V01-PB-0008(P0). The calculation uses 04’s two supplied images; the full blade interface is not needed for that one generator.

A finite scalar-weighted composition retains every derivative and mixed-source contribution (V02-PB-0016)

LAYER: 8 FORCED BY: V01-PB-0001, V02-PB-0014, V02-PB-0015, V03-PB-0017

Declares

The scalar expressions u=unity, v=void and h=u+u, and the fixed word B=e₁e₂, retain 04's meanings. Sel,D,L,F,R retain V01-PB-0006's definitions; Mix retains V02-PB-0014's definition. Each a_p is an actually supplied scalar and each X_p an actually supplied written element, on one supplied finite ordered list of pairs. Its position labels p,q and p<q refer to the order of that list, as in V02-PB-0015. Put

W_p:=a_pX_p,       S:=Σ_p W_p.

The sum is the finite accumulation starting at v. Each two-position sum uses the actual earlier/later pairs of the same list. D(a_p) is the existing seven-derivative expression on that scalar. Equal inputs at different positions remain different occurrences in this list.

What is forced

For this actual finite list, including the empty list,

R(S)
  =Σ_p a_pR(X_p)
   +Σ_p a_p(u−a_p)F(X_p)
   +hΣ_p(D(a_p))X_pB
   −Σ_(p<q) a_pa_q Mix(X_p,X_q).                     (P5)

Under the additional hypotheses R(X_p)=v at every actual position,

R(S)=v
  iff Σ_p a_p(u−a_p)F(X_p)+hΣ_p(D(a_p))X_pB
       =Σ_(p<q) a_pa_q Mix(X_p,X_q).                 (P6)

All derivatives and all mixed contributions in the displayed expressions are retained. No equality or derivative condition between the supplied scalars is an additional premise.

The forcing

1. Apply the complete scalar law at every actual position

V03-PB-0017(C6a) supplies, without an equation premise,

R(a_pX_p)=a_pR(X_p)+a_p(u−a_p)F(X_p)+h(D(a_p))X_pB.

This uses the same supplied scalar and expression at that position. The identity includes the complete derivative of the scalar. V02-PB-0014(MixOps) separately supplies, at each actual pair,

Mix(a_pX_p,a_qX_q)=a_pa_q Mix(X_p,X_q).

Only scalar factors are moved in that equality.

2. Collect the full finite composition

Apply V02-PB-0015(C1) to the actual list W_p:

R(S)=Σ_p R(a_pX_p)−Σ_(p<q)Mix(a_pX_p,a_qX_q).

Substitute both preceding identities, one at each occurrence. This gives

R(S)=Σ_p[a_pR(X_p)+a_p(u−a_p)F(X_p)+h(D(a_p))X_pB]
      −Σ_(p<q)a_pa_q Mix(X_p,X_q).

For any actual next position, additive association and commutation move its three terms next to the three existing family sums. Explicitly, if A,C,E are the already accumulated families and r_next,f_next,d_next the next three terms, then

(A+C+E)+(r_next+f_next+d_next)=(A+r_next)+(C+f_next)+(E+d_next).

Each family starts at v. This local equality applied at the actual positions produces the first three families in P5. For the third family, at its next term the fixed h is collected by

hE+h(D(a_p))X_pB=h[E+(D(a_p))X_pB].

The starting equality is v=hv by absorption. The two-position family is unchanged. Thus the whole result is P5. On an empty list all four sums are v, and V02-PB-0015(C8) gives R(v)=v, so the same statement includes its empty case.

3. Keep the solution hypotheses with the equivalence

Retain R(X_p)=v at every actual position. Each a_pR(X_p) is then void by absorption, and its finite sum is v. P5 becomes the difference of precisely the two sides written in P6. V01-PB-0001's void-difference criterion gives P6 in both directions. No complete sum is replaced by conditions on its separate terms.

What is not forced

The result concerns the supplied finite list and its complete weighted expression. The equivalence P6 retains the equation for every X_p. It uses neither an inverse for a_p nor a classification of the factors of a_p(u−a_p)F(X_p). The displayed accumulated sum, including its cross terms, is the condition tested.

What it means

The scalar law for one full input and the finite composition law combine without losing any derivative of a scalar or any mixed source between positions. The complete expression has one residual criterion; its separate summands have not been assigned independent equations.

A recovered source coefficient is the exact scalar pairing of the selector parts (V03-PB-0018)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0005, V01-PB-0006, V01-PB-0007, V01-PB-0008, V02-PB-0012, V02-PB-0014, V03-PB-0017

Declares

All expressions use the supplied finite arithmetic and the parents' equality scope. No hypothesis about which coefficients of Ψ are void is supplied.

• u,v,h,k retain 04's meanings; G,𝓑,𝓒_I retain PB-0004's definitions; 𝓑_A,𝓑_H and the reversion sign η retain PB-0008's definitions. I,J,K denote entries of the finite blade list; μ denotes a supplied generator label. A subscript {0} is the e₀ blade coefficient, not the empty-blade scalar coefficient.

• Sel,D,L,F,R retain PB-0006's definitions; T,A_W,H_W retain PB-0007's definitions; Mix retains V02-PB-0014's definition; Π_{r,t} retains V02-PB-0012's definition. Ω=Ω_all and Z(a,b)=a+bΩ, Zᶜ(a,b)=a−bΩ, n(a,b)=aa+bb retain V03-PB-0017's definitions and exact conditions.

• X,Y,W are arbitrary supplied written inputs. Use the operand-qualified recovered coefficients x_I(X):=𝓒_I(X), x_I(Y):=𝓒_I(Y), and x_I(W):=𝓒_I(W). Lowercase a,b below are supplied scalar expressions; no derivative condition is imposed on them unless stated.

• Define the written element

  g(X,Y):=𝓒_{ {0} }(X e₀ Ỹ).
  

• For the actual seven derivative terms set

  bal₀(W):=−∂₀x_{12}(W)−∂₁x_{02}(W)+∂₂x_{01}(W)
          +∂₃x_{0123}(W)+∂₄x_{0124}(W)
          +∂₅x_{0125}(W)+∂₆x_{0126}(W).
  

  Concatenated labels in coefficient subscripts stand for the increasing blade index, not numerical powers. The distinct name bal₀(W) denotes this seven-term balance. The expression ℓ_{ {0} }(W) retains PB-0008's coefficient-accumulation definition.

• Restricted sums mean actual accumulations over the named parent's fixed finite list, starting at v. The notation Σ_I x_I(X) x_I(Y) includes every actual index in 𝓑; the notation Σ_{I∈𝓑_A} includes every index in the already derived selector-active sublist. No coefficient nonvoidness test changes either list.

• r,t,s,q denote the supplied signs u or −u in G5. ε_μ retains PB-0002's generator-square value, used in PB-0008(P5). The local label i in the last sign calculation takes each actual value 3,4,5,6. Letters U,V,Q in the distributive-law example below denote actual written sum prefixes or summands at that use, not new operators or the central expression Z.

What is forced

For every actual blade, including the empty blade,

e_I e₀ ẽ_I=e₀.                                      (Sand)

For every supplied X,Y, the written g(X,Y) equals the following scalar expression:

g(X,Y)=Σ_{I∈𝓑} x_I(X) x_I(Y),                                (G1)

and therefore g is symmetric, additive in each argument, and carries scalar factors:

g(X,Y)=g(Y,X),
g(X+Y,W)=g(X,W)+g(Y,W),
g(X,Y+W)=g(X,Y)+g(X,W),
g(aX,bY)=ab g(X,Y),
g(v,Y)=g(X,v)=v.                                        (G2)

Every supplied input is recoverable from its finite blade-pairing list:

g(X,e_J)=x_J(X),
X=Σ_{J∈𝓑}g(X,e_J)e_J,
g(X,e_J)=v for all J in 𝓑 iff X=v.                      (G3)

This is the stated algebraic separation property. It is not the claim that g(X,X)=v forces X=v.

The pairings between the two complementary selector parts satisfy

g(A_X,H_Y)=g(H_X,A_Y)=v,
g(X,A_Y)=g(A_X,Y)=g(A_X,A_Y)
                  =Σ_{I∈𝓑_A}x_I(X) x_I(Y).                  (G4)

For the four sign parts, all complete sums remain present:

g(Π_{r,t}X,Π_{s,q}Y)=v if (r,t)≠(s,q),
g(X,Y)=Σ_{(r,t)}g(Π_{r,t}X,Π_{r,t}Y).                  (G5)

The actual four-pair sum in G5 is over (u,u),(u,−u),(−u,u),(−u,−u). The indices s,q are two supplied signs here; neither q nor the letter s names a new scalar value.

At the e₀ output only, the coefficients of the complete source and mixed source are exactly

𝓒_{ {0} }(F(W))=h g(A_W,A_W)
                =h Σ_{I∈𝓑_A}x_I(W) x_I(W),

𝓒_{ {0} }(Mix(X,Y))=hh g(A_X,A_Y)
                   =hh Σ_{I∈𝓑_A}x_I(X) x_I(Y).               (G6)

There are 72 entries in this active list, as the two 36-entry off-diagonal sign lists already derived in V02-PB-0012. This counts the fixed blade list, without requiring its coefficients to be nonvoid.

The pairing transforms under every supplied central multiplier by

g(ZX,ZY)=n(a,b)g(X,Y),
g(A_(ZW),A_(ZW))=n(a,b)g(A_W,A_W).                       (G7)

No derivatives of a,b are required for G7. For every supplied W, the recovered e₀ coefficients of L and R are

𝓒_{ {0} }(L(W))=h bal₀(W),
𝓒_{ {0} }(R(W))=h[bal₀(W)−g(A_W,A_W)].                  (G8)

Consequently its one exact coefficient condition is

𝓒_{ {0} }(R(W))=v
  iff bal₀(W)=Σ_{I∈𝓑_A}x_I(W) x_I(W).                         (G9)

This holds for Ψ by the full asserted equation. For an arbitrary supplied W, G9 is precisely the e₀ coefficient condition; the remaining coefficient equations belong to the full condition R(W)=v. In G6–G9 the displayed square sum remains collected. A void value of that scalar does not authorize termwise-void coefficients.

The forcing

1. The same-blade sandwich has one exact output

PB-0008(P0) writes ẽ_J=η(J)e_J. The ordered product law in PB-0005 therefore routes every occurrence in e_I e₀ ẽ_J to the blade whose index is I△{0}△J. Its coefficient can contribute to the e₀ target only when

I△{0}△J={0},

which is equivalent, by cancelling each label's two repeated toggles, to I=J. This statement concerns the supplied finite membership lists, not a test on coefficients.

Determine the same-index product locally. For the actual label 0 and for each actual spatial label i, the generator squares and PB-0003's contextual exchange give

e₀e₀e₀=e₀,
e_i e₀ e_i=−e₀(e_i e_i)=−e₀(−u)=e₀.

At the empty blade, ue₀u=e₀. Suppose the sandwich for an actual prefix E_pre of the fixed blade word has been written. For its actual next generator e_m, reversion's product rule and the just-proved generator sandwich give

(E_pre e_m)e₀(E_pre e_m)̃
  =E_pre(e_m e₀e_m)E_prẽ
  =E_pre e₀E_prẽ.

Apply this displayed equality at each member of the actual blade word. The final sandwich equals its empty-prefix value e₀. This proves Sand for every actual blade, including the empty one, without taking any sign formula from a parent's proof.

2. Expand and recover the complete pairing

The sandwich X e₀ Ỹ is a written element by 04. PB-0004 C1 applies to that complete element: its recovered coefficient 𝓒_{ {0} }(X e₀ Ỹ) equals an actual scalar-expression coefficient. Thus g(X,Y), initially defined as a written element, is proved scalar by that interface. To determine its complete value, PB-0004 reconstructs X=Σ_I x_I(X)e_I and Y=Σ_J x_J(Y)e_J. PB-0008's reversion law, followed by the finite product and collection calculations, gives

X e₀ Ỹ=Σ_{I,J}x_I(X)x_J(Y) e_I e₀ ẽ_J.

To see that every occurrence is retained, first distribute over Y's actual reversion sum for each fixed I, using U(V+Q)=UV+UQ, then distribute over X's actual list using (U+V)Q=UQ+VQ. Move only the scalar x_J(Y) past blade factors. Start each occurrence accumulator at v and append the terms in the parent's fixed paired-list order. This is the same finite collection scope proved in PB-0004 C2.

At the e₀ target, section 1 shows that exactly the occurrences I=J qualify. Each carries the scalar sign u by Sand. Recovering this coefficient therefore gives

g(X,Y)=𝓒_{ {0} }(X e₀ Ỹ)=Σ_I x_I(X)x_I(Y).

The sum is over the whole actual blade list, including coefficients that happen to be v. This proves G1. In particular it does not omit an input contribution because its selector sign is silent; the selector has not been applied in G1.

Scalar centrality gives x_I(X)x_I(Y)=x_I(Y)x_I(X) at each written occurrence, so G1 gives symmetry. The coefficient laws in PB-0004 C3 supply 𝓒_I(X+Y)=x_I(X)+x_I(Y) and 𝓒_I(aX)=a x_I(X). Substituting and distributing at each actual index gives

g(X+Y,W)=Σ_I(x_I(X)+x_I(Y))x_I(W)
         =Σ_I x_I(X)x_I(W)+Σ_I x_I(Y)x_I(W),

g(aX,bY)=Σ_I(a x_I(X))(b x_I(Y))
         =abΣ_I x_I(X)x_I(Y).

The finite regrouping is the parent's explicit accumulation, not a change of an infinite order. Symmetry gives the second-input addition law, or its identical distributed calculation. Every recovered coefficient of v is v, so absorption gives the two void-input identities. This proves G2.

3. Pairing with the finite blade list recovers the whole input

By coefficient recovery, the presented single blade e_J has coefficient u at J and v at every other entry. Substitution in G1 yields

g(X,e_J)=x_J(X)u+Σ_{I≠J}x_I(X)v=x_J(X).

Each appended v term leaves the scalar accumulator unchanged. Substituting these equalities in the already earned full coefficient reconstruction gives

X=Σ_J x_J(X)e_J=Σ_J g(X,e_J)e_J.

If every displayed pairing with a blade is v, this exact finite reconstruction is a sum of v terms and therefore equals v. Conversely X=v gives every pairing v by G2. This proves G3. The logical premise here is all the specified blade pairings. Replacing it with the single different premise g(X,X)=v would change the assertion and is not a step in this proof.

4. The disjoint part lists give void cross pairings

PB-0008 P4 reconstructs A_X on the fixed sublist 𝓑_A with its inherited coefficients x_I(X) and H_X on the complementary 𝓑_H with those coefficients. The lists have opposite T signs. At any blade index, at least one coefficient in the product contributing to g(A_X,H_Y) is therefore v. G1 and absorption at each index give

g(A_X,H_Y)=v.

The same reasoning or symmetry gives g(H_X,A_Y)=v. Since X=A_X+H_X and Y=A_Y+H_Y, distributivity of g now gives

g(X,A_Y)=g(A_X,A_Y)+g(H_X,A_Y)=g(A_X,A_Y),
g(A_X,Y)=g(A_X,A_Y)+g(A_X,H_Y)=g(A_X,A_Y).

G1 on the two retained A parts is exactly the restricted sum Σ_{I∈𝓑_A}x_I(X)x_I(Y). This proves G4. Neither argument assumes that the actual coefficients on either sublist are nonvoid.

For G5 use V02-PB-0012's proved complete coefficient membership of Π. A coefficient at I is retained in Π_{r,t}(X) exactly when its reversion and star signs are r,t; otherwise it is v. If (r,t)≠(s,q), no index has both sign pairs, so every coefficient product in the corresponding G1 sum is v. The cross pairing is therefore v. Distribute the parent's complete four-part reconstructions of X and Y in g. This initially gives all sixteen actual pairs of sign positions. The twelve unequal pairs have just been proved v; the four equal pairs remain, each once. Their finite sum is G5.

V02-PB-0012 J6s identifies 𝓑_A with the two off-diagonal sign lists: (r,t)=(u,−u) or (−u,u). That parent's explicit blade listing gives 36 entries in each. The union has no repeated blade because the two sign pairs differ, so its list length is 72. These are literal finite list lengths, not additional coefficient hypotheses or a scalar positivity statement.

5. The entire source's e₀ coefficient retains exactly the active pairing

PB-0007 proves Sel(W)=h Ã_W and therefore F(W)=hW e₀ Ã_W. Apply scalar coefficient recovery and the definition of g:

𝓒_{ {0} }(F(W))
  =h𝓒_{ {0} }(W e₀ Ã_W)
  =h g(W,A_W)
  =h g(A_W,A_W)
  =hΣ_{I∈𝓑_A}x_I(W)x_I(W).

The third equality is exactly the just-proved void cross pairing in G4. Thus the coefficient's lack of an H–A contribution is a proved cancellation/support result for this particular output, not deletion of H_W from the full source.

For the mixed source retain both terms, replacing their selectors by the same earned recovery:

Mix(X,Y)=hX e₀ Ã_Y+hY e₀ Ã_X.

The additive/scalar coefficient laws, G4 and symmetry give

𝓒_{ {0} }(Mix(X,Y))
  =h[g(X,A_Y)+g(Y,A_X)]
  =h[g(A_X,A_Y)+g(A_Y,A_X)]
  =h[h g(A_X,A_Y)]
  =hhΣ_{I∈𝓑_A}x_I(X)x_I(Y).

This proves G6, including the factor hh. The two summands are equal by the written symmetry, not omitted or averaged.

6. Central multiplication agrees with the scalar pairing

V03-PB-0017 proves that Z and Zᶜ are central, Z̃=Zᶜ, and ZZᶜ=n(a,b)u. Reverse the second input's product in its actual order and use those identities:

(ZX)e₀(ZY)̃
  =ZX e₀ỸZ̃
  =ZX e₀ỸZᶜ
  =ZZᶜX e₀Ỹ
  =n(a,b)X e₀Ỹ.

Recovering the e₀ coefficient gives the first equality G7. The parent also gives A_(ZW)=ZA_W; substitute X=Y=A_W in the just-proved equality to obtain its second assertion. This is consistent with both V03-PB-0017(C5)'s full-source scaling and G6's recovered source coefficient. No derivative terms have entered this algebraic calculation; the parent's full residual formula still retains them.

7. Recover all seven derivative contributions at the same output

PB-0008 P5 gives the complete finite derivative coefficient law. To reach output {0} after the right factor B with index {1,2}, an occurrence from derivative label μ and input blade I must have

({μ}△I)△{1,2}={0},
I={μ}△{0,1,2}.

Thus there is exactly the following one input position for each of the actual seven μ labels. The signs include the ε_μ in D, as required:

μ	Input blade index I	Complete coefficient sign in ε_μe_μe_IB
0	{1,2}	−u
1	{0,2}	−u
2	{0,1}	u
3	{0,1,2,3}	u
4	{0,1,2,4}	u
5	{0,1,2,5}	u
6	{0,1,2,6}	u

Here is the written sign calculation, including the four repeated spatial instances. PB-0006 supplies BB=−u; it also follows by the one exchange and two spatial squares. At μ=0,

ε₀e₀e_{12}B=e₀BB=−e₀.

At μ=1, using only contextual exchanges and the two spatial squares,

ε₁e₁e_{02}B
  =−e₁e₀e₂e₁e₂
  =e₀e₁e₂e₁e₂
  =e₀BB=−e₀.

At μ=2,

ε₂e₂e_{01}B
  =−e₂e₀e₁e₁e₂
  =e₂e₀e₂
  =−e₀e₂e₂=e₀.

For each actual i in the supplied four-label list 3,4,5,6, its occurrence in e_{012i} follows e₀e₁e₂. Pass the left e_i through those three distinct generators and then use its square:

ε_i e_i e_{012i}B
  =−e_i e₀e₁e₂e_iB
  =e₀e₁e₂e_i e_iB
  =−e₀BB=e₀.

Writing that equality at the four actual i positions proves all remaining rows. No other input index routes to this output for its μ, by the explicit symmetric-difference condition. To identify these local reductions with the signs in PB-0008(P5), put β={1,2}. Two applications of PB-0005's claimed product law, in their actual order, give at the qualifying I

ε_μe_μe_IB
  =ε_μχ({μ},I)χ({μ}△I,β)e₀.

The local row calculation above gives the same entire word as s_row e₀, where s_row is its displayed u or −u. Equate the two reductions and right-multiply by the actual e₀. Since e₀e₀=u, their scalar coefficients are equal:

ε_μχ({μ},I)χ({μ}△I,β)=s_row.

Thus the P5 coefficient formula has exactly the seven signs displayed here. The recovered derivative coefficient is therefore exactly

𝓒_{ {0} }(L(W))
  =h[−∂₀x_{12}(W)−∂₁x_{02}(W)+∂₂x_{01}(W)
       +∂₃x_{0123}(W)+∂₄x_{0124}(W)
       +∂₅x_{0125}(W)+∂₆x_{0126}(W)]
  =h bal₀(W).

Subtracting G6's full source coefficient, by the parent's coefficient difference law, gives the second equality in G8.

The declared inversion gives hk=u, and 04 rule 8 gives kh=hk=u. If that residual coefficient is v, multiply its G8 value by the existing k. Since kh=u, the resulting equality is bal₀(W)−g(A_W,A_W)=v; the void-difference criterion is exactly G9. Conversely the displayed scalar equality makes the bracket v and hence the recovered residual coefficient v. For Ψ the full equation supplies the premise; no equation for a generic W was used in deriving the equivalence.

The derivation leaves the entire derivative sum on one side and the entire active square sum on the other. It divides neither by an input coefficient nor by the value of the square sum.

What is not forced

The scalar pairing is reconstructed from an existing blade coefficient and the given algebraic operations. Its finite-list separation G3 uses all pairings with blades. It does not establish an ordered scalar domain, positivity, or the different implication g(X,X)=v ⇒ X=v. The finite coefficient expression Σ x_I(X)x_I(X) cannot be read as a positive quantity without an additional proved or supplied scalar property.

The e₀ source coefficient depends only on the selector-active part by G6. This is an output-specific identity. As a non-use comparison, PB-0009 G1 states that the coefficient at I_s={1,2,3,4,5,6} is identically v; that result is not a premise of the present forcing. At output indices other than {0} and I_s, the full coefficient formulas retain their H–A contributions; no universal nonvoid value of a contribution is asserted. Equality of two inputs' active square sums gives equality of precisely the e₀ output identified by G6. Full source equality retains the conditions at every other output coefficient. A void square sum is exactly this one collected coefficient condition.

The 72 positions are the derived length of the selector-active blade list. The one scalar g value in G9 remains a complete collected sum, and the seven derivative contributions remain one complete balance. Their equality is one required coefficient equation, not a replacement for the full residual.

The central multiplication law G7 is an exact relationship with V03-PB-0017. Under n(a,b)=u it preserves this pairing; the full residual still has the parent's complete derivative and source terms.

What it means

One specified output coefficient of a sandwich product gives an exact scalar pairing of all the recovered blade coefficients. Taking that pairing with the complete finite blade list recovers the entire input. The already constructed selector and four-sign decompositions divide this pairing into nonoverlapping contributions because their coefficient lists are disjoint. For the full quadratic source, its e₀ coefficient is precisely the self-pairing of the selector-active part multiplied by h; the mixed source has the corresponding two-input pairing multiplied by hh. The same output on the derivative side gives a complete seven-term balance. Their equality is exactly the e₀ coefficient condition of the full equation. Their square sums remain collected; the claims do not supply positivity.

Squaring the supplied derivative contraction retains an exact residual and a strict sufficiency boundary (V03-PB-0022)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0006, V01-PB-0008, V01-PB-0010, V02-PB-0011, V02-PB-0014, V03-PB-0020, V03-PB-0021

Declares

u,v,h,B retain 04's meanings; Sel,D,L,F,R retain PB-0006's definitions; Mix retains V02-PB-0014's definition. W is any supplied full written element. The square notation D(D(W)) means the existing expression D applied twice; it does not supply a new derivative, coordinate system or metric.

For this fixed seven-term calculation, ε_μ retains PB-0002's generator-square sign and b_{{μ}} retains that record's blade-undoing expression at the singleton {μ}. The displayed sums are fixed finite sums over these seven labels. μ<ν below means their supplied ordinary listing order and indexes each of the twenty-one distinct-label pairs once; it is not an order on scalar expressions.

Define the signed second-derivative expression only as the abbreviation

Sq(W):=∂₀∂₀W−∂₁∂₁W−∂₂∂₂W−∂₃∂₃W
              −∂₄∂₄W−∂₅∂₅W−∂₆∂₆W.

W_μ:=∂_μW retains V03-PB-0020's notation. For an actual supplied ordered derivative list P, ∂_P retains V03-PB-0021's listed-composition definition, including its empty list. The exact native expression e₀ is used at the end as an already supplied input, not as a proposed value of the distinguished solution Ψ.

What is forced

For the actual seven labels and every supplied W,

b_{{μ}}=ε_μe_μ,
D(W)=Σ_(μ=0,…,6) b_{{μ}}∂_μW.                         (Dlist)

The signed singleton factors satisfy, for the actual seven labels,

∂_ν b_{{μ}}=v,                                        (S7)
b_{{μ}}b_{{μ}}=ε_μu,                                  (S8)
b_{{μ}}b_{{ν}}+b_{{ν}}b_{{μ}}=v when μ≠ν.             (S9)

For every supplied full written W,

D(D(W))=Sq(W).                                         (S1)

Its exact residual derivative is

D(R(W))
   =hSq(W)B−D(F(W))
   =hSq(W)B−Σ_(μ=0,…,6) b_{{μ}}Mix(W,W_μ).                (S2)

Consequently R(W)=v implies the complete second-derivative balance

hSq(W)B=Σ_(μ=0,…,6) b_{{μ}}Mix(W,W_μ).                    (S3)

For a supplied arbitrary W, the exact equivalent condition for S3 is only

S3 holds iff D(R(W))=v.                               (S4)

That condition is strictly weaker than the original residual equation at the written inputs available in 04. The concrete input W=e₀ satisfies

∂_μR(e₀)=v for all seven supplied μ,
D(R(e₀))=v,
hSq(e₀)B=v=Σ_μb_{{μ}}Mix(e₀,∂_μe₀),                    (S5)

while its undifferentiated residual is

R(e₀)=−he₀,       −he₀ is nonvoid.                    (S6)

Thus neither S3 alone nor the conditions from any supplied finite collection of nonempty finite derivative lists ∂_P R(W)=v can be substituted for R(W)=v at unrestricted supplied inputs. For every actual supplied nonempty list, that derivative condition holds for e₀, while S6 does not vanish.

The forcing

1. Establish the fixed coefficient products and their derivatives

PB-0008 P5u identifies every singleton undoing as ε_μe_μ and D(W) as their seven-term derivative sum. Applied to this same W and the seven supplied labels, these are exactly Dlist.

From the definitions and the arithmetic exported in V01-PB-0001, each ε_μ is u or −u, so

ε_με_μ=u.

V02-PB-0011(D1) gives the void derivatives of both signs. The generator derivatives in 04 and 04's product rule therefore give, at each of the seven actual factors,

∂_νb_{{μ}}=(∂_νε_μ)e_μ+ε_μ∂_νe_μ
        =ve_μ+ε_μv=v.                               (S7)

Scalar centrality, association and the generator squares give

b_{{0}}b_{{0}}=(ε₀ε₀)(e₀e₀)=u,
b_{{i}}b_{{i}}=(εᵢεᵢ)(eᵢeᵢ)=−u.                            (S8)

For a fixed distinct pair μ,ν, move only the two scalar signs:

b_{{μ}}b_{{ν}}+b_{{ν}}b_{{μ}}
  =ε_με_ν(e_μe_ν)+ε_νε_μ(e_νe_μ)
  =ε_με_ν(e_μe_ν+e_νe_μ)
  =ε_με_νv=v.                                        (S9)

The last equality uses the supplied distinct-generator rule and absorption. It establishes a void value for that individual two-product sum; it is not inferred by splitting some previously void sum.

2. Expand the square before eliminating any cross term

Substitute the complete seven-term D expression into itself and use derivative additivity:

D(D(W))
  =Σ_μ b_{{μ}} ∂_μ[Σ_ν b_{{ν}}∂_νW]
  =Σ_μΣ_ν b_{{μ}}[(∂_μb_{{ν}})∂_νW+b_{{ν}}∂_μ∂_νW].

Every derivative of b_{{ν}} is void by S7. Its product is therefore void by absorption. The remaining complete expression is

D(D(W))=Σ_μΣ_ν b_{{μ}}b_{{ν}}∂_μ∂_νW.                        (S10)

This fixed double list has forty-nine positions. Regroup its seven equal-label terms and its twenty-one pairs of opposite order, using only additive association and commutation:

D(D(W))
  =Σ_μ b_{{μ}}b_{{μ}}∂_μ∂_μW
    +Σ_(μ<ν)[b_{{μ}}b_{{ν}}∂_μ∂_νW+b_{{ν}}b_{{μ}}∂_ν∂_μW].          (S11)

In each actual paired summand, 04's commuting-derivative rule gives ∂_ν∂_μW=∂_μ∂_νW. Right distribution then gives

b_{{μ}}b_{{ν}}∂_μ∂_νW+b_{{ν}}b_{{μ}}∂_ν∂_μW
  =(b_{{μ}}b_{{ν}}+b_{{ν}}b_{{μ}})∂_μ∂_νW
  =v∂_μ∂_νW=v,

where S9 supplies the particular void factor. All twenty-one paired summands have now been shown void separately by this same stated instance with their actual labels. No derivative of W has been assumed void.

Substitute S8 into the seven diagonal terms of S11. Their sum is

∂₀∂₀W−∂₁∂₁W−∂₂∂₂W−∂₃∂₃W
         −∂₄∂₄W−∂₅∂₅W−∂₆∂₆W,

which is precisely Sq(W). This proves S1 with every term accounted for.

3. Apply D to the full residual, preserving its source

The derivative addition and negation laws give D's corresponding laws by its finite definition. Hence

D(R(W))=D(L(W))−D(F(W)).                              (S12)

Expand the first term before taking any simplification:

D(L(W))
  =Σ_μ b_{{μ}}∂_μ[hD(W)B]
  =Σ_μ b_{{μ}}[(∂_μh)D(W)B
              +h∂_μ(D(W))B+hD(W)(∂_μB)].

V02-PB-0011 gives ∂_μh=v. For the other constant factor, the supplied generator derivatives and 04's product rule give locally

∂_μB=∂_μ(e₁e₂)=(∂_μe₁)e₂+e₁∂_μe₂=ve₂+e₁v=v.

The products with these two void derivative factors are void. Moving only the scalar h and distributing the common final B gives

D(L(W))
  =h[Σ_μb_{{μ}}∂_μ(D(W))]B
  =hD(D(W))B
  =hSq(W)B.                                          (S13)

For the second term in S12, retain every generator multiplication and substitute V03-PB-0020(E2) separately at each label:

D(F(W))
  =Σ_μb_{{μ}}∂_μF(W)
  =Σ_μb_{{μ}}Mix(W,∂_μW)
  =Σ_μb_{{μ}}Mix(W,W_μ).                                 (S14)

Each displayed generator remains to the left of the complete Mix expression. Equations S12–S14 prove both equalities in S2.

Now retain R(W)=v. The void derivative law implies D(v)=v by its seven-term definition. Substitution into S2 gives S3 by the void-difference criterion. For an arbitrary supplied W, that same criterion applied directly to S2 gives precisely S4, in both directions. It gives no undoing expression for D.

4. Exhibit the undifferentiated residual that this consequence loses

PB-0010(R10) supplies the actual diagnostic

D(e₀)=L(e₀)=v,    F(e₀)=he₀,    R(e₀)=−he₀≠v.

The supplied generator derivatives and V02-PB-0011(D1) also give ∂_μ∂_μe₀=∂_μv=v, hence Sq(e₀)=v. These are the same input and residual in S5–S6; no additional witness is introduced.

Nevertheless 04's product rule and the constant derivatives already proved give, for each supplied label,

∂_μR(e₀)
  =−∂_μ(he₀)
  =−[(∂_μh)e₀+h(∂_μe₀)]
  =−[ve₀+hv]=v.

Thus D(R(e₀))=v. Further supplied derivatives of this expression remain void because every derivative of v is void. At any actual nonempty finite list P, the resulting ∂_P R(e₀) is therefore v by the displayed repeated local step.

Finally, the left side of S3 at e₀ is h v B=v. Each right-side term has its second mixed input ∂_μe₀=v, so V02-PB-0014(MixOps) gives Mix(e₀,v)=v. Every term is then b_{{μ}}v=v and the finite sum is v. This verifies S5 directly as well as through S2. The same native expression therefore satisfies the second-derivative consequence and all residual derivative conditions along nonempty lists while failing the original residual equation.

What is not forced

S1 is an identity of the operations supplied in 04. Sq denotes exactly the displayed seven-term derivative expression.

S3 is a consequence of the full nonlinear equation. Its exact reverse condition is S4, not R(W)=v. The explicit e₀ instance proves that an undifferentiated residual can be lost even when its derivatives along every nonempty list are retained. Restoring the original equation requires that residual itself to be void, or some separately proved condition that forces it to be void. No such supplementary condition is silently supplied here.

The witness e₀ is not identified with Ψ. It distinguishes two conditions on arbitrary supplied inputs and demonstrates a logical non-equivalence between them. It does not contradict the axiom's asserted equation for its distinguished Ψ.

No void coefficient is imposed on the field, and no sum of second derivatives or mixed terms in S3 is split into separate void conditions. The only terms removed in the proof have their particular void factors derived explicitly in S7, S9, or the stated constant-derivative identities.

What it means

Applying the supplied derivative contraction twice gives a signed sum of its seven repeated second derivatives because the complete set of distinct-label terms cancels by the stated generator and derivative rules. Applying that identity to the equation retains a definite nonlinear source. The same calculation also identifies what this consequence can lose: the native input e₀ has a nonvoid residual whose every supplied nonempty derivative list is void. The second-derivative balance is therefore a necessary consequence with an explicit insufficiency witness, and it cannot replace the original equation without an additional condition.

Central source preservation has a complete finite differential criterion (V04-PB-0025)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0004, V01-PB-0006, V01-PB-0007, V02-PB-0011, V03-PB-0017

Declares

All scalar and written-element operations retain 04 and the parents' actual-hypothesis scope. The scalars u,v,h,k and word B retain 04's meanings; G and 𝓒_I retain PB-0004's definitions; ε_μ retains PB-0002's generator-square signs; D,L,F,R,Sel retain PB-0006's definitions; T retains PB-0007's definition. The equation is imposed only where R(W)=v is explicitly retained. The existing inverse k uses exactly the declared nonvoid guard on h; no inverse of a, b, n or an input is introduced.

• For supplied scalars a,b, write Z=Z(a,b)=a+bΩ, Zᶜ=a−bΩ and n=n(a,b)=aa+bb, exactly as in V03-PB-0017.

• For a full input W write A=A_W and H=H_W, using V01-PB-0007's part definitions. The source factor K_A(W)=e₀Ã_W retains the definition in V01-PB-0007 Declares. For the transformed full input, A'=A_(ZW), H'=H_(ZW), and the same defined factor is K_A(ZW)=e₀A'̃. The prime is a label, not a derivative.

• At each supplied derivative label μ define the scalar expressions a_{,μ}:=∂μa and b{,μ}:=∂_μb. They are not independent coefficient data. Under the retained identity n=u define

  ω_μ:=a b_{,μ}−b a_{,μ},
  K_ω:=Σ_(μ∈G) ε_μ e_μ ω_μ.                         (N0)
  

  K_ω and K_A(W) are different displayed expressions, with their subscripts retained. The first uses the actual seven derivative directions; the second uses the actual selector part.

• The condition n=u is an identity of the supplied expressions when it is differentiated. It is not a statement about values at an undeclared point. Individual equations involving W are likewise retained identities.

What is forced

For a supplied central Z, with no derivative premise,

F(ZW)=F(W) for every supplied W  iff n=u,             (N1a)
F(ZW)=F(W) for one retained W
   iff (n−u)F(W)=v.                                  (N1b)

These are different quantifier scopes. The universal one makes Zᶜ an explicit two-sided undoing factor because ZZᶜ=ZᶜZ=u.

For every Z and full W,

K_A(ZW)=ZᶜK_A(W).                                        (N2a)

Under n=u, the complete source contributions and the product condition transport as

A'K_A(ZW)=A K_A(W),       H'K_A(ZW)=H K_A(W),
ΔK_A(ZW)=Zᶜ(ΔK_A(W)),
ΔK_A(ZW)=v iff ΔK_A(W)=v                                 (N2b)

for the same supplied Δ. If T(Δ)=Δ is additionally retained, that same condition accompanies both sides of the equivalence; it is not inferred from either product.

For every supplied central Z and full W, the seven-term expression has the exact form

Σ_(μ∈G)ε_μe_μ W(∂_μZ)=(D(Z))W.                     (N2c)

Each ∂_μZ is central; the contracted D(Z) is not moved through W in this identity.

Under the retained identity n=u, every actual derivative label and pair obey

aa_{,μ}+bb_{,μ}=v,
(∂_μZ)Zᶜ=ω_μΩ,
∂_μZ=ω_μΩZ,
a_{,μ}=−bω_μ,       b_{,μ}=aω_μ,
∂_νω_μ=∂_μω_ν.                                     (N4)

The scalar ω_μ is recoverable as 𝓒_G((∂_μZ)Zᶜ).

For an arbitrary full W, still under n=u but without R(W)=v,

ZᶜR(ZW)
  =R(W)+(u−Zᶜ)F(W)+h K_ω W Ω B.                      (N5a)

Consequently, under the additional actual premise R(W)=v,

R(ZW)=v
  iff K_ω W Ω B=W K_A(W)(Zᶜ−u).                         (N5b)

This is the derivative-resolved form of the solution criterion V03-PB-0017 C7, with both n=u and R(W)=v and every seven-direction contribution retained.

The forcing

1. Apply the complete central interface

The expressions Sel,D,L,F,R here retain the definitions in V01-PB-0006 Declares. V03-PB-0017 C2 is the complete center, with unique recovered coefficients a=𝓒_∅(Z) and b=𝓒_G(Z). Its C3 gives

Z̃=Z*=Zᶜ,       T(Z)=Z,       ZZᶜ=ZᶜZ=nu.

Its C4–C6 apply to every full W, with arbitrary supplied scalar coefficients:

A'=ZA,       H'=ZH,
Sel(ZW)=ZᶜSel(W),
F(ZW)=nF(W),
R(ZW)=Z R(W)+(Z−nu)F(W)+h(D(Z))W B.                  (N7)

These results are reused at their proved scope; the complete two-term center is that parent's C2 result.

The direction from n=u in N1a follows at once from N7. To prove the reverse without arbitrary-factor cancellation, use the supplied input e₀. From its asserted generator images,

Sel(e₀)=e₀−(−e₀)=he₀,
F(e₀)=e₀e₀he₀=he₀.

The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. Universal preservation therefore gives nhe₀=he₀. Right multiplication by e₀ and left multiplication by k give n=u, using e₀e₀=u and kh=u. This is an actual diagnostic of a universal identity; e₀ is not claimed to solve R=v.

For one fixed W, subtract the two whole source values in N7. The exact difference is (n−u)F(W). The void-difference criterion proves both directions of N1b, and supplies no licence to cancel F(W).

2. Transport the actual constructed source factor

N7 supplies A'=ZA. Reversion reverses the product order; centrality of Zᶜ then gives

A'̃=ÃZ̃=ÃZᶜ,
K_A(ZW)=e₀ÃZᶜ=Zᶜe₀Ã=ZᶜK_A(W).

This proves N2a for all Z. Now retain n=u. The actual two-sided product ZZᶜ=u yields

A'K_A(ZW)=(ZA)(ZᶜK_A(W))=ZZᶜ A K_A(W)=A K_A(W),
H'K_A(ZW)=(ZH)(ZᶜK_A(W))=ZZᶜ H K_A(W)=H K_A(W).

Only the central factors Z,Zᶜ have moved; A,H and K_A(W) keep their order. PB-0007's source-factor identity F(W)=hWK_A(W), with W=A+H, gives the full source h(A+H)K_A(W), so these identities preserve its two complete contributions separately.

For the unchanged supplied Δ, centrality gives

ΔK_A(ZW)=ΔZᶜK_A(W)=Zᶜ(ΔK_A(W)).

If ΔK_A(W)=v, absorption makes the transformed product v. Conversely, if the transformed product is v, left-multiply by the supplied Z and associate:

ΔK_A(W)=ZZᶜ(ΔK_A(W))=Zv=v.

This proves both directions of N2b. Appending the same additional equality T(Δ)=Δ to both sides preserves the equivalence without deriving that equality from the product.

3. Recover the seven-term central derivative product

For every actual μ, derivative additivity and the fixed derivative ∂_μΩ=v give

∂_μZ=a_{,μ}+b_{,μ}Ω.

This is a central expression by V03-PB-0017 C2. Consequently W(∂_μZ)=(∂_μZ)W at that actual μ. Substitution and distribution over the original seven-term list give

Σ_μ ε_μe_μ W(∂_μZ)
   =Σ_μ ε_μe_μ(∂_μZ)W
   =(Σ_μ ε_μe_μ∂_μZ)W=(D(Z))W.

This is N2c. Every nonscalar e_μ retains its position; no equality W D(Z)=D(Z)W is used.

4. Derive the entire first-derivative relation from n=u

Retain the identity aa+bb=u. V02-PB-0011 gives ∂_μu=v and the required derivative-negation laws. At each actual μ, 04's product rule and scalar commutation give

v=∂_μ(aa+bb)
  =a_{,μ}a+aa_{,μ}+b_{,μ}b+bb_{,μ}
  =h(aa_{,μ}+bb_{,μ}).

Multiplying by k gives aa_{,μ}+bb_{,μ}=v. The complete two-factor expansion is

(∂_μZ)Zᶜ
  =(a_{,μ}+b_{,μ}Ω)(a−bΩ)
  =aa_{,μ}−ba_{,μ}Ω+ab_{,μ}Ω−bb_{,μ}ΩΩ
  =(aa_{,μ}+bb_{,μ})+(ab_{,μ}−ba_{,μ})Ω
  =ω_μΩ.                                             (N8)

Every contribution was retained until the first scalar sum was proved void. Right multiplication by Z and ZᶜZ=u yields ∂_μZ=ω_μΩZ.

The separate scalar derivative identities can be derived without cancelling a or b:

−bω_μ=bb a_{,μ}−ab b_{,μ}
       =a_{,μ}−a(aa_{,μ}+bb_{,μ})=a_{,μ},

aω_μ=aa b_{,μ}−ab a_{,μ}
      =b_{,μ}−b(aa_{,μ}+bb_{,μ})=b_{,μ}.                          (N9)

The second equalities insert exactly aa+bb=u and regroup scalar products. The last equalities use the already derived collected tangent condition, not termwise vanishings. This proves all first-derivative statements in N4. Coefficient recovery at G in N8 gives ω_μ, because Ω has coefficient u there and every other coefficient v.

5. Write both second-derivative expansions

Fix an actual pair μ,ν from the supplied seven-label list. Differentiation of the definition in N0 uses the product rule and the explicit negation compatibility:

∂_νω_μ
  =a_{,ν} b_{,μ}+a(∂_ν∂_μb)
     −b_{,ν} a_{,μ}−b(∂_ν∂_μa),

∂_μω_ν
  =a_{,μ} b_{,ν}+a(∂_μ∂_νb)
     −b_{,μ} a_{,ν}−b(∂_μ∂_νa).                           (N10)

Subtract these entire expressions. The terms containing second derivatives cancel in their displayed pairs because 04 gives commuting derivatives at a and at b. Scalar commutation then gives

∂_νω_μ−∂_μω_ν
  =a_{,ν} b_{,μ}−b_{,ν} a_{,μ}−a_{,μ} b_{,ν}+b_{,μ} a_{,ν}
  =h(a_{,ν} b_{,μ}−a_{,μ} b_{,ν}).                               (N11)

Substitute the four corresponding N9 identities:

a_{,ν} b_{,μ}−a_{,μ} b_{,ν}
  =(−bω_ν)(aω_μ)−(−bω_μ)(aω_ν)
  =−abω_νω_μ+abω_μω_ν=v.

The last equality moves only scalar factors, including the two actual ω expressions. Thus N11 is v, proving the final equality in N4. These written steps apply at every actual pair of the finite derivative-label list, including μ=ν. Only the two written derivatives at each actual pair are used.

6. Substitute the derivative relation into the full residual

Use the complete seven terms, now with the established N4 derivative expression:

D(Z)=Σ_μ ε_μe_μ(∂_μZ)
     =Σ_μ ε_μe_μω_μΩZ
     =K_ωΩZ.                                         (N12)

This is a finite distribution of the common right factors ΩZ. It does not set any of the seven terms to v or move a nonscalar K_ω through W.

Under n=u, left-multiply N7 by Zᶜ. Centrality of Zᶜ and the two-sided identity give

ZᶜR(ZW)
  =R(W)+(u−Zᶜ)F(W)+h Zᶜ(D(Z))W B.

Substitute N12. Since Ω and Z,Zᶜ are central,

Zᶜ(D(Z))W B
  =ZᶜK_ωΩZ W B
  =K_ω W Ω B.

This proves N5a on every full W. Under R(W)=v, the transformed residual is v iff its left product by Zᶜ is v, because Z undoes that factor. Insert PB-0007's source-factor identity F(W)=hW K_A(W) in the right-hand side of N5a, commute only Zᶜ through W K_A(W), and undo h using k. The resulting equality is exactly N5b.

For the exact comparison with V03-PB-0017 C7, use its equivalent form (D(Z))WB=WK_A(W)(u−Z) under n=u and R(W)=v. Substitute N12 and right-multiply that whole equality by Zᶜ:

K_ωΩZ W B Zᶜ=K_ω W Ω B,
W K_A(W)(u−Z)Zᶜ=W K_A(W)(Zᶜ−u).

Conversely, right multiplication by the actual undoing factor Z returns that C7 equality. Thus both complete solution conditions agree in both directions under precisely the displayed premises.

What is not forced

The identity aa+bb=u is retained where N2b, N4 and N5 require it. The single-input source equality N1b is the displayed product condition; N1a instead quantifies over every supplied W. Neither statement cancels F(W).

The relation ∂_νω_μ=∂_μω_ν is a finite derivative identity of the supplied scalars. The seven contributions in K_ω stay collected. No individual term is removed from N5a or N5b.

The product transport N2b concerns the same supplied Δ under the same central change. Its T sign, if needed by a consumer, remains a separate hypothesis. K_A(W) is constructed from A_W, while the full source retains both A_W and H_W through F(W)=hW K_A(W).

The complete residual criterion in N5b requires n=u and R(W)=v. It concerns the retained input W; a universal residual identity has a different quantifier scope.

What it means

The complete central source law determines universal source preservation and transports the same right source factor. Under the identity aa+bb=u, each actual derivative of Z is expressed through one constructed scalar ω_μ, and the written two-derivative calculation proves their compatibility. Under the same n=u, substitution into the complete residual gives N5a for every input; its solution criterion N5b additionally retains R(W)=v. The different all-input condition R(ZW)=ZR(W) for every supplied W is V05-PB-0037(C1); under n=u its exact restriction is that record's C3. This is a non-use reference locating the universal result, not a premise of N1–N5.

A retained source-factor core determines an exact quadratic remainder and transfers every completion with its derivatives (V04-PB-0028)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0006, V01-PB-0007, V01-PB-0004, V02-PB-0011, V04-PB-0026, V04-PB-0031, V05-PB-0035, V07-PB-0042

Declares

Retain actual supplied factors K,V satisfying

Γ(K)=K, Γ(V)=V, KVK=K, VKV=V.                       (H0)

Γ retains V04-PB-0026’s definition and T retains V01-PB-0007’s. Put p=KV,r=VK and E_L=E_L(K,V),E_R=E_R(K,V). These are V05-PB-0035's parameterized complements at the same supplied K,V. For another supplied Γ-fixed factor K′, the prime is a label and the additional retained-core condition is

pK′r=K.                                             (H1)

Define, on that same full K′,

W_c:=E_LK′r,           C_c:=E_LK′E_R,
S_c:=C_c−W_cVΓ(W_c),
M:=u+W_cV,             M_undo:=u−W_cV,
K_c:=K+S_c.                                         (H2)

S_c is the corrected complementary remainder. These are finite expressions, constructed by the displayed products and sums. K_c retains this meaning throughout the proof. For a proposed mixed output P_out set

P_M:=T(M)P_out Γ(M_undo).

For any supplied pair M,M_undo, including H2’s constructed pair, and each of the seven supplied derivative labels μ, define

Mdrv_μ:=M_undo(∂_μM),
e_μ^M:=T(M)e_μT(M_undo),
Der_(M,μ)(Θ):=∂_μΘ−T(Mdrv_μ)Θ−ΘMdrv_μ.                   (H3)

These names abbreviate the displayed actual products. ε_μ retains V01-PB-0002’s definition carried by V01-PB-0004; D and R retain V01-PB-0006’s definitions, and K_A retains V01-PB-0007’s. Every sum on μ below is the fixed seven-entry accumulation. Scalar guards retain 04.

What is forced

Under H0–H1, the full factor is exactly

K′=K+W_c+Γ(W_c)+W_cVΓ(W_c)+S_c
   =M(K+S_c)Γ(M),
W_c=E_L W_c r, S_c=E_L S_c E_R, Γ(S_c)=S_c,
M M_undo=M_undo M=u,       M_undo K′Γ(M_undo)=K+S_c.                       (R1)

Conversely every supplied W_c,S_c satisfying the displayed sandwich and Γ conditions constructs a Γ-fixed K′ with pK′r=K by R1, and H2 recovers those very W_c,S_c uniquely relative to the fixed K,V.

Under H0–H1 there is an exact constructive equivalence of supplied witness availability:

K′U′K′=K′  ⇔  S_c U_s S_c=S_c.                    (R2)

The two directions mean the following finite constructions. From U′ take

U_s=E_R Γ(M)U′M E_L.                               (R2a)

From U_s use V05-PB-0035 S6a at ι=Γ,σ_fac=u on S_c to obtain V₀, and set

V_s=E_RV₀E_L,          V′=Γ(M_undo)(V+V_s)M_undo.             (R2b)

Then

Γ(V_s)=V_s, S_cV_sS_c=S_c, V_sS_cV_s=V_s,
Γ(V′)=V′, K′V′K′=K′, V′K′V′=V′,
u−K′V′=M(E_L−S_cV_s)M_undo,
u−V′K′=Γ(M_undo)(E_R−V_sS_c)Γ(M).                      (R2c)

At S_c=v one may take V_s=v. If the stronger actual products S_cV_s=E_L and V_sS_c=E_R hold, V′ undoes K′ in both orders.

For either σ=u or σ=−u, V05-PB-0035 S7b and S7e apply with ι=T,ϑ=Γ, K_c=K+S_c and its actual H0 products. Thus, for the same Θ,

T(H)=σH, HK′=P_out
  iff T(Θ)=σΘ, ΘK=P_Mr, ΘS_c=P_ME_R,
Θ=T(M)HM, H=T(M_undo)ΘM_undo.                              (R3)

At P_out=v its complete null presentation is S7f with that same K,V. At S_c=v its two tests and all completions are precisely S7g. These applications keep H0–H1, both operations and the same actual factor transformation; no active-part realization is assumed.

For arbitrary actual M,M_undo obeying M M_undo=M_undo M=u, not only H2's particular construction, the entire derivative transforms as

∂_μ[T(M_undo)ΘM_undo]=T(M_undo)Der_(M,μ)(Θ)M_undo,
D[T(M_undo)ΘM_undo]=T(M_undo)[Σ_με_μe_μ^M Der_(M,μ)(Θ)]M_undo.         (R6a)

Thus its vanishing is equivalent to vanishing of that one complete middle expression. The transformed generators and actual Mdrv expressions obey

e_μ^M e_μ^M=ε_μ,
e_μ^M e_ν^M+e_ν^M e_μ^M=v (μ≠ν),
∂_μMdrv_ν−∂_νMdrv_μ+Mdrv_μ Mdrv_ν−Mdrv_ν Mdrv_μ=v.                   (R6b)

For an actual X with K_A(X)=K′ and an actual T-fixed Δ with ΔK′=v, V07-PB-0042 FC4 supplies the source-preserving residual comparison. Under the additional R(X)=v, FC4s gives the new-solution condition as exactly the single middle expression in R6a being v, with Δ=T(M_undo)ΘM_undo. (R6c)

There is a finite diagnostic for the distinct C_c and S_c. Use V04-PB-0031's pair (3,4), take K=V=P_{34,+}, and define

K_bare=P_{34,+}+e₁I_{34}, K_complete=u+e₁I_{34}.

Both are Γ-fixed and retain pK′r=P_{34,+}. For the H2 expressions on each factor separately,

C_c[K_bare]=v, S_c[K_bare]=−P_{34,−},
C_c[K_complete]=P_{34,−}, S_c[K_complete]=v,
K_bare(e₁I_{34}−P_{34,−})
   =(e₁I_{34}−P_{34,−})K_bare=u,
(−I_{34}−e₁)K_complete=v, −I_{34}−e₁≠v.           (R8)

The bracketed argument indicates the actual K′ used in H2; it defines no second remainder map.

The forcing

1. Recover the full finite change from its supplied core

V05-PB-0035 S6b at ι=Γ,σ_fac=u supplies all ordered products from H0. In particular p+E_L=u, r+E_R=u, E_LK=KE_R=VE_L=E_RV=v, Γ(p)=r and Γ(E_L)=E_R. Insert both complete identities into the full K′:

K′=(p+E_L)K′(r+E_R)
   =pK′r+pK′E_R+E_LK′r+E_LK′E_R
   =K+Γ(W_c)+W_c+C_c.

The equality pK′E_R=Γ(W_c) follows by V04-PB-0026 F1’s product reversal applied to E_LK′r, using Γ(K′)=K′ and the S6b images Γ(E_L)=E_R and Γ(r)=p. The products give

E_L W_c r=W_c,
E_LC_cE_R=C_c,
Γ(C_c)=Γ(E_R)K′Γ(E_L)=E_LK′E_R=C_c.

Since E_LW_c=W_c and Γ(W_c)E_R=Γ(W_c), subtraction in H2 gives E_L S_c E_R=S_c. Product reversal gives Γ(W_cVΓ(W_c))=W_cVΓ(W_c), so Γ(S_c)=S_c.

Moreover VW_c=VE_LK′r=v. Therefore the actual square of W_cV is

(W_cV)(W_cV)=W_c(VW_c)V=v.

Expanding all four terms of each product gives

M M_undo=u−W_cV+W_cV−(W_cV)(W_cV)=u,
M_undo M=u+W_cV−W_cV−(W_cV)(W_cV)=u.

Applying Γ to both supplies Γ(M)Γ(M_undo)=Γ(M_undo)Γ(M)=u.

Now Γ(M)=u+VΓ(W_c), and the complete four-term product is

MKΓ(M)=K+W_cVK+KVΓ(W_c)+W_cVKVΓ(W_c)
        =K+W_c+Γ(W_c)+W_cVΓ(W_c).

Here W_cVK=W_cr=W_c; KVΓ(W_c)=pΓ(W_c)=Γ(W_c); VKV=V supplies the final reduction. Also VS_c=VE_L S_c E_R=v and S_cV=E_L S_c E_RV=v, hence the full expansion

MS_cΓ(M)
  =S_c+W_cVS_c+S_cVΓ(W_c)+W_cVS_cVΓ(W_c)=S_c.

Adding gives K′=M(K+S_c)Γ(M). Actual multiplication by M_undo and Γ(M_undo) recovers K+S_c. No term was dropped because it was small.

2. Prove the converse and uniqueness of the recovered remainder

Take supplied W_c=E_L W_c r and S_c=E_L S_c E_R with Γ(S_c)=S_c. Define K′ by the five-term expression in R1. Γ exchanges W_c with Γ(W_c), fixes K,S_c, and fixes W_cVΓ(W_c), so Γ(K′)=K′. Multiplication by p on the left and r on the right gives

pK′r=pKr+pW_cr+pΓ(W_c)r
             +pW_cVΓ(W_c)r+pS_cr=K.

The four removed terms have respectively pE_L, E_Rr, pE_L and pE_L as factors. For the recovery of W_c,

E_LK′r=E_LKr+E_L W_c r+E_LΓ(W_c)r
                +E_LW_cVΓ(W_c)r+E_LS_cr=W_c.

The removed terms use E_LK=v, E_Lp=v or E_Rr=v, as appropriate. The complementary recovery is

E_LK′E_R=W_cVΓ(W_c)+S_c.

Here the K term uses E_LK=v; W_cE_R=E_L W_c rE_R=v; E_LΓ(W_c)=E_LpΓ(W_c)=v; the two retained terms have their defining outer sandwiches. Subtracting the recovered W_cVΓ(W_c) returns exactly S_c. Hence any two such descriptions of this K′ with the fixed K,V have the same recovered W_c and S_c. The square and transformed-product calculations of §1 apply to these supplied sandwich conditions as well, proving all of R1 in both directions.

3. Transfer an actual witness to the corrected remainder

Retain K′U′K′=K′ and use the already established K′=MK_cΓ(M). Multiplying the full equality by M_undo on the left and Γ(M_undo) on the right gives

K_c[Γ(M)U′M]K_c=K_c.

Put U_c=Γ(M)U′M. Since E_LK_c=S_c and K_cE_R=S_c, left multiplication by E_L and right multiplication by E_R give S_cU_cS_c=S_c. As S_cE_R=S_c and E_LS_c=S_c,

S_c(E_RU_cE_L)S_c=S_cU_cS_c=S_c.

This proves R2a without requiring a sign on U′.

Conversely retain S_cU_sS_c=S_c. Because Γ(S_c)=S_c, V05-PB-0035 S6a at ι=Γ,σ_fac=u applies to that actual U_s:

W_s=k(U_s+Γ(U_s)), V₀=W_sS_cW_s,
Γ(V₀)=V₀, S_cV₀S_c=S_c, V₀S_cV₀=V₀.

With V_s=E_RV₀E_L, reversal and Γ(E_L)=E_R give Γ(V_s)=V_s. The ordered sandwiches and the two preceding identities give

S_cV_sS_c=S_cE_RV₀E_LS_c=S_cV₀S_c=S_c,
V_sS_cV_s=E_RV₀E_L S_c E_RV₀E_L
            =E_RV₀S_cV₀E_L=E_RV₀E_L=V_s.

The four cross products vanish for individually displayed reasons:

KV_s=(KE_R)V₀E_L=v,
V_sK=E_RV₀(E_LK)=v,
VS_c=(VE_L)S_cE_R=v,
S_cV=E_LS_c(E_RV)=v.

Let V_c=V+V_s. The two-factor products are therefore K_cV_c=p+S_cV_s and V_cK_c=r+V_sS_c. Their complete subsequent products are

K_cV_cK_c=(p+S_cV_s)(K+S_c)
          =pK+pS_c+S_cV_sK+S_cV_sS_c=K+S_c=K_c,
V_cK_cV_c=(r+V_sS_c)(V+V_s)
          =rV+rV_s+V_sS_cV+V_sS_cV_s=V+V_s=V_c.

The removed terms have pE_L, rE_R or one of the four cross products as factors. Γ(V_c)=V_c by V04-PB-0026 F1’s additivity. For the defined V′=Γ(M_undo)V_cM_undo, F1’s product reversal and the actual adjacent undoings give both products directly,

Γ(V′)=Γ(M_undo)Γ(V_c)M_undo=V′,
K′V′K′=M K_cV_cK_c Γ(M)=M K_c Γ(M)=K′,
V′K′V′=Γ(M_undo)V_cK_cV_cM_undo=Γ(M_undo)V_cM_undo=V′.

Finally use K_cV_c=p+S_cV_s and V_cK_c=r+V_sS_c, and write each exterior unity by its actual undoing products:

u−K′V′=M[u−p−S_cV_s]M_undo=M(E_L−S_cV_s)M_undo,
u−V′K′=Γ(M_undo)[u−r−V_sS_c]Γ(M)
          =Γ(M_undo)(E_R−V_sS_c)Γ(M).

At S_c=v, V_s=v meets every stated condition. Under the stronger S_cV_s=E_L, V_sS_c=E_R, both complementary expressions vanish, so K′V′=V′K′=u. If a supplied Δ then satisfies ΔK′=v, its actual right undoing gives Δ=ΔK′V′=v. This conclusion is a consequence of the full construction, not an assumed cancellation by a nonvoid factor.

4. Apply the complete product spine to one common unknown

R1 has supplied K′=M(K+S_c)Γ(M),M M_undo=M_undo M=u. Apply V05-PB-0035 S7b with ι=T,ϑ=Γ and the same K_c=K+S_c to obtain T(H)=σH,HK′=P_out iff T(Θ)=σΘ,Θ(K+S_c)=P_M. H0 and the recovered sandwich for S_c give Kr=K,KE_R=v,S_cr=v and S_cE_R=S_c. These are exactly S7e's hypotheses, so its coupled same-Θ criterion proves R3. S7f and S7g apply using the actual KVK=K,VKV=V from H0. Their positive-sign instance is the T-fixed completion and null class; their negative-sign instance is the same product theorem with the other retained sign. No duplicate completion proof or separate choice of Θ is required.

5. Transfer all derivative contributions, including both exterior factors

This paragraph uses only actual M M_undo=M_undo M=u; it consequently proves the broader derivative claim as stated. V02-PB-0011 D1 gives ∂_μu=v, and D3 gives ∂_μT(Y)=T(∂_μY). Differentiate M_undo M=u, multiply by M_undo on the right, and use the actual M M_undo=u:

(∂_μM_undo)M+M_undo(∂_μM)=v,
∂_μM_undo=−M_undo(∂_μM)M_undo=−Mdrv_μ M_undo,
∂_μM=M Mdrv_μ,
∂_μT(M_undo)=−T(M_undo)T(Mdrv_μ).

04’s product rule on every factor of H=T(M_undo)ΘM_undo now gives

∂_μH=−T(M_undo)T(Mdrv_μ)ΘM_undo+T(M_undo)(∂_μΘ)M_undo−T(M_undo)ΘMdrv_μ M_undo
      =T(M_undo)[∂_μΘ−T(Mdrv_μ)Θ−ΘMdrv_μ]M_undo.

For each of the seven generator multiplications,

e_μT(M_undo)=T(M_undo)[T(M)e_μT(M_undo)]=T(M_undo)e_μ^M.

Multiply the previous derivative equality by its actual ε_μe_μ on the left and append all seven terms in 04’s order. The definition of ε_μ gives ε₀=u and εᵢ=−u, exactly the seven written signs in D. Distribution gives the second equality of R6a. Multiplication on the left by T(M) and on the right by M reverses those exterior factors; hence the complete D vanishes exactly when that single middle expression vanishes.

The generator products follow from the actual adjacent undoing:

e_μ^M e_ν^M=T(M)e_μ[T(M_undo)T(M)]e_νT(M_undo)
             =T(M)e_μe_νT(M_undo).

At equal indices this is ε_μ by scalar centrality and T(M)T(M_undo)=u. At distinct indices add the two orders and substitute the supplied e_μe_ν+e_νe_μ=v. This yields the first two lines of R6b. It does not make ∂_νe_μ^M vanish.

Finally, on the actual written M and the supplied commuting derivatives,

∂_μMdrv_ν=(∂_μM_undo)(∂_νM)+M_undo∂_μ∂_νM
        =−Mdrv_μ Mdrv_ν+M_undo∂_μ∂_νM,
∂_νMdrv_μ=−Mdrv_ν Mdrv_μ+M_undo∂_ν∂_μM.

Subtracting proves the last identity of R6b. The proof starts with M,M_undo; it does not run backward to integrate arbitrary Mdrv values.

6. Cite the full source comparison and verify the diagnostic

At actual K_A(X)=K′,T(Δ)=Δ,ΔK′=v, V07-PB-0042 FC4 gives the source and residual comparison. At the additional R(X)=v, FC4s equates R(X+Δ)=v with D(Δ)=v. R6a at the actual Δ=T(M_undo)ΘM_undo is an equivalence through its actual outer undoings, so it supplies precisely R6c.

For R8, V04-PB-0031 BC5 gives the ordered complementary products and I_{34}I_{34}=u; BC5a gives I_{34}e₁=−e₁I_{34} and P_{34,σ}I_{34}=σP_{34,σ}, e₁P_{34,σ}=P_{34,−σ}e₁, P_{34,σ}e₁=e₁P_{34,−σ}, and e₀I_{34}=I_{34}e₀. In particular

(e₁I_{34})(e₁I_{34})=−e₁e₁I_{34}I_{34}=u,
P_{34,+}e₁I_{34}=−e₁P_{34,−},
(e₁I_{34})P_{34,+}=e₁P_{34,+},
(e₁I_{34})P_{34,−}=−e₁P_{34,−},
P_{34,−}e₁I_{34}=e₁P_{34,+}.

Γ(I_{34})=e₀T(I_{34})e₀=e₀I_{34}e₀=I_{34} by BC5 and BC5a. V04-PB-0026 F1 therefore gives Γ(P_{34,+})=k[u+Γ(I_{34})]=P_{34,+} and Γ(e₁I_{34})=Γ(I_{34})Γ(e₁)=−I_{34}e₁=e₁I_{34}. Its additive law now makes both factors Γ-fixed. The preceding products give pK_bare r=pK_complete r=P_{34,+}; their common W_c is e₁P_{34,+}. Its Γ image is −e₁P_{34,−}, and the quadratic correction is

W_cVΓ(W_c)=−e₁P_{34,+}e₁P_{34,−}=P_{34,−}.

The two full complementary products are P_{34,−}K_bareP_{34,−}=v and P_{34,−}K_completeP_{34,−}=P_{34,−}. Subtracting the same quadratic correction gives the two S_c values. The complete undoing products are

(P_{34,+}+e₁I_{34})(e₁I_{34}−P_{34,−})
   =−e₁P_{34,−}−v+u+e₁P_{34,−}=u,
(e₁I_{34}−P_{34,−})(P_{34,+}+e₁I_{34})
   =e₁P_{34,+}+u−v−e₁P_{34,+}=u.

Finally

(−I_{34}−e₁)(u+e₁I_{34})
  =−I_{34}−I_{34}e₁I_{34}−e₁−e₁e₁I_{34}
  =−I_{34}+e₁−e₁+I_{34}=v.

Its singleton-{1} coefficient is −u, which is nonvoid: u=v would imply h=v, contrary to the retained guard. V01-PB-0004 recovery therefore proves the stated nonvoid diagnostic.

What is not forced

H0 and H1 are genuine retained conditions. The theorem does not supply a core/witness for an unrestricted Γ-fixed factor, impose this decomposition on Ψ, or furnish the witness of S_c without its displayed premise. R1 recovers the full finite change at those conditions; it is not an iterative solver or a small-change approximation. S_c retains its quadratic correction. R8 computes the different C_c and S_c values on two actual factors with the same core and mixed product.

The two product equations in R3 share one expression and are equivalent in both directions to the original product. Their separate solution with unrelated expressions is not licensed. R6 preserves every derivative term and the transformed generator placement. Its identity for Mdrv does not provide an integration theorem. R6c supplies solution preservation only with an actual base solution and the entire derivative condition.

What it means

For an actual Γ-fixed pair K,V satisfying both H0 products, every supplied Γ-fixed K′ retaining pK′r=K has uniquely recovered W_c and S_c. Finite products remove W_c while retaining its quadratic contribution in S_c. Witness availability for K′ is equivalent to actual witness availability for S_c. The constructed transport is also the instance of V05-PB-0035 S7c at ϑ=Γ, stated here as a relationship rather than a step in its proof.

A prescribed output, including P_out=v, keeps both equations ΘK=P_Mr and ΘS_c=P_ME_R on the same signed Θ. The core equation has not been replaced by the complementary one. The derivative identity holds for any actual M,M_undo with both undoing products, and keeps its two exterior-factor terms and transformed generator multiplications. Its use for a new solution additionally retains K_A(X)=K′, R(X)=v, T(Δ)=Δ and ΔK′=v as in R6c.

Two even operands give a second complete writing of the same residual (V04-PB-0030)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0006, V01-PB-0007, V02-PB-0011, V02-PB-0013, V03-PB-0017, V07-PB-0043

Declares

All operands are supplied full written elements. Scalars u,v,h,k retain 04 and its guard; Sel,D,L,F,R retain V01-PB-0006’s definitions; T,A_X,H_X retain V01-PB-0007’s. Ω=e₀e₁e₂e₃e₄e₅e₆ retains V03-PB-0017’s definition; Ω_s=e₁e₂e₃e₄e₅e₆ retains V01-PB-0009’s definition carried by V02-PB-0013. Every input W retains its full scope.

Keep V02-PB-0013's α(Z)=e₀Ze₀ and V07-PB-0043's Gr(Z)=e₀Z*e₀. Even means Gr(Z)=Z and odd means Gr(Z)=−Z. For this supplied W put

U_ev:=k[W+Gr(W)],    O_ev:=k[W−Gr(W)],
V_ev:=−O_evΩ.                                         (EC1)

For these operands U_ev and V_ev, define

C_ev(U_ev,V_ev):=V_ev e₀Sel(U_ev)−U_ev e₀Sel(V_ev),
M_ev(W):=L(U_ev)−F(U_ev)−F(V_ev),
N_ev(W):=L(V_ev)−C_ev(U_ev,V_ev).                            (EC2)

The name C_ev denotes the displayed ordered mixed expression. For a spatial written Z put Z⁺=k[Z+α(Z)] and Z⁻=k[Z−α(Z)]. Superscripts here name the two displayed expressions; they are not powers.

What is forced

Every full W has the unique representation

W=U_ev+V_evΩ,        Gr(U_ev)=U_ev,        Gr(V_ev)=V_ev,              (EC4)

with exactly the operands in EC1. Its full source and residual are

F(W)=F(U_ev)+F(V_ev)+C_ev(U_ev,V_ev)Ω,
R(W)=M_ev(W)+N_ev(W)Ω.                             (EC5)

Both M_ev and N_ev are odd. They are recovered from the same residual by

M_ev(W)=k[R(W)−Gr(R(W))],
N_ev(W)=−k[R(W)+Gr(R(W))]Ω.                        (EC6)

In particular, retaining both equations,

R(W)=v
  iff D(U_ev)(−B)=(U_ev A_(U_ev)+V_ev A_(V_ev))e₀
   and D(V_ev)(−B)=(V_ev A_(U_ev)−U_ev A_(V_ev))e₀.                  (EC7)

No self-source or crossed source term has been discarded.

The exact bridge to V02-PB-0013 uses its unique W=X+e₀Y with spatial X,Y:

U_ev=X⁺+e₀Y⁻,
V_ev=−Y⁺Ω_s+e₀X⁻Ω_s.                               (EC8)

Conversely, write the already constructed U_ev,V_ev in their index-0 writings U_ev=U_ev0+e₀U_ev1 and V_ev=V_ev0+e₀V_ev1. Then

X=U_ev0−V_ev1Ω_s,       Y=U_ev1+V_ev0Ω_s.                   (EC9)

These are recovery maps between two complete descriptions of W, not two equations added together. If E₀,E₁ are V02-PB-0013's residual operands, so R(W)=E₀+e₀E₁ with E₀*=E₀ and E₁*=E₁ by V5, EC6 gives explicitly

M_ev=(E₀)⁻+e₀(E₁)⁺,
N_ev=(E₁)⁻Ω_s−e₀(E₀)⁺Ω_s.                       (EC10)

Thus either complete pair of residual operands vanishes exactly when the other does. Under the further condition V_ev=v the second equation in EC7 becomes v=v; the first remains D(U_ev)(−B)=U_ev A_(U_ev) e₀. That is an explicitly conditioned specialization.

The forcing

1. Use the all-generator sign operation

V07-PB-0043(G1–G3) supplies Gr's additive, scalar, product and double-application laws, its generator signs, its commutation with the two involutions, and Gr(DZ)=−D(GrZ). It also gives Gr(B)=B and Gr(Ω)=−Ω. Each is used on the same supplied expression in the following reconstruction.

2. Construct and recover the two operands without a support choice

The guarded inversion gives hk=u, and rule 8 gives kh=hk=u. Use Gr(Gr(W))=W in the actual two expressions in EC1:

U_ev+O_ev=k[W+Gr W+W−Gr W]=khW=W,
Gr(U_ev)=k[Gr W+W]=U_ev,
Gr(O_ev)=k[Gr W−W]=−O_ev.

These are the three concrete sum-and-difference calculations; no generic splitting theorem is presumed. Since Gr(Ω)=−Ω,

Gr(−O_evΩ)=−[−O_ev][−Ω]=−O_evΩ=V_ev.

V03-PB-0017 gives ΩΩ=−u. Consequently

V_evΩ=−O_evΩΩ=O_ev,

and EC4 reconstructs W. Conversely suppose supplied even operands U_ev′,V_ev′ satisfy W=U_ev′+V_ev′Ω. Apply Gr to obtain Gr W=U_ev′−V_ev′Ω. Addition and multiplication by k give U_ev′=U_ev. Subtraction and multiplication by k give V_ev′Ω=O_ev; right multiplication by the written undoing −Ω gives V_ev′=−O_evΩ. Thus EC1 is the unique pair under its stated sign conditions.

3. Retain all four ordered source products

V03-PB-0017 C1–C2 give centrality of Ω, Ω̃=Ω*=−Ω and its void derivatives. Therefore

Sel(V_evΩ)=−Sel(V_ev)Ω,       D(V_evΩ)=D(V_ev)Ω.

The first follows by expanding the two involutions; the second follows from each product derivative, since its ∂_μΩ term is void. Expand the full source in the original order:

(U_ev+V_evΩ)e₀[Sel(U_ev)−Sel(V_ev)Ω]
  =U_ev e₀Sel(U_ev)−U_ev e₀Sel(V_ev)Ω
    +V_evΩe₀Sel(U_ev)−V_evΩe₀Sel(V_ev)Ω
  =F(U_ev)+F(V_ev)+[V_ev e₀Sel(U_ev)−U_ev e₀Sel(V_ev)]Ω.

Only the central Ω moves. Its square changes the last displayed negative term into +F(V_ev). V01-PB-0006 (1) gives D’s additivity, hence L(W)=L(U_ev)+L(V_ev)Ω. Subtracting the entire source gives EC5 with the expressions EC2.

Because Gr commutes with reversion and star, Sel preserves the even sign of U_ev,V_ev. The single e₀ in each source term changes that sign to odd. Also D(U_ev),D(V_ev) are odd by V07-PB-0043(G3) and B is even. Hence both M_ev,N_ev are odd. Applying Gr to their full residual reconstruction gives

Gr(R)=−M_ev+N_evΩ.

Subtract it from R and multiply by k to recover M_ev. Add it to R, multiply by k and then by −Ω to recover N_ev. This proves EC6. Necessity of M_ev=N_ev=v from R=v follows by these recoveries; sufficiency follows by substituting both voids in EC5. No summand of an arbitrary void sum has been cancelled separately.

4. Resolve the complete equation on the even operands

If Gr(Y)=Y, multiplication by the exterior e₀ factors gives Y*=e₀Ye₀. Its reverse is also even by §1. The same holds for A_Y=k[Y−T(Y)] since Gr commutes with T and fixes k. Applying these facts to A_Y and using T(A_Y)=−A_Y from V01-PB-0007 gives

T(A_Y)=(Ã_Y)*=e₀Ã_Ye₀=−A_Y,
e₀Ã_Y=−A_Ye₀.

V01-PB-0007’s stated selector identity Sel(Y)=hÃ_Y then yields

F(Y)=−hY A_Y e₀.

Use this identity at Y=U_ev,V_ev and in the two crossed terms:

C_ev(U_ev,V_ev)=−h(V_ev A_(U_ev)−U_ev A_(V_ev))e₀,
M_ev=h[D(U_ev)B+(U_ev A_(U_ev)+V_ev A_(V_ev))e₀],
N_ev=h[D(V_ev)B+(V_ev A_(U_ev)−U_ev A_(V_ev))e₀].

The already guarded k removes h in either void condition. Additive inverse laws convert each remaining equality into exactly EC7. Every transformation is reversible and §3 proves the conjunction is equivalent to R(W)=v. If V_ev=v, V02-PB-0011 D1 gives ∂_μV_ev=v; V01-PB-0007’s T(v)=v gives A_v=k[v−T(v)]=v, and its M1 gives Sel(v)=v. Substituting these voids proves the claimed particular case without using it for an unrestricted W.

5. Recover the index-0 writing and its residual pair

V02-PB-0013 V3 gives the unique W=X+e₀Y with X,Y spatial; V3a identifies that scope with X*=X and Y*=Y. Its V0 gives α(α(Z))=Z by the literal product e₀e₀=u. Apply the defining Gr to the full reconstruction:

Gr(W)=e₀(X−e₀Y)e₀=α(X)−e₀α(Y).

Its sum and difference with W prove

U_ev=X⁺+e₀Y⁻,      O_ev=X⁻+e₀Y⁺.

The definitions give α(X⁻)=−X⁻ and α(Y⁺)=Y⁺ by the same two explicit sum-and-difference calculations. Because Ω=e₀Ω_s, use Ze₀=e₀α(Z) from V02-PB-0013(V0):

−(X⁻+e₀Y⁺)Ω
  =−X⁻e₀Ω_s−e₀Y⁺e₀Ω_s
  =e₀X⁻Ω_s−Y⁺Ω_s.

This is EC8, with its actual signs. For every spatial Z, star's supplied product rule gives

(α(Z))*=(e₀Ze₀)*=(−e₀)Z*(−e₀)=e₀Ze₀=α(Z),

because Z*=Z. Also Ω_s*=Ω_s by its six spatial generators, so the product of any spatial Z with Ω_s is star-fixed. These establish the spatial scope of each operand used in the following uniqueness arguments.

Conversely apply the same unique index-0 representation to U_ev and V_ev. Their even conditions imply α(U_ev0)=U_ev0, α(U_ev1)=−U_ev1 and the corresponding two conditions for V_ev0,V_ev1: apply Gr to U_ev0+e₀U_ev1, then use uniqueness of the index-0 pair on its now verified spatial operands. Hence

U_ev+V_evΩ
  =U_ev0+e₀U_ev1+V_ev0e₀Ω_s+e₀V_ev1e₀Ω_s
  =(U_ev0−V_ev1Ω_s)+e₀(U_ev1+V_ev0Ω_s).

The two parentheses are spatial: V_ev1Ω_s and V_ev0Ω_s are star-fixed by the displayed product argument, and the other terms already are the spatial operands of their index-0 writings. Uniqueness therefore identifies them as X,Y, proving EC9 in both directions.

For the residual, V02-PB-0013 V5 gives R(W)=E₀+e₀E₁ together with E₀*=E₀ and E₁*=E₁. These are the spatial premises of the preceding Gr calculation, which therefore applies to this same residual. The EC6 difference gives M_ev=(E₀)⁻+e₀(E₁)⁺. Its sum gives N_ev=−[(E₀)⁺+e₀(E₁)⁻]Ω; use α((E₀)⁺)=(E₀)⁺ and α((E₁)⁻)=−(E₁)⁻ to obtain EC10. This is an exact map between the already derived full residuals. M_ev=N_ev=v iff R=v iff E₀=E₁=v, by their separate written recoveries. No claim that an individual member of one pair equals an individual member of the other is made.

What is not forced

Neither U_ev nor V_ev is asserted to satisfy a residual equation on its own. Setting one to v is an extra condition; the full writing first reconstructs every supplied input. Even here is the stated Gr sign, distinct from star-fixed spatial content and from T-fixed selector silence. The names Gr and α keep their displayed definitions at each use.

The two complete pairs express one nonlinear equation. Their component equations retain all the displayed mixed products. The only scalar division is by the already guarded h. Every multiplication by −Ω or e₀ is a displayed written undoing product.

What it means

Every supplied written W admits two reversible writings: two spatial operands, and two even operands. Their source and derivative balances retain the same complete residual. The explicit bridge permits later work in either writing while carrying every component and every coupling, instead of treating similar-looking equations as interchangeable without a recovery map.

Actual factor derivatives have complete algebraic compatibility and an exact source-derivative recovery (V04-PB-0032)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0006, V01-PB-0007, V02-PB-0011, V02-PB-0014, V03-PB-0020, V04-PB-0026, V05-PB-0035, V07-PB-0042

Declares

Scalar guards retain 04; Sel,D,F,R retain V01-PB-0006’s definitions; T,A_X,H_X,K_A retain V01-PB-0007’s; Γ retains V04-PB-0026’s; Mix retains V02-PB-0014’s. The actual supplied K,V obey

Γ(K)=K, Γ(V)=V, KVK=K, VKV=V.                     (FD1)

Put p=KV,r=VK and use V05-PB-0035’s same complement definitions E_L=E_L(K,V),E_R=E_R(K,V). For an actual derivative μ put K_{,μ}=∂μK,V{,μ}=∂_μV. In the separate algebraic compatibility problem, K_d,V_d,S_d,Z_d are supplied written expressions; their names do not assert that they are derivatives of anything.

Where a supplied Δ has T(Δ)=Δ,ΔK=v, its Comp_μ and Rem_μ are the same expressions defined in V05-PB-0035 S8b at ι=T,σ=u,U=V.

What is forced

Every actual derivative of FD1 has Γ(K_{,μ})=K_{,μ},Γ(V_{,μ})=V_{,μ} and obeys both complete identities

K_{,μ}VK+KV_{,μ}K+KVK_{,μ}=K_{,μ},
V_{,μ}KV+VK_{,μ}V+VKV_{,μ}=V_{,μ}.                (FD4)

For any supplied Γ-fixed K_d, these three conditions are equivalent:

1. E_L K_d E_R=v.
2. A supplied Γ-fixed V_d obeys both FD4 with K_d,V_d in its two slots.
3. A supplied S_d obeys K_d=KS_d+Γ(S_d)K.

Their actual constructions and complete recovery are

K_d=pK_d+K_d r−pK_d r,
S_d=VK_d−kVK_d VK,
V_d=−VK_d V+E_R Z_d p+rΓ(Z_d)E_L.                    (FD5)

Under condition 1 every supplied Z_d constructs an allowed V_d; every allowed V_d is recovered by Z_d=V_d. Thus Z_d=v supplies V_d=−VK_dV. These are algebraic equivalences, not a derivative-realization assertion.

For any other supplied V′ with KV′K=K, put E′_L=E_L(K,V′), E′_R=E_R(K,V′). Without a Γ condition on V′,

E′_L K_d E′_R=E′_L(E_L K_d E_R)E′_R,
E_L K_d E_R=E_L(E′_L K_d E′_R)E_R.                (FD6)

Their voidness criteria are therefore equivalent for the same K,K_d. For actual derivatives use Z_{d,μ}=V_{,μ} in FD5. Then

∂_μE_L=−E_LK_{,μ}V−KΓ(Z_{d,μ})E_L,
∂_μE_R=−VK_{,μ}E_R−E_RZ_{d,μ}K,                  (FD7)
E_L(∂_μE_L)E_L=v, p(∂_μE_L)p=v,
E_R(∂_μE_R)E_R=v, r(∂_μE_R)r=v.                  (FD8)

For the same actual K,V, any supplied T(Δ)=Δ,ΔK=v satisfies every identity of V05-PB-0035 S4b and S8c–S8d with the S8b definitions at ι=T,σ=u,U=V. In particular (∂μΔ)K=−ΔK{,μ}; its complete derivative contains both Comp_μ and Rem_μ. When additionally K=K_A(X) for the same actual X, the exact source comparison V07-PB-0042 FC4 has the derivative recovery

∂_μ[F(X+Δ)−F(X)]
  =Mix(Δ,∂_μX)+Mix(X,∂_μΔ)
  =h[ΔK_{,μ}+(∂_μΔ)K]=v.                       (FD12)

The base residual R(X) is unrestricted in this derivative identity.

The forcing

1. Differentiate the actual witness identities, not a nearby substitute

V04-PB-0026(F1) gives the individual derivative's commutation with Γ. Applying it to Γ(K)=K and Γ(V)=V yields the two Γ signs of their actual derivatives. Differentiating KVK=K with the product rule of 04 gives the three contributions

(∂_μK)VK+K(∂_μV)K+KV(∂_μK)=∂_μK.

Differentiating VKV=V gives the other three. With the names in the declaration these are FD4 exactly. No finite addition to K or V has been substituted for these derivative identities, and no quadratic remainder has been discarded.

The retained witness identities, as proved in V05-PB-0035 S6b, give pp=p,rr=r, E_LK=KE_R=VE_L=E_RV=v and pE_L=E_Lp=rE_R=E_Rr=v. Multiplying the first FD4 equation on the left by E_L and on the right by E_R makes each left summand void: its three respective terminal factors are KE_R, E_LK and E_LK. Therefore E_LK_{,μ}E_R=v.

The full distribution for a supplied K_d is

E_L K_d E_R=K_d−pK_d−K_d r+pK_d r.

Its voidness is exactly FD5's first reconstruction by additive rearrangement. No factor is cancelled.

2. Construct precisely the compatible Γ-fixed factor changes

Retain Γ(K_d)=K_d and E_L K_d E_R=v. For the actual S_d=VK_d−kVK_d VK, V04-PB-0026 F1’s product reversal gives

Γ(S_d)=K_d V−kKVK_d V,
KS_d=pK_d−kpK_d r,
Γ(S_d)K=K_d r−kpK_d r.

The guarded inversion gives hk=u, and rule 8 gives kh=hk=u. Since k+k=k(u+u)=u, their sum is pK_d+K_d r−pK_d r=K_d. This constructs condition 3. Conversely any supplied S_d gives Γ(KS_d+Γ(S_d)K)=Γ(S_d)K+KS_d=KS_d+Γ(S_d)K. Moreover

E_L[KS_d+Γ(S_d)K]E_R
   =(E_LK)S_dE_R+E_LΓ(S_d)(KE_R)=v.

Thus conditions 1 and 3 are equivalent on the same full expressions.

3. Construct and recover every compatible witness slot

Continue with Γ(K_d)=K_d and E_L K_d E_R=v. For supplied Z_d put C_d=E_R Z_d p; then Γ(C_d)=rΓ(Z_d)E_L. Γ fixes −VK_d V and exchanges these two summands, so V_d=−VK_d V+C_d+Γ(C_d) is Γ-fixed. The ordered products in S6b give

KC_dK=v, KΓ(C_d)K=v,
C_dp=C_d, rC_d=v,
Γ(C_d)p=v, rΓ(C_d)=Γ(C_d).

Their individual reasons are KE_R=v,E_LK=v,pp=p,rE_R=v,E_Lp=v and rr=r in the displayed order. Hence KV_d K=−pK_d r and

K_d VK+KV_d K+KVK_d=K_d r−pK_d r+pK_d=K_d.

For the second compatibility equation, VKV=V gives (−VK_d V)p=−VK_d V and r(−VK_d V)=−VK_d V. Consequently

V_d KV+VK_d V+VKV_d
  =[−VK_d V+C_d]+VK_d V+[−VK_d V+Γ(C_d)]
  =−VK_d V+C_d+Γ(C_d)=V_d.

This proves sufficiency for every supplied Z_d, including Z_d=v.

Conversely let an actual Γ-fixed V_d satisfy both displayed compatibility equations. Multiply the second on the left by E_R and right by E_L. Its three left terms vanish respectively by pE_L=v,E_RV=v,E_Rr=v, giving E_R V_d E_L=v. Multiply that same equation on the left by r and right by p; rV=V=Vp gives

rV_d p+VK_d V+rV_d p=rV_d p,
rV_d p=−VK_d V.

Insert both full unity decompositions around that same V_d:

V_d=(r+E_R)V_d(p+E_L)
   =rV_d p+E_R V_d p+rV_d E_L+E_R V_d E_L
   =−VK_d V+E_R V_d p+rV_d E_L.

Since Γ(V_d)=V_d, this is FD5 with Z_d=V_d. It recovers every allowed slot, completing both directions of condition 2 as well as the parameter recovery. These statements do not supply derivatives of arbitrary inputs.

4. Prove the same obstruction with any available witness

For V′ satisfying KV′K=K, both V and V′ qualify for V05-PB-0035 S5a's ordered left and right complement products. With exactly E′_L=E_L(K,V′),E′_R=E_R(K,V′), they give

E′_LE_L=E′_L, E_RE′_R=E′_R,
E_LE′_L=E_L, E′_RE_R=E_R.

Associate these products around K_d to obtain both equations in FD6. If either inner product is void, its surrounding multiplication is void; the two equations therefore give both implications. No derivative of V′ and no Γ(V′)=V′ was used. The specific expressions in FD5 may depend on which witness is supplied; the stated voidness criterion does not.

5. Preserve every term of the complement derivatives

04’s product rule on their actual definitions first gives

∂_μE_L=−K_{,μ}V−KV_{,μ},
∂_μE_R=−V_{,μ}K−VK_{,μ}.

Section 1 gives Γ(K_{,μ})=K_{,μ}, Γ(V_{,μ})=V_{,μ} and both FD4 identities. These discharge FD5’s condition 2 for this actual pair, so its recovery applies with Z_{d,μ}=V_{,μ}. Multiplication by K on the left and on the right, using KE_R=E_LK=v, pK=K and Kr=K, gives separately

KV_{,μ}=−pK_{,μ}V+KΓ(Z_{d,μ})E_L,
V_{,μ}K=−VK_{,μ}r+E_RZ_{d,μ}K.

Substitute these two distinct ordered identities in the two direct product-rule equations. The terms −K_{,μ}V+pK_{,μ}V collect to −E_LK_{,μ}V; the terms VK_{,μ}r−VK_{,μ} collect to −VK_{,μ}E_R. This proves both FD7 identities. In particular the right multiplication route uses E_LK=v and pK=K, not an unrelated annihilation in another order.

Sandwich the first FD7 identity by E_L. Its first term contains VE_L=v and its second contains E_LK=v. Sandwich it instead by p. Its first contains pE_L=v and its second E_Lp=v. This proves the first two void identities of FD8. Apply Γ to these two whole identities, using Γ(E_L)=E_R, Γ(p)=r and derivative commutation; this gives exactly the other two. No constancy of either complement is assumed.

6. Recover the source derivative through the common interfaces

At the actual K,V, the hypothesis KVK=K qualifies for V05-PB-0035 S4b and S8c–S8d to the retained T(Δ)=Δ,ΔK=v. With ι=T,σ=u,U=V, S0b gives Part_(T,u)(Y)=H_Y; its complement is exactly E_L(K,V), and its K_{,μ} is the derivative used here. Thus its complete null derivative recovery is obtained by that direct citation, without requiring a separate proof or independently chosen derivative values. At the additional K=K_A(X), V07-PB-0042 FC4 gives F(X+Δ)=F(X) for the same T-fixed Δ right-null at K_A(X), before any base-equation assumption.

Differentiating the equality F(X+Δ)=F(X) and using V02-PB-0011 D1's ∂_μv=v already gives ∂_μ[F(X+Δ)−F(X)]=v. To identify its complete mixed-source form, V03-PB-0020 E2, derivative additivity and V02-PB-0014 MixOps give

∂_μ[F(X+Δ)−F(X)]
  =Mix(X+Δ,∂_μX+∂_μΔ)−Mix(X,∂_μX)
  =Mix(Δ,∂_μX)+Mix(X,∂_μΔ)+Mix(Δ,∂_μΔ).

V02-PB-0011 D3 applied to T(Δ)=Δ gives T(∂_μΔ)=∂_μΔ. V01-PB-0007’s selector criterion therefore makes both selectors void, so Mix(Δ,∂_μΔ)=v by its two-term definition. V02-PB-0011 D3–D4 also give ∂_μK=e₀(∂μA_X)̃=e₀Ã{∂μX}=K{,μ}. Using V01-PB-0007’s stated Sel(Y)=hÃ_Y at X and ∂μX, the remaining two mixed expressions are hΔK{,μ} and h(∂_μΔ)K. Their sum is void by the actual product derivative already proved. This establishes FD12 as an exact recovery of the general derivative identity at this stated comparison with Δ right-null at K_A(X).

What is not forced

FD5 classifies the algebraic equations obtained from the retained witness identities. It does not identify all compatible K_d,V_d with actual ∂K,∂V. The seven actual derivatives concern the same K,V, satisfy the commuting-derivative rules, and retain the undifferentiated FD1.

The derivative recovery preserves the changing product as (∂μΔ)K=−ΔK{,μ}. Its seven-term sum is the full criterion in V05-PB-0035 S8d. The finite source comparison and its quadratic term remain V07-PB-0042's identities; differentiation has not substituted a finite-change approximation for them.

What it means

For actual Γ(K)=K, Γ(V)=V and both FD1 products, the coupled algebraic derivative conditions are fully recoverable. The common product theorem also describes the full derivative of a supplied T-fixed right-null difference, T(Δ)=Δ and ΔK=v. Its separate right-product recoveries need only KVK=K; its reconstruction and seven-term identity have the weaker scopes stated in V05-PB-0035 S8c–S8d.

These interfaces distinguish compatible algebraic slots from actual derivatives. When K=K_A(X), FD12 recovers the derivative of the complete source comparison; the equation for an actual new solution still retains the base residual and the full finite-change residual condition.

Universal multiplication of the residual by its central input factor is exactly a scalar idempotent condition (V05-PB-0037)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V02-PB-0011, V03-PB-0017

Declares

The symbols Sel,D,L,F,R retain PB-0006's definitions; in particular R(W)=hD(W)B−F(W) and F(W)=We₀Sel(W). No equation is assumed for the supplied input W. A universal identity below means the same supplied multiplier is retained as W ranges over supplied written expressions, including the actual inputs u and e₀. It does not mean an identity assumed only on solutions.

The scalar expressions u,v,h,k and word B retain 04's meanings; G and 𝓒_I retain PB-0004's definitions; ε_μ retains PB-0002's generator-square signs; ρ retains PB-0003's finite sign accumulation. Retain V03-PB-0017's central expressions

Z=a+bΩ,       Zᶜ=a−bΩ,       n=aa+bb,

with the two recovered scalar coefficients a,b. Neither n=u nor derivative constancy is assumed. The inverse k retains only the existing guard on h=u+u.

A supplied scalar a is called idempotent here precisely when the actual identity aa=a is retained. This word supplies no list of such scalars and no new scalar-domain premise. Under that identity, the proof uses the written expression

t_a:=ha−u.                                            (C0)

This is an ordinary scalar expression, not an inverse notation or an extra nonvoid guard.

What is forced

For every supplied central Z, the following are equivalent:

R(ZW)=Z R(W) for every supplied W;                     (C1a)

b=v and aa=a,
equivalently Z=a for a supplied scalar with aa=a.       (C1b)

For each supplied scalar satisfying aa=a, the actual written products and all seven derivative identities are

t_a t_a=u,
∂_μa=v for every μ in G.                              (C2)

Thus no derivative condition is required in advance in C1b. It follows from the scalar identity using its displayed undoing expression.

For an arbitrary central Z=a+bΩ, independently of the idempotent hypothesis or the universal identity C1a, there is also the exact whole-derivative criterion

D(Z)=v
  iff ∂_μa=∂_μb=v for every μ in G
  iff ∂_μZ=v for every μ in G.                       (C2a)

Its proof recovers the seven singleton and seven complementary blade coefficients. This is a central-input result, not a criterion for arbitrary written W.

Under the additional source-preservation condition n=u, C1 reduces exactly to

R(ZW)=Z R(W) for every supplied W iff Z=u.             (C3)

The unrestricted criterion C1 also retains the scalar self-product cases without asserting that any case other than u or v is supplied.

The forcing

1. Retain the arbitrary-coefficient residual before restricting the multiplier

V03-PB-0017 C5–C6 supplies, for every full supplied W and arbitrary supplied scalar expressions a,b,

F(ZW)=nF(W),
R(ZW)=Z R(W)+(Z−nu)F(W)+h(D(Z))WB.                   (C4)

These are whole-expression identities. They do not require n=u or the equation for W. Every use of them below keeps the full two-term difference from the desired identity R(ZW)=ZR(W) until a separate argument removes it.

Suppose C1a holds. At the supplied input W=u, both involutions fix u, so Sel(u)=u−u=v and F(u)=ue₀v=v. All ∂_μu=v by V02-PB-0011, giving D(u)=v and R(u)=v. Substituting W=u into C4 and using the universal identity therefore gives

v=R(Zu)−Z R(u)=h(D(Z))B.

Left multiplication by the existing k and right multiplication by the actual −B yield

D(Z)=k[h(D(Z))B](−B)=v,                              (C5)

because hk=u by the declared inversion, kh=hk=u by 04 rule 8, and B(−B)=u by PB-0006 B0. No individual derivative has been inferred void from C5, and none is needed for the next step.

2. Use a second actual input to recover the whole central coefficients

The generator images give the complete local source value

Sel(e₀)=e₀−(−e₀)=he₀,
F(e₀)=e₀e₀he₀=he₀.                                  (C6)

Apply C1a and C4 at this supplied e₀, retaining C5. The complete difference is now

v=R(Ze₀)−Z R(e₀)=(Z−nu)he₀.

Right multiplication by e₀ and multiplication by k undo exactly the displayed factors:

Z−nu=k[(Z−nu)he₀]e₀=v,
Z=nu.                                                (C7)

Here scalar h moved by scalar commutation; the only nonscalar cancellation used e₀e₀=u. No premise that e₀ solves the equation was made.

Reconstruct the two sides of C7 on the complete blade list. The empty-index coefficient of Z=a+bΩ is a, and its G coefficient is b. The scalar n has empty-index coefficient n and G coefficient v. V01-PB-0004's recovered coefficients therefore give

b=v,        a=n.

Substitute b=v in the retained definition n=aa+bb, using void absorption. It gives n=aa and hence aa=a. This proves C1b as a necessary condition. It rules out no scalar cases by an unprovided order or permission to invert arbitrary nonvoid scalar expressions.

3. Derive all derivatives from the idempotent identity itself

Now retain a supplied scalar identity aa=a. The complete product of the written scalar t_a is

t_a t_a
  =(ha−u)(ha−u)
  =hh aa−ha−ha+u
  =hh a−ha−ha+u.                                    (C8)

The two equal occurrences satisfy ha+ha=(u+u)ha=hh a by the definition of h and distribution. Thus the three a-containing terms in the last line of C8 cancel in full, leaving

t_a t_a=u.                                          (C9)

This supplies an actual two-sided undoing because both factors are the same displayed scalar. No inversion operation or guard on t_a has been introduced.

For each actual μ, differentiate the retained identity aa=a. V02-PB-0011 and the supplied product rule give

(∂_μa)a+a(∂_μa)=∂_μa.

Scalar commutation groups the left side as ha(∂_μa). Move the right side to the left, retaining both original occurrences:

(ha−u)(∂_μa)=t_a(∂_μa)=v.                            (C10)

Multiply this actual equality by t_a and associate, using C9:

∂_μa=u(∂_μa)=(t_a t_a)(∂_μa)
       =t_a[t_a(∂_μa)]=t_av=v.                       (C11)

Write these steps at each of the seven supplied derivative labels. They prove every identity in C2. Differentiation here applies to the retained expression identity aa=a; agreement only at an unspecified point would not supply C10.

4. Prove sufficiency for the complete residual

Assume C1b. Then Z=a is scalar and n=aa=a. By C2, every derivative of a is v, so D(Z)=D(a)=v. Return to the full arbitrary-coefficient identity C4:

R(ZW)
  =aR(W)+(a−au)F(W)+h v W B
  =aR(W)
  =Z R(W)

for every supplied W. The middle terms are void by the right-unity and absorption laws, not by an equation for W. This proves C1a and completes both directions of C1.

Equivalently, at these same conditions the original two sides separately obey

L(aW)=aL(W),       F(aW)=aa F(W)=aF(W),

whose full difference gives the same residual identity. The statement is R(aW)=aR(W) for every supplied W; it is not an assertion that the factor has an inverse or preserves distinguishability of inputs.

5. Recover every central derivative coefficient before comparing conditions

For an arbitrary central Z=a+bΩ, with neither aa=a nor n=u imposed, the derivative-constant complete word Ω gives

∂_μZ=a_{,μ}+b_{,μ}Ω,
D(Z)=Σ_μ ε_μa_{,μ}e_μ+Σ_μ ε_μb_{,μ}e_μΩ,                 (C12)

where a_{,μ}:=∂μa and b{,μ}:=∂_μb are the actual supplied derivative expressions. No derivative coefficient has been specified independently.

For each actual μ, let G_{<μ} and G_{>μ} be the supplied sublists before and after μ. The complete word is Ω=e_(G_{<μ}) e_μ e_(G_{>μ}). By V01-PB-0003, moving the leading e_μ in e_μΩ across the actual prefix e_(G_{<μ}) contributes exactly the finite sign ρ(G_{<μ}), with value u at the empty prefix. Removing the adjacent pair gives

e_μΩ=ρ(G_{<μ})e_(G_{<μ})e_μe_μe_(G_{>μ})
      =ρ(G_{<μ})ε_μe_(G∖{μ}),
ε_μe_μΩ=ρ(G_{<μ})e_(G∖{μ}).                             (C13)

The second equality uses only ε_με_μ=u. The seven prefix sign products are obtained by starting at u and appending the next actual negative exchange sign. They are u,−u,u,−u,u,−u,u in the supplied label order. Thus the complete output routes and coefficients of C12 are exactly:

μ	Singleton output	Its coefficient	Complement output	Its coefficient
0	{0}	a_{,0}	{1,2,3,4,5,6}	b_{,0}
1	{1}	−a_{,1}	{0,2,3,4,5,6}	−b_{,1}
2	{2}	−a_{,2}	{0,1,3,4,5,6}	b_{,2}
3	{3}	−a_{,3}	{0,1,2,4,5,6}	−b_{,3}
4	{4}	−a_{,4}	{0,1,2,3,5,6}	b_{,4}
5	{5}	−a_{,5}	{0,1,2,3,4,6}	−b_{,5}
6	{6}	−a_{,6}	{0,1,2,3,4,5}	b_{,6}

Each singleton is distinct, each complement is distinct, and no listed singleton equals any listed complement. C12 is the full distributed presentation on precisely these fourteen positions; every other recovered coefficient is v because no occurrence routes there. V01-PB-0004 C2–C4 therefore gives D(Z)=v if and only if all fourteen displayed coefficients are v. Each displayed sign is its own undoing, so this is equivalent to a_{,μ}=b_{,μ}=v at every μ. Conversely those identities make all fourteen contributions and hence D(Z) void.

For the individual derivative at each actual μ, the first C12 equality has coefficient a_{,μ} at the empty index and b_{,μ} at G. Recovery at these two distinct indices shows ∂μZ=v if and only if a{,μ}=b_{,μ}=v. Applying that equality at all seven supplied labels proves the other equivalence in C2a.

The idempotent derivative argument in §3 is independent of this coefficient route. It proves a scalar consequence of aa=a through the explicit t_a undoing, even before that scalar is used as a central multiplier. Conversely C2a needs no idempotent premise. Neither calculation replaces a generic contracted derivative condition by individual vanishings.

6. Apply the source-preservation condition to the complete criterion

Under C1's necessary condition, b=v and n=aa=a. Adding the retained n=u gives a=u and therefore Z=u. Conversely Z=u has n=u and C1 applies because uu=u. This proves both directions of C3.

What is not forced

The scalar condition aa=a is left intact. Neither 04 nor this proof supplies a rule that a void scalar product has a void factor, from which one could replace it by a=u or a=v. The actual supplied scalars u and v satisfy the identity, but no assertion that they exhaust it is made, and no nontrivial idempotent is asserted to exist.

The quantifier in C1a covers every supplied W with the same Z. Both necessity tests use that premise. The result does not classify central maps required only to take actual solutions of R=v to solutions, or maps chosen separately for each input. In particular neither u nor e₀ is being adopted as a nonvoid solution of R=v by using it to test the universal identity.

C2 differentiates an actual scalar identity. It does not infer componentwise derivative vanishing from an arbitrary contracted derivative equation, and it does not infer a derivative identity from equality of values at an unprovided location. No arbitrary scalar factor is divided or cancelled.

C2a uses the already complete two-coefficient description of the center and all fourteen distinct derivative-output routes. This distinctness belongs to central Z; no corresponding separation is supplied here for a general written input.

The two-term central representation is the complete result of V03-PB-0017 C2. The proof retains the full source and derivative terms until their exact diagnostic consequences have been established.

What it means

The universal identity R(ZW)=ZR(W) imposes a specific condition on the same central Z for every supplied W, beyond the source-scaling identity F(ZW)=nF(W). The full universal condition first determines its derivative contraction and then recovers the multiplier as a scalar whose self-product equals itself. That scalar identity in turn makes all seven derivatives void through one explicit undoing product.

This completes the universal central question without assuming source preservation first. Its source-preserving part is the exact n=u specialization C3. The unsolved existence or classification of nontrivial scalar idempotents remains visible as a scalar-domain question, rather than being silently removed by adding a rule for void scalar products.

The full right-null class has two recovered corners and a complete native corner algebra (V05-PB-0038)

LAYER: 8 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0007, V01-PB-0008, V04-PB-0026, V04-PB-0029, V04-PB-0031

Declares

Retain 04's finite written operations, h=u+u, k=h⁻¹ and the guard h≠v. The coefficient maps and full blade list are those of V01-PB-0004/0008. Γ is V04-PB-0026 F1; T is V01-PB-0007. The same fixed-pair expressions of V04-PB-0029 and V04-PB-0031 are

I=I_{34}=e₁e₂e₃e₄,  P_+=P_{34,+}, P_−=P_{34,−},
W₄=e₁P_+,  M_t=u+tW₄, N_t=u−tW₄,
K_t=M_tP_+Γ(M_t),  Δ_t=T(N_t)P_−N_t.

I and P_± abbreviate only this actual pair. The scalar t is supplied without a further guard or derivative condition.

Initially Z is an arbitrary full written expression; no T condition or support is assumed. A corner H means precisely H=P_−HP_−. Before any T condition, let

𝒪=(empty,0,5,6,05,06,56,056),
R₀=empty, R₂=12, R₃=13, R₄=14,
F_out=(F_(out,0),F_(out,2),F_(out,3),F_(out,4)),
Enc(F_out):=[F_(out,0)+B F_(out,2)+e₁e₃ F_(out,3)+e₁e₄ F_(out,4)]P_−.       (N1)

Each supplied F_(out,r) is a full eight-position expression on 𝒪; products of these outside expressions keep their order. A second tuple G_out=(G_(out,0),G_(out,2),G_(out,3),G_(out,4)) uses the same encoder, not a new map.

A T-fixed corner H_c means T(H_c)=H_c and H_c=P_−H_cP_−. (CN0)

For a supplied scalar list c on the twelve positions 𝒥 below, define

E_J(t)=T(N_t)e_JP_−N_t,
Z_c=Σ_{J∈𝒥}c_J E_J(t),
Θ_t(Z)=T(M_t)ZM_t,  Û_L=T(M_t)U_LT(N_t).             (CN1)

U_L denotes an arbitrary supplied left multiplier. In the following table J′=J△{1,2,3,4}. Write J=O∪R_r with O∈𝒪. The table defines the scalar signs s_J,λ_J,ξ_J and the index M_J by their displayed entries. Each displayed +/− means the actual scalar sign u/−u.

J	J′	s_J	λ_J	M_J	ξ_J
empty	1234	+	+	1	+
5	12345	+	−	2345	+
6	12346	+	−	2346	+
05	012345	+	+	015	−
06	012346	+	+	016	−
056	0123456	+	−	023456	−
012	034	−	+	02	+
013	024	+	+	03	+
014	023	−	+	04	+
1256	3456	−	−	13456	+
1356	2456	+	−	12456	−
1456	2356	−	−	12356	+

What is forced

For every one of the twelve actual positions J in the table,

e_JI=s_Je_(J′),  e₁e_J=λ_Je_Je₁,
e_Je₁=ξ_Je_(M_J)       when λ_J=u,
e_Je₁I=ξ_Je_(M_J)      when λ_J=−u.                    (CN1s)

For the whole supplied Z, before any T condition,

ZK_t=v iff Z=Y−tYe₁ for a supplied Y with YP_−=Y,
Y=ZP_−,     ZM_t=Y.                                  (N3)

The full class YP_−=Y has exactly two free corners:

Y=H+e₁G_c,       H=P_−HP_−,  G_c=P_−G_cP_−,
H=P_−Y=P_−ZP_−,
G_c=−e₁P_+Y=−e₁P_+ZP_−.                               (CA1)

Conversely every supplied pair H,G_c of such corners gives exactly one right-null Z=(H+e₁G_c)(u−te₁), with those recoveries. The two corners are unique for the same t,Z.

Every supplied corner H is Enc(F_out) for exactly one full outside tuple. Its recovery and void criterion are

𝓒_J(F_(out,r))=h𝓒_(J∪R_r)(H)   for all J∈𝒪, r∈{0,2,3,4},
Enc(F_out)=v iff F_(out,0)=F_(out,2)=F_(out,3)=F_(out,4)=v.                           (N8,N16)

Thus the unrestricted right-null class has the complete coordinate expression

Z=[Enc(F_out)+e₁Enc(G_out)](u−te₁),
𝓒_J(F_(out,r))=h𝓒_(J∪R_r)(P_−ZP_−),
𝓒_J(G_(out,r))=−h𝓒_(J∪R_r)(e₁P_+ZP_−).                   (CA2)

Here H=Enc(F_out) and G_c=Enc(G_out). Both four-by-eight tuples are free supplied coefficients, and the displayed expression and recoveries are mutual inverses. They introduce no new scalar realization or derivative rule.

The corner has the following entire four-unit multiplication. Each entry means the displayed product followed by P_−; its unity is P_−.

Product with final P_−	u	B	e₁e₃	e₁e₄
u	u	B	e₁e₃	e₁e₄
B	B	−u	e₁e₄	−e₁e₃
e₁e₃	e₁e₃	−e₁e₄	−u	B
e₁e₄	e₁e₄	e₁e₃	−B	−u

The four units commute with each outside expression. The outside expressions retain their own ordered products. Therefore

Enc(F_out)Enc(G_out)=Enc(F_out⋆G_out),
(F_out⋆G_out)₀=F_(out,0)G_(out,0)−F_(out,2)G_(out,2)−F_(out,3)G_(out,3)−F_(out,4)G_(out,4),
(F_out⋆G_out)₂=F_(out,0)G_(out,2)+F_(out,2)G_(out,0)+F_(out,3)G_(out,4)−F_(out,4)G_(out,3),
(F_out⋆G_out)₃=F_(out,0)G_(out,3)+F_(out,3)G_(out,0)+F_(out,4)G_(out,2)−F_(out,2)G_(out,4),
(F_out⋆G_out)₄=F_(out,0)G_(out,4)+F_(out,4)G_(out,0)+F_(out,2)G_(out,3)−F_(out,3)G_(out,2).                     (N9,CA3)

The full T action is

T(Enc(F_out))=Enc((T(F_(out,0)),−T(F_(out,2)),−T(F_(out,3)),−T(F_(out,4)))).           (N10)

It follows that the full T-fixed corner has precisely

F_(out,0)=c_empty+c_5e₅+c_6e₆+c_05e₀e₅+c_06e₀e₆+c_056e₀e₅e₆,
F_(out,2)=c_012e₀+c_1256e₅e₆,
F_(out,3)=c_013e₀+c_1356e₅e₆,
F_(out,4)=c_014e₀+c_1456e₅e₆.                              (N2)

These are exactly CN1's twelve positions and coefficients, after the unrestricted corner has been recovered.

For the same full Z, the following are equivalent:

T(Z)=Z and ZK_t=v;
T(H_c)=H_c and H_c=P_−H_cP_−, with
   Z=T(N_t)H_cN_t=(u−te₁)H_c(u−te₁);
Z=Z_c for exactly one twelve-entry scalar list c.        (CN2)

Their constructive recoveries are

H_c=Θ_t(Z)=P_−ZP_−,
Y=(u−te₁)H_c,
c_J=h𝓒_J(H_c).                                         (CN3,N6)

In CA1 this T-fixed condition is exactly G_c=−tH and T(H)=H. Conversely any corner H, without a T condition, still gives a right-null expression Z=(u−te₁)H(u−te₁); here the same algebraic recovery H=P_−ZP_− holds.

Every U_L has (U_LΔ_t)K_t=v. Its output is T-fixed exactly when Δ_tT(U_L)=U_LΔ_t. All and only such multipliers are

U_L=T(N_t)[H_c+Y_LP_+]T(M_t),                            (CN4)

where H_c is a T-fixed corner and Y_L is any full supplied expression. Their output is T(N_t)H_cN_t. For an actual qualifying U_L take H_c=Û_LP_− and Y_L=Û_L. The full term Y_LP_+ has void output; every T-fixed right-null Z is obtained already with Y_L=v.

For each actual J∈𝒥, the complete E_J(t) has exactly the coefficients

𝓒_J(E_J)=k(u−λ_J tt),
𝓒_(J′)(E_J)=−s_Jk(u+λ_J tt),
𝓒_(M_J)(E_J)=−ξ_Jt,                                   (CN5)

and void coefficients elsewhere. The three index columns contain thirty-six distinct positions. Consequently the whole Z_c has

𝓒_J(Z_c)=k(u−λ_Jtt)c_J,
𝓒_(J′)(Z_c)=−s_Jk(u+λ_Jtt)c_J,
𝓒_(M_J)(Z_c)=−ξ_Jtc_J,
𝓒_K(Z_c)=v at every remaining blade index K,             (CN6)

with the exact direct recovery

c_J=𝓒_J(Z_c)−s_J𝓒_(J′)(Z_c).                           (CN7)

The two specializations are literal expression identities:

H=P_− gives Z=Δ_t and c_empty=u with the other c_J=v;
t=v gives Z=H_c=P_−H_cP_−.                              (CA4)

The latter is exactly BC9's negative-corner null description at the pair (3,4): its active pair sign is σ=−u. No scalar t, coefficient, or u±tt is undone.

The forcing

1. Recover the unrestricted product and its T-fixed corner directly

The full factor is K_t=M_tP_+Γ(M_t). V04-PB-0026 F1 and V04-PB-0029 E1 give the actual Γ(M_t),Γ(N_t) undoings. Thus

ZK_t=v iff ZM_tP_+=v.

V04-PB-0031 BC5 gives P_++P_−=u and P_−P_+=v. Thus YP_+=v iff Y=Y(P_++P_−)=YP_−; the converse is YP_+=YP_−P_+=v. Apply this to the whole Y=ZM_t. Undoing M_t gives

Z=YN_t=Y−tYe₁P_+=Y−tYe₁,

because YP_−=Y and V04-PB-0031(BC5a) gives P_−e₁=e₁P_+. Conversely that actual expression multiplied by M_t gives Y, since

(Y−tYe₁)(u+te₁P_+)=Y+tYe₁−tYe₁+ttYP_+=Y.

Its right product by K_t is therefore void. Multiplication by P_− gives ZP_−=Y; this proves N3 with both directions and unique recovery.

For the additional T condition put Θ=T(M_t)ZM_t. The actual undoings give Z=T(N_t)ΘN_t. Applying V01-PB-0007's product reversal to these two full products shows T(Θ)=T(M_t)T(Z)M_t, so T(Z)=Z iff T(Θ)=Θ. The same two undoings give ZM_tP_+=v iff ΘP_+=v. By the preceding unity calculation this is Θ=ΘP_−. When T(Θ)=Θ, reversing that identity gives Θ=P_−Θ; hence Θ=P_−ΘP_−. Conversely a T-fixed corner Θ has ΘP_+=v. This proves both directions of CN2's corner condition, with H_c=Θ and the same inverse Z=T(N_t)H_cN_t.

V04-PB-0029(E1a) and the T laws in V01-PB-0007 give T(N_t)=T(u−tW₄)=u−tT(W₄)=u−te₁P_−. On a corner H,

T(N_t)H=(u−te₁)H,
HN_t=H−tHe₁P_+=H−tHe₁.

Hence Z=(u−te₁)H(u−te₁). Multiplying this entire expression by P_− on both sides gives H: its three other terms contain P_−e₁P_−=v. Also ZP_−=(u−te₁)H. The actual T(M_t),M_t undoings give Θ_t(Z)=H. This establishes every algebraic equality in CN3/N6, for all corners where no T premise is used, and for the whole signed class where that premise is retained.

2. Recover both unrestricted free corners

For any YP_−=Y, insert the actual left unity P_−+P_+:

Y=P_−Y+P_+Y.

Set H=P_−Y and G_c=−e₁P_+Y. Then HP_−=H and P_−H=H. Further,

G_cP_−=G_c,
P_−G_c=−P_−e₁P_+Y=−e₁P_+Y=G_c,
e₁G_c=−e₁e₁P_+Y=P_+Y.

Thus both are corners and Y=H+e₁G_c. Conversely any two such corners have (H+e₁G_c)P_−=H+e₁G_c. Multiplying by P_− on the left recovers H, since P_−e₁G_c=e₁P_+G_c=v; multiplying by −e₁P_+ recovers G_c, since P_+H=v and −e₁P_+e₁G_c=G_c. These actual recoveries prove both existence and uniqueness in CA1. N3 gives the stated formulas directly from Z.

The full corner result N8 proved next applies separately to those actual H and G_c. Substituting its two unique encoder presentations into N3 gives CA2; applying the two CA1 recoveries and then N8 recovers all sixty-four coefficients. Under CN2, Y=H−te₁H, so the unique second corner is G_c=−tH. Conversely that relation and T(H)=H give CN2's exact expression.

3. Recover the full 32-coordinate corner before its T-fixed part

Begin with any actual H=P_−HP_−, without a T condition. Its full 128-blade presentation is V01-PB-0004(C1,C4)'s coefficient recovery. V04-PB-0031(BC5a) gives IP_−=P_−I=−P_−, hence IH=HI=−H and IHI=H. For a blade with m inside labels from {1,2,3,4}, the actual I exchanges give the signs +,−,+,−,+ at m=0,1,2,3,4. Applying coefficient recovery to IHI=H therefore makes each odd-inside coefficient void: its negative equals itself, so h times it is v. The guarded inverse in 04 gives hk=u, and scalar commutation gives kh=u; multiplication by k therefore recovers v.

Every remaining index has one of the eight even inside subsets, together with one of the eight outside subsets in 𝒪. The eight inside subsets are paired by multiplication with I as follows:

Representative inside index R_r	Complementary inside index R'_r	Sign s_r in e_(R_r)I=s_r e_(R'_r)
empty	1234	+u
12	34	−u
13	24	+u
14	23	−u

The three nonempty products are the explicit finite word identities

e₁e₂I=−e₃e₄,
e₁e₃I=e₂e₄,
e₁e₄I=−e₂e₃.

They follow by moving the repeated e₁ next to itself, applying its square, and then doing the same for the other repeated spatial label, by the generator exchanges of V01-PB-0003. Multiplying each equality once more by I and using II=u gives the reverse paired product with the same sign. For every O∈𝒪, each of R_r and R′_r has an even number of inside letters. Moving a possible outside e₀ to the front therefore has sign u, and the outside 5,6 letters already follow the inside letters. Thus

e_(O∪R_r)=e_(R_r)e_O, e_(R′_r)e_O=e_(O∪R′_r).

Each outside letter also commutes with I through four exchanges. Combining these actual products gives, at every one of the eight outside indices,

e_(O∪R_r)I=e_(R_r)e_OI=e_(R_r)I e_O
           =s_r e_(R′_r)e_O=s_r e_(O∪R′_r).

This proves CN1s's s_J relation for all twelve J, and the same pairing for every unrestricted corner coefficient.

If the two actual recovered coefficients in one pair are x and x', HI=−H gives s_rx'=−x, hence x'=−s_rx. Its entire paired contribution is therefore

x e_(J∪R_r)−s_rx e_(J∪R'_r)
   =(hx)e_(J∪R_r)P_−.

The word e_(J∪R_r) equals e_(R_r)e_J: an outside e₀ crosses the zero or two inside letters with sign u, and the possible 5,6 letters are already after the inside letters. Thus all eight paired occurrences at a fixed r collect into e_(R_r)F_(out,r)P_− with

F_(out,r)=Σ_(J∈𝒪)[h𝓒_(J∪R_r)(H)]e_J.

Appending the four r contributions gives exactly Enc(F_out). This is the whole corner: all 128 original coefficients were either removed by the proved odd-inside equality or included in one of these 32 pairs.

Conversely every outside letter commutes with each two-letter inside word through two exchanges. Each inside representative also commutes with I, hence with P_−. Therefore every expression Enc(F_out) is unchanged by P_− on both sides. In its full expansion the representative and complementary inside lists are disjoint, and each representative appears only once at its own outside index. Its recovered coefficient at J∪R_r is consequently k𝓒_J(F_(out,r)). Multiplication by h proves N8 and uniqueness. It also proves

Enc(F_out)=v iff F_(out,0)=F_(out,2)=F_(out,3)=F_(out,4)=v.                       (N16)

This argument used all four arbitrary eight-position outside inputs. No T sign or six/two restriction has entered it.

4. Derive the complete corner multiplication and its T-fixed recovery

All outside expressions commute with B,e₁e₃,e₁e₄ by the two actual exchanges just used; they need not commute with other outside expressions. For r=2,3,4, the inside square is

(e₁e_r)(e₁e_r)=−(e₁e₁)(e_re_r)=−u.

The three cyclic products before the corner reduction are

(e₁e₂)(e₁e₃)=e₂e₃,
(e₁e₃)(e₁e₄)=e₃e₄,
(e₁e₄)(e₁e₂)=−e₂e₄.

Use the paired identities in §3 and IP_−=−P_− to obtain

e₂e₃P_−=e₁e₄P_−,
e₃e₄P_−=BP_−,
−e₂e₄P_−=e₁e₃P_−.

Reversing either two distinct inside factors gives the opposite sign by the same generator products. This proves the entire finite table:

Product, with final P_− retained	u	B	e₁e₃	e₁e₄
u	u	B	e₁e₃	e₁e₄
B	B	−u	e₁e₄	−e₁e₃
e₁e₃	e₁e₃	−e₁e₄	−u	B
e₁e₄	e₁e₄	e₁e₃	−B	−u

The table entries mean equality only after the final P_−; no identity e₂e₃=e₁e₄ is asserted outside the corner. In the product Enc(F_out)Enc(G_out), the middle P_− commutes with every displayed outside and inside factor and is idempotent. Distribute all sixteen ordered component products, move only outside factors past inside factors, and apply that table. The scalar-inside occurrences are F_(out,0)G_(out,0) and the three negative diagonal products. The B occurrences are F_(out,0)G_(out,2),F_(out,2)G_(out,0),F_(out,3)G_(out,4),−F_(out,4)G_(out,3); the other two rows are their explicitly listed cyclic table entries. This gives precisely N9, retaining the order F_(out,r)G_(out,s) in every outside product.

V04-PB-0031(BC5) gives T(P_−)=P_−. The supplied star fixes each spatial generator, and one reversal exchange negates each of B,e₁e₃,e₁e₄. Applying V01-PB-0007's complete product reversal and the outside/inside commutations therefore gives N10. By injectivity N16, T(H)=H is exactly

T(F_(out,0))=F_(out,0), T(F_(out,2))=−F_(out,2), T(F_(out,3))=−F_(out,3), T(F_(out,4))=−F_(out,4).

On all eight outside blades, the native whole-blade signs give

Outside index	empty	0	5	6	05	06	56	056
T sign	+	−	+	+	+	+	−	+

The single e₀ has the supplied negative star image; the spatial singletons are fixed. Reversion negates each two-letter word, so a word containing e₀ has the two negatives and a spatial pair has only one. The three-letter word 056 has three reversal exchanges and one negative star occurrence, hence is fixed. The empty word is u. These are all eight positions. Coefficient recovery and the same guarded h calculation therefore give six positions for F_(out,0) and precisely 0,56 for each of F_(out,2),F_(out,3),F_(out,4). Comparing the representative indices J∪R_r with the twelve J displayed here gives exactly CN1 and its c_J, with no new coefficient choices. This proves the stated T-fixed specialization and its complete twelve-position list. No T condition entered the preceding 32-coordinate reconstruction.

Substitute that entire recovered presentation of H_c into T(N_t)H_cN_t and distribute the twelve actual summands. Each becomes c_J E_J(t), so the result is Z_c. Conversely the same undoings recover H_c from every Z_c, and N8 recovers each c_J=h𝓒_J(H_c). This proves the last equivalence and unique list in CN2.

For H_c=P_− the definition gives Z=Δ_t, and N8 gives c_empty=u with the other entries v. At t=v, N_t=u, so Z=H_c. To identify the latter with BC9's negative corner, use BC8 at σ=−u:

k Prod_{A^{pair}_{34,−}}(Z)Be₀=ZP_+.

The written undoings of k and Be₀ make the source product void exactly when ZP_+=v. Under the same T(Z)=Z, BC9 identifies that class as Z=P_−H_ZP_−. This is precisely the corner just recovered, proving CA4's identification without equating its two different factors.

5. Describe every allowed left multiplier and its complete lost part

For arbitrary U_L, associate (U_LΔ_t)K_t=U_L(Δ_tK_t)=v. V01-PB-0007's product reversal and V04-PB-0029(E1)'s T(Δ_t)=Δ_t give

T(U_LΔ_t)=Δ_tT(U_L).

Hence its exact T-fixedness condition is the one preceding CN4. Transform the actual output using both undoings:

Θ_t(U_LΔ_t)
  =T(M_t)U_LT(N_t)P_−N_tM_t
  =Û_LP_−.

By CN2–CN3 the output is T-fixed exactly when H_c=Û_LP_− is T-fixed and satisfies CN0. The latter sandwich then also follows directly by applying T to H_cP_+=v.

The full multiplier, before any restriction, has the exact split

Û_L=Û_L(P_−+P_+)=H_c+Û_LP_+.

Undo the transform to obtain CN4 with Y_L=Û_L. Conversely supply any H_c with CN0 and any Y_L. For the multiplier in CN4,

U_LΔ_t
  =T(N_t)(H_c+Y_LP_+)T(M_t)T(N_t)P_−N_t
  =T(N_t)[H_cP_−+Y_LP_+P_−]N_t
  =T(N_t)H_cN_t.

This is T-fixed and right-null by CN2–CN3. All multipliers and all their output classes have therefore been recovered without assuming the multiplier itself is T-fixed or commutes with Δ_t.

6. Expand every transported basis expression in full

V04-PB-0029(E1a) gives T(W₄)=e₁P_−, and V04-PB-0031(BC5a) gives P_−e₁=e₁P_+; W₄=e₁P_+ is its definition. Every e_J in the recovered corner list commutes with P_±. Therefore the complete four-term expansion of E_J is

E_J=(u−tT(W₄))e_JP_−(u−tW₄)
   =e_JP_−−tT(W₄)e_JP_−−te_JP_−W₄
               +(tt)T(W₄)e_JP_−W₄
   =e_JP_−−te₁e_JP_−−te_Je₁P_+
               +(tt)e₁e_Je₁P_+.

The actual exchange sign of e₁ with e_J is λ_J from the declaration. For an index excluding 1, moving e₁ through its actual letters gives one minus sign per letter; for an index including 1, the two orders have the same e₁ square, and only its other letters are crossed. Applying these actual finite products to the twelve displayed J gives exactly the listed λ_J. Thus

e₁e_J=λ_Je_Je₁,
e₁e_Je₁=−λ_Je_J,
E_J=e_JP_−−te_Je₁(P_++λ_JP_−)−λ_Jtt e_JP_+.

If λ_J=u the middle factor is u, and its product is e_Je₁. If λ_J=−u it is I, and the middle product is e_Je₁I. These are the two complete alternatives forced by the actual signs, with no parameter restriction. The ordered products in these alternatives are:

J	Actual middle product	Reduced product
empty	e₁	e₁
5	e₅e₁I	e₂e₃e₄e₅
6	e₆e₁I	e₂e₃e₄e₆
05	e₀e₅e₁	−e₀e₁e₅
06	e₀e₆e₁	−e₀e₁e₆
056	e₀e₅e₆e₁I	−e₀e₂e₃e₄e₅e₆
012	e₀e₁e₂e₁	e₀e₂
013	e₀e₁e₃e₁	e₀e₃
014	e₀e₁e₄e₁	e₀e₄
1256	e₁e₂e₅e₆e₁I	e₁e₃e₄e₅e₆
1356	e₁e₃e₅e₆e₁I	−e₁e₂e₄e₅e₆
1456	e₁e₄e₅e₆e₁I	e₁e₂e₃e₅e₆

The first needed word is obtained directly from its supplied square:

e₁I=e₁e₁e₂e₃e₄=−e₂e₃e₄.

Use V01-PB-0003's contextual generator exchanges to move the outside 5,6 letters past the inside letters, and remove each repeated spatial square. The middle three rows with 012,013,014 use e₁e_je₁=−e₁e₁e_j=e_j for j=2,3,4. The final three rows then use the inside-pair products −e₃e₄,+e₂e₄,−e₂e₃ proved in §3 after the initial e₁ square is removed. These written products give all twelve ξ_J and complete CN1s's three relations.

The first and last terms of E_J give

e_J(P_−−λ_JttP_+)
  =k(u−λ_Jtt)e_J−s_Jk(u+λ_Jtt)e_{J′}.

Adding the complete middle term −ξ_Jt e_{M_J} proves CN5. All three index columns were written in the declaration; comparison of those actual lists shows no repetition within or between them. Hence no term from another E_J enters a row's three positions. Distributing the actual twelve-term scalar sum proves CN6 and proves voidness at every remaining index by full reconstruction, rather than assuming it.

Finally

𝓒_J(Z_c)−s_J𝓒_(J′)(Z_c)
  =k[(u−λ_Jtt)+(u+λ_Jtt)]c_J
  =khc_J=c_J.

This establishes CN7 without any scalar factor of t being cancelled.

What is not forced

CA1–CA2 describe the whole right-null class at the supplied K_t before a T sign. N8 describes a single full corner, and N2 is its T-fixed twelve-coordinate part. These distinct domains retain their own recovery maps. The derivative equation is treated separately; no arbitrary derivative data or nonconstant scalar realization is supplied by this algebraic classification.

CN4 classifies multipliers through their output, without a sign condition on the multiplier itself. The right-null class does not supply a base solution whose actual source factor is K_t. No coefficient or scalar parameter is cancelled unless the written k or a proved sign provides the relevant undoing.

What it means

One native corner algebra organizes the entire right-null input at K_t. Two independently supplied and uniquely recovered corners describe the unrestricted class. T(Z)=Z holds exactly when T(H)=H and G_c=−tH, with the same scalar t. The encoder's paired-blade recovery gives all thirty-two corner coordinates; its T action then gives the twelve-coordinate refinement. The four-unit table separately supplies their complete ordered multiplication. These interfaces use one reduction of the full blade list.

After coefficient recovery, the single-word reductions in §6 also give their corresponding instances of V01-PB-0008(P1). The local proof uses the words themselves and their generator exchanges; P1 identifies the coefficient of a product, not the whole product.

Two recovered tuples identify exactly two residual coefficients (V04-PB-0024)

LAYER: 9 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0005, V01-PB-0006, V01-PB-0009, V02-PB-0011, V03-PB-0017, V03-PB-0018

Declares

The supplied full input W has recovered coefficients x_I(W)=𝓒_I(W). No support, derivative or equation condition is imposed unless stated. The scalars u,v,h,k and word B retain 04's meanings; 𝓑,G,𝓒_I retain PB-0004's definitions; ε_μ retains PB-0002's generator-square signs; D,L,F,R retain PB-0006's definitions; A_W retains PB-0007's definition; I_s,Ω_s retain PB-0009's definitions; Ω retains V03-PB-0017's complete word; g retains V03-PB-0018's pairing definition.

Put P_perp=e₃e₄e₅e₆, the existing word on I_perp={3,4,5,6}. Define the two seven-entry scalar tuples

j_μ(W):=𝓒_{ {μ} }(W P_perp),
j^c_μ(W):=𝓒_{ {μ} }(W e₀B).                         (J0)

The superscript c labels the companion tuple. For any supplied seven-entry scalar tuple 𝒕=(t_μ), define the native finite derivative sum

Bal(𝒕):=Σ_(μ∈G)∂_μt_μ.

The sum starts at v and follows the original seven-label order. The distinct scalar bal₀(W) retains the seven-term expression of V03-PB-0018; it is not the name of this tuple operation.

What is forced

For every supplied W, the complete tuple presentations are

j(W)=(x_{03456}(W), x_{13456}(W), x_{23456}(W),
      −x_{456}(W), x_{356}(W), −x_{346}(W), x_{345}(W)),

j^c(W)=(−x_{12}(W), −x_{02}(W), x_{01}(W),
        x_{0123}(W), x_{0124}(W), x_{0125}(W), x_{0126}(W)).                (J2)

Concatenated subscripts name increasing blade-index lists. For every supplied W, with no full-equation premise,

Bal(j(W))=𝓒_(I_s)(D(W)B)=k𝓒_(I_s)(R(W)),
Bal(j^c(W))=bal₀(W)=𝓒_{ {0} }(D(W)B)
             =g(A_W,A_W)+k𝓒_{ {0} }(R(W)),
j^c(W)=j(WΩ),       j^c(WΩ)=−j(W).                    (J3)

Thus Bal(j(W))=v is equivalent to 𝓒_(I_s)(R(W))=v, and Bal(j^c(W))=g(A_W,A_W) is equivalent to 𝓒_{ {0} }(R(W))=v. The complete equation R(W)=v supplies both coefficient conditions.

The fixed products used by these tuples are

P_perp P_perp=u,                                     (J9)
Ω_s=B P_perp=P_perp B,
Ω=e₀B P_perp,
B(−Ω_s)=P_perp,
ΩP_perp=e₀B.                                         (J10)

For every full supplied Y and each μ in G, the coefficient extractions are

𝓒_∅(Y(−Ω_s))=𝓒_(I_s)(Y),                            (J11)
𝓒_∅(e_μY)=ε_μ𝓒_{ {μ} }(Y),                         (J12)
𝓒_∅(Ye₀)=𝓒_{ {0} }(Y).                             (J13)

Consequently the tuples are exactly the two complete coefficient contractions

ε_μ𝓒_(I_s)(e_μWB)=j_μ(W),                            (J14)
ε_μ𝓒_{ {0} }(e_μWB)=j^c_μ(W).                       (J14c)

No derivative constancy, source condition or equation on W is required by J9–J14c.

The forcing

1. Fixed words and whole-presentation coefficient extraction

V01-PB-0006 gives BB=−u. Reversion of P_perp=e₃e₄e₅e₆ reverses a four-position list, with successive crossing signs −u,u,−u,u; their product is u. Thus P_perp̃=P_perp. Removing the innermost equal generator pairs in P_perp P_perp̃ gives four spatial squares. Their product is ((−u)(−u))((−u)(−u))=u, hence

P_perp P_perp=u.                                     (J9)

By V01-PB-0003, each of e₀,e₁,e₂ commutes with P_perp by its four distinct contextual exchanges. Moving e₁ and e₂ across that word gives

Ω_s=B P_perp=P_perp B,
Ω=e₀B P_perp,
B(−Ω_s)=−BB P_perp=P_perp,
Ω P_perp=e₀B.                                        (J10)

V03-PB-0017 gives ΩΩ=−u and centrality. Since Ω_s=e₀Ω, e₀e₀=u and Ω commutes with e₀,

Ω_sΩ_s=e₀Ωe₀Ω=ΩΩ=−u.

Therefore Ω_s(−Ω_s)=(−Ω_s)Ω_s=u. These are actual products, without introducing a nonscalar inverse operation.

The following coefficient extractions concern full inputs, not only a blade. For a supplied Y reconstruct Y=Σ_I y_I e_I with V01-PB-0004. At an actual prefix U and its next summand y_Ie_I, the distributive law gives

(U+y_Ie_I)(−Ω_s)=U(−Ω_s)+y_I[e_I(−Ω_s)].

The initial product is v(−Ω_s)=v. This step at every prescribed I yields the entire right-product presentation. V01-PB-0005 routes a summand to the empty index exactly when I△I_s=∅, which means I=I_s. The actual product at that index is u, as just proved. Coefficient collection and recovery consequently give

𝓒_∅(Y(−Ω_s))=𝓒_(I_s)(Y).                            (J11)

For left multiplication, the analogous actual prefix step is

e_μ(U+y_Ie_I)=e_μU+y_I(e_μe_I).

A scalar output requires I={μ} and its sign is e_μe_μ=ε_μ. The full recovered equality is

𝓒_∅(e_μY)=ε_μ𝓒_{ {μ} }(Y).                         (J12)

The same right-product presentation, this time for Y e₀, has an empty output only from I={0}, whose square is u. Thus

𝓒_∅(Y e₀)=𝓒_{ {0} }(Y).                            (J13)

Every nonqualifying index in these computations contributes to a different blade or carries a void coefficient; none has been omitted by a support assumption. The empty blade and empty accumulation are included in the reconstructed list.

Combining J10–J12, with ε_με_μ=u, yields for each actual μ

ε_μ𝓒_(I_s)(e_μWB)
  =ε_μ𝓒_∅(e_μWB(−Ω_s))
  =ε_μ𝓒_∅(e_μW P_perp)
  =ε_με_μ𝓒_{ {μ} }(W P_perp)=j_μ(W).                (J14)

This is the whole-element recovery map from the canonical coefficient-contraction tuple to the compact tuple J0.

2. Determine all fourteen tuple signs

The complete right-product distribution used above also gives W P_perp=Σ_I x_I(W)(e_I P_perp). Its output {μ} has exactly one possible input index, I={μ}△I_perp. For μ=0,1,2 the input word is e_μP_perp, so its product with P_perp is e_μ by J9.

For the four remaining signs, a word on three different spatial labels r<s<t has square u: passing the second e_r left through e_t and e_s gives sign u, its square contributes −u, and the remaining e_s e_t e_s e_t contributes −u by one exchange and two spatial squares. Their product is u. Apply this written calculation to precisely the four triples 456,356,346,345. Their placements in P_perp are

P_perp=e₃e456,
P_perp=e356e₄,
P_perp=−e346e₅,
P_perp=e345e₆.

The second equality moves e₄ left through e₆,e₅; the third moves e₅ left through e₆; the fourth is already ordered. The first gives e456 e₃=−e₃e456 through three exchanges. Hence

e456 P_perp=−e₃(e456e456)=−e₃,
e356 P_perp=(e356e356)e₄=e₄,
e346 P_perp=−(e346e346)e₅=−e₅,
e345 P_perp=(e345e345)e₆=e₆.                           (J15)

Together these are all seven input/sign entries of j in J2.

For the companion, distribute the complete W e₀B. Its output {μ} requires I={μ}△{0,1,2}. Its first three products are

e12 e₀B=e₀BB=−e₀,
e02 e₀B=−e₂B=−e₁,
e01 e₀B=−e₁B=e₂.                                    (J16)

For the middle identity e₂B=e₂e₁e₂=−e₁e₂e₂=e₁; for the last e₁B=e₁e₁e₂=−e₂. The first uses the two exchanges commuting e₀ with B. Also (e₀B)(e₀B)=BB=−u. Each actual i in 3,4,5,6 crosses the three distinct generators in e₀B with negative sign, giving

e012i e₀B=(e₀B)e_i(e₀B)
          =−(e₀B)(e₀B)e_i=e_i.                      (J17)

These are the remaining four entries. By coefficient recovery in the entire distributed product, J16–J17 prove the companion formula in J2. No coefficient of W is chosen or removed.

3. Cite the two whole coefficient identities at their exact scope

Differentiate the seven entries of J2 and accumulate them in the declared G order. At each negative entry, V02-PB-0011(D1) gives ∂_μ(−x_I(W))=−∂_μx_I(W). Thus the first complete sum is exactly the bracket in V01-PB-0009(G0). Apply that parent's claimed G0r identity, with G0's explicit derivative coefficient, to this same full W:

𝓒_(I_s)(L(W))=h Bal(j(W)),
𝓒_(I_s)(R(W))=h Bal(j(W)).

Coefficient scalar compatibility and L(W)=hD(W)B also give 𝓒_(I_s)(L(W))=h𝓒_(I_s)(D(W)B). The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. Multiplication by this k proves J3's first line. This uses the already proved seven-route coefficient identity, with no equation imposed on W.

The second differentiated tuple in J2 is exactly bal₀(W) by its definition, again using V02-PB-0011(D1) at its two negative entries. V03-PB-0018 G8 gives

𝓒_{ {0} }(L(W))=h bal₀(W),
𝓒_{ {0} }(R(W))=h[bal₀(W)−g(A_W,A_W)].

Again the coefficient of L is h times that of D(W)B. Multiplying the two displayed identities by k proves the companion line of J3. These are citations of the complete-input coefficient results, not second derivations of their source terms.

From J10, right multiplication by Ω has the exact tuple action

j_μ(WΩ)=𝓒_{ {μ} }(WΩP_perp)
        =𝓒_{ {μ} }(We₀B)=j^c_μ(W).

Apply the same identity at WΩ. V03-PB-0017 C1 supplies ΩΩ=−u, and coefficient negation gives j^c(WΩ)=−j(W). This proves the remaining J3 identities.

For the last exported recovery, B commutes with e₀ by the two contextual exchanges of its factors. Combining J12 and J13 gives

ε_μ𝓒_{ {0} }(e_μWB)
  =ε_μ𝓒_∅(e_μWB e₀)
  =ε_μ𝓒_∅(e_μW e₀B)
  =ε_με_μ𝓒_{ {μ} }(We₀B)=j^c_μ(W).

This is J14c, with every output coefficient recovered on the full supplied W. J14 was obtained in §1 from the corresponding Ω_s extraction.

What is not forced

Each balance in J3 characterizes its indicated residual coefficient. Their claims contain no equivalence to the complete equation. The expression g(A_W,A_W) remains a collected scalar sum; its value supplies no individual coefficient condition without a separate proved premise.

The tuples are defined by fixed-word multiplication and coefficient recovery. Their seven entries remain functions of the same full supplied input. The index I_s is an identically void source output; it is not an excluded input coefficient.

What it means

Two existing scalar coefficient equations have exact seven-entry tuple presentations. One source coefficient is identically void; the e₀ source coefficient is h g(A_W,A_W). Multiplication by Ω connects their tuples with a fixed sign. The exported word and coefficient identities allow general contraction results to use these two tuples directly, while each balance keeps its one-coefficient scope.

Prescribed source completion has an unrestricted coefficient criterion and an actual-witness application (V04-PB-0027)

LAYER: 9 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V01-PB-0008, V03-PB-0017, V04-PB-0025, V04-PB-0026, V04-PB-0031, V05-PB-0035

Declares

The scalar guards retain 04. F and R retain V01-PB-0006’s definitions; T, A_X, H_X and K_A retain V01-PB-0007’s definitions. Γ retains V04-PB-0026’s definition. For supplied A put K:=e₀Ã. Coefficient recovery and the blade list 𝓑 retain V01-PB-0004; χ and △ retain V01-PB-0005’s definitions carried by V01-PB-0008. A proposed whole-source output is hZ_out; put P_out=Z_out−AK.

Where an actual U with KUK=K is supplied, use V05-PB-0035 at ι=T,σ=u: E_L=E_L(K,U), H₀=C_part, and N_Y=N_Y^(T,u). These are that record's same expressions, with no further definitions. X₀=A+H₀ and X_Y=X₀+N_Y. The fixed-sign coefficient list 𝓑_H retains V01-PB-0008. Z=Z(a,b), Zᶜ and n=aa+bb retain V03-PB-0017 and are distinct from Z_out. For the scalar-factor specialization c_fac is a supplied scalar. For the central transport define K^Z:=ZᶜK, U^Z:=UZ, V^Z:=VZ and A^Z:=ZA. Given σ_tr and P_tr, C_part^Z and (N^(T,σ_tr))^Z mean V05-PB-0035’s same S0 expressions at (K^Z,U^Z,P_tr), with ι=T and sign σ_tr.

For the ordered-complement diagnostic only, put

J_diag=e₁e₂e₃, K₃=u+J_diag, U₃=kkK₃,
E₃=k(u−J_diag), E₃^T=k(u+J_diag),
A₃=e₀+e₀J_diag, Δ₃=e₄E₃.

E₃^T names the displayed expression; its equality to T(E₃) is proved.

What is forced

For supplied T(A)=−A, its constructed active part satisfies A_A=A, so K=e₀Ã=K_A(A). For an actual source completion X=A+H, T(H)=H, XK=Z_out, the necessary source consistency is

T(K)Z_out−T(Z_out)K=hT(K)AK.                      (C1)

At the asserted Ψ equation, take A=A_Ψ, K=K_A(Ψ), H=H_Ψ and Z_out=D(Ψ)B. These are the same actual operands, and C1 reads T(K)D(Ψ)B−T(D(Ψ)B)K=hT(K)A_ΨK. No commutation of T with D is used.

For supplied scalars d_I on the full 𝓑_H, put H(d)=Σ_(I∈𝓑_H)d_Ie_I and κ_J=𝓒_J(K). Then, with actual T(A)=−A and K=e₀Ã,

F(A+H(d))=hZ_out
  iff Σ_(I∈𝓑_H,J∈𝓑,I△J=J_out)d_Iκ_Jχ(I,J)
                 =𝓒_(J_out)(P_out) for every J_out∈𝓑.       (C2a)

Every actual completion with active part A is recovered from its own H_X coefficients. Under the retained source equality in C2a, the same expression satisfies R(A+H(d))=v exactly when

D(A+H(d))B=Z_out                                  (C2b)

holds. No partial-undo witness is needed for C1–C2.

For supplied T(A)=−A,K=e₀Ã and actual KUK=K, V05-PB-0035 S3a is exactly the two conditions

Z_out U K=Z_out,
T(K)Z_out−T(Z_out)K=hT(K)AK.                      (C5a)

Under precisely these conditions every source completion is X_Y=A+H₀+N_Y, and its full residual is

R(X_Y)=h[D(X₀)B+D(N_Y)B−Z_out],
R(X_Y)=v iff D(N_Y)B=Z_out−D(X₀)B.               (C5b)

At the additional R(X₀)=v this last condition is D(N_Y)=v. For K=c_fac u and T(Δ)=Δ, without a witness condition,

ΔK=v iff c_fac 𝓒_I(Δ)=v for every I∈𝓑_H.          (C5c)

The general product transport has the following separate scope. Take arbitrary actual K,U,P_tr, retain KUK=K and n(a,b)=u, and let σ_tr be either u or −u. Use V05-PB-0035's C_part and N_Y^(T,σ_tr) at that actual (K,U,P_tr). Then

K^Z=ZᶜK, U^Z=UZ,
K^ZU^Z=KU, U^ZK^Z=UK, K^ZU^ZK^Z=K^Z,
E_L(K^Z,U^Z)=E_L(K,U), E_R(K^Z,U^Z)=E_R(K,U),
C_part^Z=C_part Z,
Part_(T,σ_tr)(YZ)=Part_(T,σ_tr)(Y)Z,
(N^(T,σ_tr))^Z_(YZ)=N_Y^(T,σ_tr)Z.               (C8a)

The same P_tr satisfies both S3a tests at (K,U) iff it satisfies them at (K^Z,U^Z). X↦XZ carries every signed completion in both directions, with reverse X↦XZᶜ. Both signs belong to this product statement; neither is imposed on a source active part.

For every central Z, without an n=u condition,

Γ(Z)=Z, Γ(Ω)=Ω.                                 (C8b)

For actual Γ(K)=K, Γ(V)=V, KVK=K, VKV=V and n=u, the transported witness obeys

Γ(K^Z)=K^Z, Γ(V^Z)=V^Z,
K^ZV^ZK^Z=K^Z, V^ZK^ZV^Z=V^Z.                   (C8c)

Return now to the prescribed-source scope T(A)=−A,K=e₀Ã, and retain n(a,b)=u. The full source and residual identities are

A^Z=ZA, K^Z=ZᶜK,
A_(ZX)=ZA, F(ZX)=hZ_out whenever A_X=A,F(X)=hZ_out,
R(ZX)=ZR(X)+(Z−u)F(X)+h(DZ)XB.                   (C8)

These source identities need no KUK=K. Their source correspondence is reversed by Zᶜ, and the residual's first term is removed only at R(X)=v. When an actual U with KUK=K is also supplied, C8a at σ_tr=u and P_tr=P_out applies to precisely the T-fixed completion part H.

For each of V04-PB-0031's six listed pairs and both signs, K=K_{ij,σ_pair}, U=kBe₀ and E_L(K,U)=P_{ij,σ_pair}. V04-PB-0031 BC10 gives the actual KUK=K for these same operands. Hence V05-PB-0035 S3–S4 at ι=T and positive sign gives the complete null family P_{ij,σ_pair}H_YP_{ij,σ_pair}, and the complete prescribed output is

H=P_out U+T(P_out U)P_{ij,σ_pair}+P_{ij,σ_pair}H_YP_{ij,σ_pair},   (C9)

with both S3a tests retained. At A=B use K=−e₀B,U=Be₀; at A=e₀ use K=U=u. Their actual two-sided products are u, so the same theorem gives unique T-fixed completions when its output tests hold.

For the declared diagnostic,

T(A₃)=−A₃, K_A(A₃)=K₃, Γ(K₃)=K₃,
K₃U₃K₃=K₃, E_L(K₃,U₃)=E₃, T(E₃)=E₃^T,
E₃^TE₃=E₃E₃^T=v,
T(Δ₃)=Δ₃, Δ₃K₃=v, Δ₃≠v,
T(E₃)Δ₃E₃=Δ₃, E₃Δ₃E₃=v.                        (C10)

The forcing

1. Recover the unrestricted source condition before any witness method

The guarded inversion gives hk=u, and scalar commutation gives kh=hk=u. V01-PB-0007 gives the unique T-active/T-fixed split of every X and the source identity F(X)=hXK_A(X). If T(A)=−A and T(H)=H, then for X=A+H,

T(X)=−A+H=X−hA,
A_X=k[X−T(X)]=k(hA)=A,
H_X=k[X+T(X)]=k(hH)=H,
F(X)=hXK, K=e₀Ã.

At H=v the same computation gives A_A=A and K_A(A)=e₀Ã. At Z_out=XK, V01-PB-0007’s product reversal gives

T(Z_out)K=T(K)T(X)K
         =T(K)(X−hA)K
         =T(K)Z_out−hT(K)AK.

Additive rearrangement is exactly C1. For an actual solution, R(X)=v and the guarded k give D(X)B=XK; substitution supplies the same necessary identity at Z_out=D(X)B. T is applied to that whole derivative product. No commutation of T with D is used.

2. A complete coefficient criterion without a witness

V01-PB-0008 P3 says that the supplied H(d) presentation on 𝓑_H is selector-silent, and V01-PB-0007 gives T(H(d))=H(d). The sign-characterized split above therefore makes A_(A+H(d))=A. Its exact source is

F(A+H(d))=h(A+H(d))K=hAK+hH(d)K.

Thus F(A+H(d))=hZ_out is equivalent, using k, to H(d)K=P_out. V01-PB-0004 C1 reconstructs K=Σ_Jκ_Je_J and recovers the coefficients of H(d): d_I on 𝓑_H and v elsewhere. At each output index J_out, V01-PB-0008 P1 therefore gives precisely

𝓒_(J_out)(H(d)K)
  =Σ_(I∈𝓑_H,J∈𝓑,I△J=J_out)d_Iκ_Jχ(I,J).

V01-PB-0004 C4 then says that comparing all recovered coefficients is equivalent to the complete product equality, including empty occurrence lists. Conversely each supplied X with A_X=A has H_X T-fixed and, by V01-PB-0008 P4, exactly H_X=Σ_(I∈𝓑_H)𝓒_I(H_X)e_I. Taking those actual coefficients returns that X and the same product equality. This proves the positive completeness claim without independently assigning roots.

Once the source equality holds, the residual of that same X is h[D(A+H(d))B−Z_out]. Multiplication by k gives both directions of C2b. The source and derivative equations concern the same actual coefficients and their derivatives, so neither can be solved with an independently chosen list and then substituted for the other.

3. Convert the complete product result to the whole source

At the actual A and K_A, source completion is HK=P_out=Z_out−AK by §1. V05-PB-0035 S3a–S3b at ι=T,σ=u supplies the two exact product tests and all completions, under actual KUK=K. For its first test,

P_out U K=Z_out U K−AKUK=Z_out U K−AK.

Thus it equals P_out exactly when Z_out U K=Z_out. For the second, T(A)=−A gives T(P_out)=T(Z_out)+T(K)A, hence

T(K)P_out−T(P_out)K
  =T(K)Z_out−T(K)AK−T(Z_out)K−T(K)AK
  =T(K)Z_out−T(Z_out)K−hT(K)AK.

This is C5a, with both directions preserved by additive cancellation. Applying that S3b now gives exactly X_Y=A+H₀+N_Y. For an actual completion, Δ=H_X−H₀ is T-fixed and ΔK=P_out−P_out=v. V05-PB-0035 S2’s uniqueness at Δ=Δ+v gives Part_(T,u)(Δ)=Δ, and S4b gives Δ=T(E_L)ΔE_L. Consequently the actual choice Y=H_X−H₀ gives N_Y=T(E_L)Part_(T,u)(Δ)E_L=Δ and returns X without replacing its active part.

For all these expressions F(X_Y)=hZ_out. V01-PB-0006 (1), applied to X₀+N_Y, gives D(X₀+N_Y)=D(X₀)+D(N_Y), hence

R(X_Y)=hD(X₀+N_Y)B−hZ_out
       =h[D(X₀)B+D(N_Y)B−Z_out].

Multiplication by k gives C5b. If additionally R(X₀)=v, its own source equality gives D(X₀)B=Z_out, leaving D(N_Y)B=v. V01-PB-0006 B0 gives B(−B)=u, so right multiplication by −B gives D(N_Y)=v; the reverse uses absorption. No individual derivative is erased by this equivalence.

At the scalar factor K=c_fac u, the same right-product condition is particularly explicit: by V01-PB-0008 P3–P4, Δ=Σ_(I∈𝓑_H)δ_Ie_I, and ΔK=v is exactly c_fac δ_I=v for every listed I. This is a coefficient criterion, with no cancellation of c_fac or additional scalar guard.

4. Transport the complete product at both signs, then apply it to the source

For C8a retain the actual KUK=K, n=u and either σ_tr. V03-PB-0017 C2–C3 gives centrality of Z,Zᶜ, T(Z)=Z,T(Zᶜ)=Zᶜ and, under n=u, ZZᶜ=ZᶜZ=u. Therefore

K^ZU^Z=ZᶜKUZ=KU,
U^ZK^Z=UZZᶜK=UK,
K^ZU^ZK^Z=ZᶜKUK=ZᶜK=K^Z.

Subtracting the two ordered products from u proves both complement identities. The first test is unchanged. The second test's complete difference is

T(K^Z)P_tr−σ_trT(P_tr)K^Z
  =Zᶜ[T(K)P_tr−σ_trT(P_tr)K].

Multiplication by the actual Z reverses multiplication by Zᶜ, so its voidness is equivalent to the original test's. Centrality and T(Z)=Z also give

C_part^Z=P_tr UZ+σ_trT(P_tr UZ)E_L
     =[P_tr U+σ_trT(P_tr U)E_L]Z=C_part Z,
Part_(T,σ_tr)(YZ)=k[YZ+σ_trZT(Y)]=Part_(T,σ_tr)(Y)Z,
(N^(T,σ_tr))^Z_(YZ)=T(E_L)Part_(T,σ_tr)(Y)ZE_L=N_Y^(T,σ_tr)Z.

The actual companion product reverses each expression and its parameter change. For every signed X, T(XZ)=ZT(X)=σ_trXZ and (XZ)K^Z=XK; multiplying by Zᶜ proves the converse. This proves every part of C8a.

For any central Z, Γ’s definition and V03-PB-0017 C2–C3 give Γ(Z)=e₀T(Z)e₀=e₀Ze₀=Z. Taking Z=Ω gives C8b. At the actual Γ(K)=K, Γ(V)=V, KVK=K, VKV=V and n=u, V04-PB-0026 F1’s product reversal then yields

Γ(K^Z)=Γ(K)Γ(Zᶜ)=KZᶜ=K^Z,
Γ(V^Z)=Γ(Z)Γ(V)=ZV=V^Z,
K^ZV^ZK^Z=ZᶜKVK=K^Z,
V^ZK^ZV^Z=VKVZ=VZ=V^Z.

Both witness identities are used separately. These are C8c, with no condition on derivatives of a or b inserted.

For C8, return to T(A)=−A,K=e₀Ã. V04-PB-0025 N2a–N2b at n=u gives K_A(ZA)=ZᶜK and (ZA)K^Z=AK. Consequently the prescribed P_out=Z_out−AK is unchanged. V03-PB-0017 C4 gives A_(ZX)=ZA when A_X=A; C5 at n=u gives F(ZX)=F(X); C6 gives the entire displayed residual. The source completion can therefore use C8a with the additional actual KUK=K, σ_tr=u and P_tr=P_out. The residual identity itself uses neither a witness nor R(X)=v.

5. Recover the actual complementary-pair and elementary applications

V04-PB-0031 BC10 supplies, for all six written pairs and each σ_pair=u or −u, the complete ordered products at U=kBe₀:

K_{ij,σ_pair}U=UK_{ij,σ_pair}=P_{ij,−σ_pair},
K_{ij,σ_pair}P_{ij,−σ_pair}=P_{ij,−σ_pair}K_{ij,σ_pair}=K_{ij,σ_pair}.

Their association gives K_{ij,σ_pair}UK_{ij,σ_pair}=K_{ij,σ_pair}, the exact antecedent of V05-PB-0035 S3a at ι=T and positive unknown sign. BC5 gives P_{ij,σ_pair}+P_{ij,−σ_pair}=u and T(P_{ij,σ_pair})=P_{ij,σ_pair}. Consequently E_L=u−K_{ij,σ_pair}U=P_{ij,σ_pair}. V05-PB-0035 S0b gives Part_(T,u)(Y)=H_Y, so every term of S3b becomes

H=P_out U+T(P_out U)P_{ij,σ_pair}+P_{ij,σ_pair}H_YP_{ij,σ_pair}.

No sign case is omitted: at σ_pair=u the witness product is P_{ij,−}; at σ_pair=−u it is P_{ij,+}. Their respective complements are P_{ij,+} and P_{ij,−}. At P_out=v the first two terms are v, and this exact specialization recovers the entire BC9 null family. At general P_out the two S3a tests still apply; the null-family recovery alone would not supply a nonvoid prescribed output.

For the elementary A=B, V04-PB-0031 BC5 gives B̃=−B, BB=−u and e₀B=Be₀. Star fixes both spatial generators, so T(B)=−B. Thus K=−e₀B. Use U=Be₀ and compute both orders:

KU=−e₀BBe₀=u,
UK=−Be₀e₀B=−BB=u.

It follows that KUK=K and E_L=v. Thus H₀=P_out U, N_Y=v, and the unique completion is H=P_out U. Its necessary and sufficient condition is T(P_out U)=P_out U: every completion obeys H=HKU=P_out U, and conversely that signed expression has right product P_out UK=P_out. This is also the S3a criterion at these two undoing products. For A=e₀, 04 gives T(e₀)=(ẽ₀)*=−e₀; K=e₀e₀=u and U=u give both ordered products u. Again E_L=v, but now H₀=P_out and the remaining condition is T(P_out)=P_out. These are actual algebraic applications, with no assertion that B or e₀ is the active part of a supplied solution.

6. Derive the diagnostic where the two sandwich sides differ

For J_diag=e₁e₂e₃, apply V01-PB-0003’s contextual exchanges and then 04’s declared squares at the actual repeated labels:

J_diag J_diag=e₁e₂e₃e₁e₂e₃
     =−e₁e₂e₁e₃e₂e₃
     =e₁e₁e₂e₃e₂e₃
     =−(e₁e₁)(e₂e₂)(e₃e₃)
     =−(−u)(−u)(−u)=u,
J̃_diag=e₃e₂e₁
     =−e₂e₃e₁
     =e₂e₁e₃
     =−e₁e₂e₃=−J_diag.

Star fixes the three spatial generators, so T(J_diag)=−J_diag. For each actual g in the two-entry list e₀,e₄, all three labels are distinct from g, and the complete ordered chain is

gJ_diag=ge₁e₂e₃
    =−e₁ge₂e₃
    =e₁e₂ge₃
    =−e₁e₂e₃g=−J_diag g.

This supplies e₀J_diag=−J_diag e₀ and e₄J_diag=−J_diag e₄ at their two actual uses. V04-PB-0026 F1 supplies Γ’s additive and scalar laws, so

Γ(J_diag)=e₀(−J_diag)e₀=J_diag,
Γ(K₃)=Γ(u)+Γ(J_diag)=u+J_diag=K₃,
(e₀J_diag)̃=J̃_diag e₀=−J_diag e₀=e₀J_diag,
Ã₃=e₀+e₀J_diag=A₃,
A₃*=−e₀−e₀J_diag=−A₃,
T(A₃)=−A₃, e₀Ã₃=u+J_diag=K₃.

Expand K₃K₃=u+J_diag+J_diag+J_diag J_diag=hK₃. With U₃=kkK₃ this gives

K₃U₃=kkK₃K₃=kK₃,
U₃K₃=kkK₃K₃=kK₃,
K₃U₃K₃=kK₃K₃=khK₃=K₃,
u−K₃U₃=u−k(u+J_diag)=k(u−J_diag)=E₃.

The unity identity u−k=k follows from kh=u and h=u+u, so no scalar division beyond k occurs in the last line. T(E₃)=k(u+J_diag)=E₃^T. The two complete ordered products are

E₃^TE₃=kk[u−J_diag+J_diag−J_diag J_diag]=v,
E₃E₃^T=kk[u+J_diag−J_diag−J_diag J_diag]=v.

The same plus/minus square expansions as displayed explicitly here give

E₃E₃=kk[u−J_diag−J_diag+J_diag J_diag]=k(u−J_diag)=E₃,
E₃^TE₃^T=kk[u+J_diag+J_diag+J_diag J_diag]=k(u+J_diag)=E₃^T.

The e₄ exchanges yield both E₃^Te₄=k(e₄+J_diag e₄)=e₄E₃ and E₃e₄=k(e₄−J_diag e₄)=e₄E₃^T. Therefore

T(Δ₃)=T(E₃)T(e₄)=E₃^Te₄=e₄E₃=Δ₃,
Δ₃K₃=e₄k(u−J_diag)(u+J_diag)=e₄kv=v,
T(E₃)Δ₃E₃=E₃^Te₄E₃E₃=e₄E₃=Δ₃,
E₃Δ₃E₃=E₃e₄E₃E₃=e₄E₃^TE₃=v.

The empty coefficient of E₃ is k≠v by the following guarded argument: if k=v, hk=u gives u=v, and then h=u+u=v, contradicting its retained guard, so E₃≠v. If Δ₃=e₄E₃ were v, left multiplication by −e₄ would give E₃=v, since (−e₄)e₄=u. Thus Δ₃≠v. This completes C10 with the actual active operand, factor, witness and both ordered sandwiches.

What is not forced

C1 is a necessary condition on an actual completion. C2 is an equivalence on the same actual coefficient list and its supplied derivatives. C5a–C5b use an actual KUK=K, while C2 and C5c do not. Algebraic source completion retains the further full residual condition.

The diagnostic C10 supplies a positive example showing why the left T(E_L) in the completion construction must remain in that position: the displayed replacement by E_L erases that particular nonvoid Δ₃. It makes no claim that these two expressions differ at every input. C8a keeps n=u and actual KUK=K for both signs. C8c additionally keeps both Γ signs and both witness products. C8’s source identity F(ZX)=F(X) under n=u needs no witness; its full residual is retained. None of these statements supplies a base solution or independent derivative data.

What it means

One full coefficient system describes source completion before a witness is supplied. When an actual witness is available, the common signed product theorem constructs precisely the compatible remainder, and this record identifies its whole-source and remaining derivative equations. At the diagnostic’s nonvoid T-fixed Δ₃ with Δ₃K₃=v, T(E₃)Δ₃E₃=Δ₃ while E₃Δ₃E₃=v. Thus replacing the left T(E_L) by E_L erases this actual completion; no claim of a difference at every input is made.

The full right-null derivative has two recovered outputs and a simultaneous two-corner criterion (V06-PB-0040)

LAYER: 9 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V01-PB-0008, V02-PB-0011, V04-PB-0031, V05-PB-0038, V07-PB-0042

Declares

Retain 04's seven derivatives, finite expressions, h=u+u, k=h⁻¹ and h≠v. D,L,F,R retain V01-PB-0006's definitions; T and K_A retain V01-PB-0007's. K_t, P_±=P_{34,±}, Enc and the outside index list 𝒪 have V05-PB-0038's meanings. The same record's right-null notation is Z=Y−tYe₁ with YP_−=Y and Y=H+e₁G_c, where H and G_c are P_− corners. No T condition is imposed until explicitly stated. ε_μ is the generator-square sign of V01-PB-0002, carried in V01-PB-0004.

The derivative collections of V04-PB-0031 at (3,4), and the following additional written combinations, are

D_out(X)=e₀∂₀X−e₅∂₅X−e₆∂₆X,
D_in(X)=−e₁∂₁X−e₂∂₂X−e₃∂₃X−e₄∂₄X,
ℬ(X)=B∂₂X+e₁e₃∂₃X+e₁e₄∂₄X,
D_+(X)=∂₁X+ℬ(X),  D_−(X)=∂₁X−ℬ(X),
t_μ=∂_μt,
q_t=D(t),  q_out=D_out(t),  b_+=D_+(t), b_−=D_−(t),
κ_t=t₀t₀−t₁t₁−t₂t₂−t₃t₃−t₄t₄−t₅t₅−t₆t₆.            (N0)

The + and − on D_± label those displayed expressions, not T signs. All coefficient derivatives concern the same supplied expressions.

For a full outside tuple F_out=(F_(out,0),F_(out,2),F_(out,3),F_(out,4)), define

(𝒟_out F_out)_r=D_out(F_(out,r)),
(𝒨_out(t;F_out))_r=q_out F_(out,r).

The other four tuple operations are defined by these complete rows, where χ_s=u for the + subscript and χ_s=−u for the − subscript:

r	(𝒟_(χ_s)F_out)_r	(𝒨_(χ_s)(t;F_out))_r
0	∂₁F_(out,0)−χ_s(∂₂F_(out,2)+∂₃F_(out,3)+∂₄F_(out,4))	t₁F_(out,0)−χ_s(t₂F_(out,2)+t₃F_(out,3)+t₄F_(out,4))
2	∂₁F_(out,2)+χ_s(∂₂F_(out,0)+∂₃F_(out,4)−∂₄F_(out,3))	t₁F_(out,2)+χ_s(t₂F_(out,0)+t₃F_(out,4)−t₄F_(out,3))
3	∂₁F_(out,3)+χ_s(∂₃F_(out,0)+∂₄F_(out,2)−∂₂F_(out,4))	t₁F_(out,3)+χ_s(t₃F_(out,0)+t₄F_(out,2)−t₂F_(out,4))
4	∂₁F_(out,4)+χ_s(∂₄F_(out,0)+∂₂F_(out,3)−∂₃F_(out,2))	t₁F_(out,4)+χ_s(t₄F_(out,0)+t₂F_(out,3)−t₃F_(out,2))

The tuple entries are the displayed expressions on the supplied outside inputs.

What is forced

The seven actual contributions recombine without a restriction:

D(X)=D_out(X)+D_in(X).                                  (N0a)

For every supplied right-null input of V05-PB-0038 N3, before a T condition,

D(Z)=D(Y)−D(tY)e₁,
D(Y)=D(Z)P_−,
D(tY)=D(Z)P_+e₁,
q_tY=D(Z)P_+e₁−tD(Z)P_−.                            (N4)

Thus the whole derivative criterion is

D(Z)=v iff D(Y)=v and q_tY=v.                          (N5)

For its entire two-corner representation Y=H+e₁G_c, the four recovered corners are

D(Y)=A_D−e₁C_D,
A_D=D_out(H)+D_+(G_c),    C_D=D_−(H)+D_out(G_c),
q_tY=A_q−e₁C_q,
A_q=q_out H+b_+G_c,       C_q=b_−H+q_out G_c,
A_D=P_−D(Y),           C_D=e₁P_+D(Y),
A_q=P_−q_tY,           C_q=e₁P_+q_tY.                  (RD1)

Every A_D,C_D,A_q,C_q is a P_− corner. Consequently N5 is equivalent to all four being void. No term within one of them is required separately void.

On the full thirty-two-coordinate encoder, the declared operations satisfy

D_out(Enc(F_out))=Enc(𝒟_out F_out),
D_±(Enc(F_out))=Enc(𝒟_±F_out),
q_out Enc(F_out)=Enc(𝒨_out(t;F_out)),
b_± Enc(F_out)=Enc(𝒨_±(t;F_out)).                             (N17)

For the full sixty-four-coordinate right-null input CA2 of V05-PB-0038, D(Z)=v is therefore equivalent to the simultaneous tuple equations

𝒟_out F_out+𝒟_+G_out=(v,v,v,v),
𝒟_−F_out+𝒟_out G_out=(v,v,v,v),
𝒨_out(t;F_out)+𝒨_+(t;G_out)=(v,v,v,v),
𝒨_−(t;F_out)+𝒨_out(t;G_out)=(v,v,v,v).                       (RD2)

For any supplied corner H, put Y=(u−te₁)H and Z=Y−tYe₁. This is the actual specialization G_c=−tH, already right-null by V05-PB-0038. No T condition is needed for the exact four equations

D_out(H)−D_+(tH)=v,
D_−(H)−D_out(tH)=v,
(q_out−tb_+)H=v,
(b_−−tq_out)H=v.                                    (N7)

They are together equivalent to D(Z)=v. Writing H=Enc(F_out), their equivalent complete tuple form is

𝒟_out F_out−𝒟_+(tF_out)=(v,v,v,v),
𝒟_−F_out−𝒟_out(tF_out)=(v,v,v,v),
𝒨_out(t;F_out)−t𝒨_+(t;F_out)=(v,v,v,v),
𝒨_−(t;F_out)−t𝒨_out(t;F_out)=(v,v,v,v).                       (N11)

This holds on every corner. Under T(H)=H, substitute exactly V05-PB-0038 N2's complete twelve-coordinate tuple. It then gives a necessary-and-sufficient simultaneous criterion on all twelve c_J and the same t.

For the T-fixed input Z_c, the recovered corner is H=P_−Z_cP_− with T(H)=H, the same object named H_c in V05-PB-0038 CN0. Write c_(J,μ)=∂_μc_J. For each actual derivative μ, the exact individual-derivative criterion is

∂_μZ_c=v iff c_(J,μ)=v and t_μc_J=v for every J∈𝒥.    (CN8)

The whole product identity and its derivative consequences are

q_tq_t=κ_tu,
D(Z)=v ⇒ κ_tY=v,
D(Z)=v ⇒ κ_tH=κ_tG_c=v     for CA1's recovered corners,
D(Z_c)=v ⇒ κ_tc_J=v for every J∈𝒥 on the T-fixed class. (N12)

No product in the last three lines is presented as a substitute for N5.

Under the additional actual identities t_μ=v at all seven labels, N7 reduces exactly to

D_out(H)=tD_+(H),
D_−(H)=tD_out(H),

or, equivalently, to

D_out(H)=tD_+(H),
(u−tt)∂₁H−(u+tt)ℬ(H)=v.                          (N13)

At the actual t=v, these are precisely V04-PB-0031 BC12's negative-corner balances D_out(H)=D_in(H)=v.

On the T-fixed class only, the full original equation has

Sel(Z_c)=F(Z_c)=v,
R(Z_c)=hD(Z_c)B,
R(Z_c)=v iff D(Y)=v and q_tY=v,
           equivalently all four equations N7 or N11.                (N18)

For any separate supplied X with K_A(X)=K_t, V07-PB-0042(FC4) gives A_(X+Z_c)=A_X, F(X+Z_c)=F(X), and R(X+Z_c)−R(X)=hD(Z_c)B. Under exactly that factor premise, N5 decides residual preservation. Its solution corollary FC4s additionally retains the separate actual R(X)=v.

The forcing

1. Recover both derivative outputs from the full derivative

The definitions of D_out and D_in list all seven terms of V01-PB-0006's D, proving N0a. V02-PB-0011(D1) supplies the negative and scalar-constant derivative rules. Differentiate the actual identity YP_−=Y; V04-PB-0031(BC5)'s ∂μP−=v gives (∂μY)P−=∂μY. The same holds for tY, because (tY)P−=tY. Left multiplication by each original derivative generator and addition of all seven terms preserve these right products. Therefore

D(Y)P_−=D(Y), D(Y)P_+=v,
D(tY)P_−=D(tY), D(tY)P_+=v.

Since e₁ is derivative-constant, the full derivative of Z=Y−tYe₁ is D(Z)=D(Y)−D(tY)e₁. Its two actual right products are

D(Z)P_−=D(Y)−D(tY)e₁P_−=D(Y),
D(Z)P_+=−D(tY)e₁P_+=−D(tY)e₁.

The first removed product uses V04-PB-0031(BC5a)'s e₁P_−=P_+e₁; the second retained one uses its e₁P_+=P_−e₁. Right multiplication of the second equality by e₁, and e₁e₁=−u, gives D(tY)=D(Z)P_+e₁. This proves the first three N4 identities in their full scope.

For each actual μ, 04's product rule gives

ε_μe_μ∂_μ(tY)
  =ε_μe_μ(t_μY+t∂_μY)
  =ε_μe_μt_μY+tε_μe_μ∂_μY.

Only the scalar t moved through e_μ. Appending all seven equalities gives D(tY)=(D(t))Y+tD(Y). Substitute the two just-recovered outputs and rearrange to prove the final N4 equality.

If D(Z)=v, every displayed recovery is void, proving D(Y)=v and (D(t))Y=v. Conversely these two actual equalities make D(tY)=v by its full product rule, then D(Z)=v by the original complete sum. This proves both directions of N5 with the same Y,t throughout.

2. Recover all four corners before the specialization

From the actual generator products of V01-PB-0003, with all derivatives of the generators void, every supplied expression X satisfies

D_in(X)=−e₁D_−(X),
D_in(e₁X)=D_+(X),
D_out(e₁X)=−e₁D_out(X).

For the first identity multiply its four original terms on the left by e₁: e₁(−e₁)=u, e₁(−e₂)=−B, e₁(−e₃)=−e₁e₃, e₁(−e₄)=−e₁e₄; the left undoing is −e₁. For the second, the corresponding right factors −e_μe₁ are u,B,e₁e₃,e₁e₄. For the third, each outside label 0,5,6 exchanges once with e₁. These are all seven labels.

Apply those identities to Y=H+e₁G_c:

D(Y)=D_out(H)−e₁D_−(H)−e₁D_out(G_c)+D_+(G_c)
    =[D_out(H)+D_+(G_c)]−e₁[D_−(H)+D_out(G_c)].

The same corner G_c=Enc(G_out) is distinct from its outside tuple G_out. Each collected bracket is a corner by V04-PB-0031(BC11) and the commutation of the two-inside words with P_−. This proves RD1's first decomposition. Write q_t=q_out+q_in with q_in=D_in(t). The same actual products give

q_in=−e₁b_−,  q_ine₁=b_+,  q_oute₁=−e₁q_out.

The complete distributed product is consequently

q_t(H+e₁G_c)=q_out H−e₁b_−H−e₁q_out G_c+b_+G_c
          =[q_out H+b_+G_c]−e₁[b_−H+q_out G_c].

Both brackets are again corners. If A,C are any such corners, then

P_−(A−e₁C)=A,
e₁P_+(A−e₁C)=C.

The omitted terms contain P_+P_−=v; the retained second term uses −e₁e₁=u. Thus A−e₁C=v exactly when A=C=v, which proves all four recoveries in RD1 and their combined equivalence with N5.

For the special actual G_c=−tH, substitute that expression into every term of RD1. The full product rule remains inside D_+(tH) and D_out(tH). The two algebraic brackets become (q_out−tb_+)H and (b_−−tq_out)H. This is exactly N7, without a T premise. N3/CN2 of V05-PB-0038 already identifies every T-fixed input with this specialization.

3. Write every row of the complete simultaneous criterion

Every outside expression uses only the eight positions on {0,5,6}. An individual derivative retains those positions, since every generator has void derivative. Left multiplication by e₀,−e₅,−e₆ also stays on those positions: the native word reductions reorder those same letters and remove repeated squares. Each such outside generator commutes with u,B,e₁e₃,e₁e₄ and P_− by V04-PB-0031 BC5a and the two exchanges past an inside pair. Consequently differentiation of the four encoder terms, and each of the three outside generator multiplications, give

D_out(Enc(F_out))=Enc(𝒟_out F_out),
q_out Enc(F_out)=Enc(𝒨_out(t;F_out)).

Distributing the same four defining terms also gives Enc(F_out+G_out)=Enc(F_out)+Enc(G_out) and Enc(cF_out)=cEnc(F_out) for a supplied scalar c. Their generator factors and P_− have void derivatives, so ∂_μEnc(F_out)=Enc(∂_μF_out). These identities justify each tuple sum and individual derivative used below.

For the other rows apply the complete multiplication of V05-PB-0038 N9/CA3 at the three actual outside tuples (v,u,v,v), (v,v,u,v), (v,v,v,u). Their encoder values are BP_−, e₁e₃P_−, e₁e₄P_−, so that multiplication gives all four components in each product:

B Enc(F_out)=Enc((−F_(out,2),F_(out,0),−F_(out,4),F_(out,3))),
e₁e₃ Enc(F_out)=Enc((−F_(out,3),F_(out,4),F_(out,0),−F_(out,2))),
e₁e₄ Enc(F_out)=Enc((−F_(out,4),−F_(out,3),F_(out,2),F_(out,0))).

Apply these three exact rows to ∂₂F_out,∂₃F_out,∂₄F_out, respectively, and add Enc(∂₁F_out), with the actual sign χ_s on all three rows. Component 0 is ∂₁F_(out,0)−χ_s(∂₂F_(out,2)+∂₃F_(out,3)+∂₄F_(out,4)); the other three are, in order, exactly rows 2,3,4 of the declared 𝒟_(χ_s). Thus D_±(Enc(F_out))=Enc(𝒟_±F_out) on the complete tuple.

For the product, put B_out=(t₁,χ_s t₂,χ_s t₃,χ_s t₄). Its encoder is b_(χ_s)P_−. The same N9/CA3 gives

b_(χ_s)Enc(F_out)=Enc(B_out)Enc(F_out)=Enc(B_out⋆F_out).

Substituting those four entries in each of N9's four product rows gives precisely the declared 𝒨_(χ_s)(t;F_out). This establishes N17 using the same native multiplication; the outside order in every product is retained.

For V05-PB-0038 CA2 put H=Enc(F_out),G_c=Enc(G_out) in RD1. Each of its four recovered corners is the encoder of the corresponding tuple in RD2. V05-PB-0038 N16's injectivity proves all four equivalences. For the corner-built class the actual tuple G_out=−tF_out gives N11, with each ∂μ(tF(out,r))=t_μF_(out,r)+t∂μF(out,r) retained. Its T-fixed instance is precisely V05-PB-0038 N2's complete twelve-coordinate tuple, without changing any operation row.

4. Recover each individual derivative from the full coefficient interface

For the T-fixed coefficient presentation, V05-PB-0038 CN7 gives c_J=𝓒_J(Z_c)−s_J𝓒_(J′)(Z_c). V01-PB-0008 P2 commutes each derivative with these coefficient maps. Therefore ∂μZ_c=v implies every c(J,μ)=v. The coefficient at M_J in V05-PB-0038 CN6 then gives

v=−ξ_J[t c_(J,μ)+t_μc_J]=−ξ_Jt_μc_J.

Multiplication by the actual sign ξ_J yields t_μc_J=v. Conversely, if both conditions hold at every J, the derivatives of V05-PB-0038 CN6's first two coefficients contain only c_(J,μ) and (ht t_μ)c_J, and the third contains only tc_(J,μ)+t_μc_J. The full product rule gives ∂_μ(tt)=ht t_μ, so all these terms are void. Every remaining coefficient is identically void. V01-PB-0004(C1,C4)'s full coefficient reconstruction proves ∂_μZ_c=v. This proves CN8 in both directions.

5. Keep each scalar consequence and fixed-t application at its scope

The full product q_tq_t has the seven diagonal products ε_μt_μt_μu. For each distinct pair of original labels, its two ordered contributions combine to ε_με_νt_μt_ν(e_μe_ν+e_νe_μ)=v. Appending every diagonal and every paired contribution gives exactly κ_tu. No individual derivative was cancelled.

Under N5, left multiplication of q_tY=v by q_t gives κ_tY=v. The actual CA1 recoveries H=P_−Y and G_c=−e₁P_+Y give κ_tH=κ_tG_c=v by scalar commutation. In the T-fixed instance, V05-PB-0038 N8's coefficient recovery gives κ_tc_J=h𝓒_J(κ_tH)=v. This proves all of N12 with its explicit sign scope.

For the separate fixed-t application, retain t_μ=v at all seven labels. Then q_out=b_+=b_−=v and the last two N7 equations are automatic. Under these actual identities the full product rule gives

D_+(tH)=tD_+(H),
D_out(tH)=tD_out(H).

The first two equations reduce to D_out(H)=tD_+(H) and D_−(H)=tD_out(H). Substitute the first into the second to obtain D_−(H)=ttD_+(H). By N0 its complete difference is

(u−tt)∂₁H−(u+tt)ℬ(H)=v.

Conversely this equality recovers D_−=ttD_+; together with the retained first equation it gives D_−=tD_out, and hence both original fixed-t equations. This proves both directions of N13 without dividing t or either displayed quadratic scalar.

At the further actual t=v identity, Y=H and Z=H. The two equations become D_out(H)=v and D_−(H)=v. Since e₁D_in(H)=D_−(H) and −e₁ undoes the left e₁, these are exactly the earlier two complete constant-corner balances of V04-PB-0031(BC12) at the negative P sign. That application is recovered after, rather than imposed before, the general simultaneous criterion.

For N18 use V01-PB-0007's exact T-fixed selector criterion: Sel(Z_c)=v, hence F(Z_c)=Z_ce₀v=v. V01-PB-0006 then gives R(Z_c)=hD(Z_c)B. The guarded inverse gives hk=u, scalar commutation gives kh=u, and the two supplied spatial squares give B(−B)=−e₁e₂e₁e₂=u. Thus k and −B undo the exterior factors, proving the two-way void criterion. For the separate actual-factor comparison, apply V07-PB-0042(FC4) to T(Z_c)=Z_c and Z_cK_A(X)=Z_cK_t=v for all three displayed identities. The solution corollary FC4s additionally retains R(X)=v, as N18 states.

What is not forced

RD2 is the derivative criterion on the full two-corner right-null class. The corner-built class N7 and its T-fixed twelve-coordinate restriction are explicitly retained specializations. N18 applies the latter sign to the original nonlinear equation; RD2 alone concerns the derivative expression and carries no selector premise. Outside the T-fixed class the original residual retains its whole source term; RD2 is not presented there as its equation criterion.

All rows concern the same actual scalar expressions and their commuting derivatives. They do not construct realizations of freely assigned derivative values. Each product κ_tc_J and t_μc_J remains a whole product unless a separately supplied undoing is actually used. N12 is a consequence, while N5, RD2, N7 and N11 are the complete criteria at their stated domains.

What it means

The full right-null derivative at the supplied K_t is organized by two recovered outputs and two free corners. Its simultaneous equations keep every derivative and product, with the T-fixed twelve-coordinate system obtained by an explicit specialization of the unrestricted sixty-four-coordinate expression. The full coefficient formulas and their direct recovery separately give the individual-derivative criterion CN8. The derivative pair yields the fixed-t cases, while the original-equation reading retains the T-fixed selector premise.

The direct B(−B)=u calculation used at N18 is the ordered product also stated in V01-PB-0006(B0). It undoes only that exterior factor; the N5 criterion still retains both whole derivative outputs.

A whole coefficient representative gives the exact varying-contraction and universal-balance criteria (V05-PB-0036)

LAYER: 10 FORCED BY: V01-PB-0001, V01-PB-0004, V01-PB-0005, V01-PB-0006, V01-PB-0007, V01-PB-0008, V02-PB-0011, V03-PB-0017, V03-PB-0018, V04-PB-0023, V04-PB-0024

Declares

The supplied scalar list 𝛇=(ζ_K) and its contraction Λ_𝛇 retain V04-PB-0023 S1. The tuples j,j^c and the finite derivative sum Bal retain V04-PB-0024 J0. Define the full list tuple and its scalar derivative remainder by

j_(𝛇,μ)(W):=ε_μΛ_𝛇(e_μWB),
Drem_𝛇(W):=Σ_(μ,K)ε_μ(∂_μζ_K)𝓒_K(e_μWB).              (U0a)

The paired sum follows the original G listing and then the full blade list 𝓑, starting at v. The same actual list is retained when an input W changes. Its entries may have arbitrary supplied derivatives; it is not an input-dependent rule for choosing a different list. All identities below concern supplied expressions and their actual derivatives, not values at an unprovided point. No equation is imposed on W unless stated.

The scalars u,v,h,k, blades e_I and word B retain 04's meanings; G,𝓑,𝓒_I retain PB-0004's definitions; ε_μ retains PB-0002's generator-square signs; χ retains PB-0005's blade-product sign; D,F,R retain PB-0006's definitions; I_s,Ω_s retain PB-0009's definitions; Ω,Z,Zᶜ,n retain V03-PB-0017's definitions; g retains V03-PB-0018's pairing definition. In particular Ω is the complete seven-generator word, Ω_s=e₁e₂e₃e₄e₅e₆, and Z=a+bΩ with Zᶜ=a−bΩ. The scalar b in Z and the complete list 𝛇 are different objects; ζ_K is not the blade-undoing expression b_K. For each supplied input U, use x_I(U):=𝓒_I(U).

Define two actual written elements, on the complete fixed lists:

𝓤_𝛇 := Σ_(K∈𝓑) ζ_K e_K,
𝓥_𝛇 := −Σ_(μ∈G) e_μ(∂_μ𝓤_𝛇)B.                     (U0)

They represent respectively the full contraction list and its full derivative remainder. The symbol 𝓥_𝛇 does not name a field or a selected part of W. Every sum starts at v. Neither definition restricts the coefficient list or uses an inverse of a written element.

For every actual output index I put

p_I(𝛇) := Σ_(μ∈G) χ({μ},{μ}△I)
                     ∂_μ ζ_({μ}△I).                  (U1)

This is one complete seven-term scalar expression. Its terms are not separately required to vanish.

For a supplied central Z define the new complete scalar list by coefficient recovery:

ζ^Z_K := 𝓒_K(Zᶜ𝓤_𝛇),       𝛇^Z := (ζ^Z_K).          (U2)

No derivative constancy or condition n=u is part of U2. The old list remains 𝛇; the superscript records the defined change rather than silently replacing it.

What is forced

The representative recovers exactly the contraction and all its coefficients:

𝓒_K(𝓤_𝛇)=ζ_K,
Λ_𝛇(Y)=g(𝓤_𝛇,Y).                                    (U3)

For arbitrary supplied X,Y and each actual μ, the finite pairing has the exact product-transfer identities

g(X,e_μY)=ε_μ g(e_μX,Y),
g(X,Ye_μ)=ε_μ g(Xe_μ,Y),
g(X,YB)=−g(XB,Y),
g(X,ZY)=g(ZᶜX,Y).                                   (U4)

For arbitrary supplied 𝛇 and W,

j_(𝛇,μ)(W)=−g(e_μ𝓤_𝛇B,W),
Drem_𝛇(W)=g(𝓥_𝛇,W),
𝓥_𝛇=[D(𝓤_𝛇*)]*B,                                   (U5)

Bal(j_𝛇(W))=g(𝓤_𝛇,D(W)B)+g(𝓥_𝛇,W)
             =Λ_𝛇(D(W)B)+Drem_𝛇(W),
h Bal(j_𝛇(W))=Λ_𝛇(F(W))+Λ_𝛇(R(W))+h g(𝓥_𝛇,W).        (U6)

For the same supplied list, the following conditions are equivalent:

h Bal(j_𝛇(W))=Λ_𝛇(F(W))+Λ_𝛇(R(W))
                              for every supplied W;
Drem_𝛇(W)=v                     for every supplied W;
𝓥_𝛇=v;
D(𝓤_𝛇*)=v;
p_I(𝛇)=v                     for every I in 𝓑.        (U7)

Thus the exact condition for removing the varying-list remainder on every supplied input is a whole derivative equality. Its complete scalar form is the list U1; no separate coefficient-derivative assumptions are inserted.

There is also a stronger complete classification without assuming source annihilation or coefficient constancy in advance:

h Bal(j_𝛇(W))=Λ_𝛇(R(W)) for every supplied W            (U8a)

holds if and only if, with c:=ζ_(I_s),

ζ_K=v for every K≠I_s,
∂_μc=v for every μ in G.                              (U8b)

Equivalently its representative is 𝓤_𝛇=cΩ_s with those seven derivative identities. Under these equivalent conditions,

j_𝛇(W)=c j(W),
Λ_𝛇(R(W))=c𝓒_(I_s)(R(W)).                            (U8c)

This is an all-input identity classification. It does not replace its premise with a statement required only when R(W)=v.

The central change U2 gives, for every full supplied W,Y,

𝓤_(𝛇^Z)=Zᶜ𝓤_𝛇,
Λ_𝛇(ZY)=Λ_(𝛇^Z)(Y),
j_𝛇(ZW)=j_(𝛇^Z)(W),                                 (U9)

𝓥_(𝛇^Z)=Zᶜ𝓥_𝛇−Σ_μ e_μ(∂_μZᶜ)𝓤_𝛇B,
Drem_(𝛇^Z)(W)=Drem_𝛇(ZW)
                   +Σ_μ j_(𝛇,μ)((∂_μZ)W).           (U10)

The seven terms in the last line combine exactly to Λ_𝛇((D(Z))WB). In particular U6 at the transformed tuple is

h Bal(j_𝛇(ZW))
  =Λ_(𝛇^Z)(F(W))+Λ_(𝛇^Z)(R(W))
     +hDrem_𝛇(ZW)+hΛ_𝛇((D(Z))WB).                      (U11)

All these transport identities allow varying a,b and 𝛇. They require neither n=u nor a base-solution premise. Their source and residual contributions are retained separately.

The list operations also have an exact finite recovery. For supplied lists 𝛇,𝛇′ and supplied scalars a_mix,b_mix, define the combined list 𝛇_mix by ζ_(mix,K)=a_mix ζ_K+b_mix ζ′_K at every K. Then, for every full W,Y,

𝓤_(𝛇_mix)=a_mix 𝓤_𝛇+b_mix 𝓤_𝛇′,
Λ_(𝛇_mix)(Y)=a_mix Λ_𝛇(Y)+b_mix Λ_𝛇′(Y),
j_(𝛇_mix)(W)=a_mix j_𝛇(W)+b_mix j_𝛇′(W).                           (U25)

No derivative condition on a_mix,b_mix or either list is required in U25.

If 𝛇 satisfies the equivalent source-annihilation conditions V04-PB-0023 S6, put c=ζ_(I_s). Then, without any derivative-constancy premise,

𝓤_𝛇=cΩ_s,
j_𝛇(W)=c j(W),
Drem_𝛇(W)=g(𝓥_𝛇,W)=Σ_μ(∂_μc)j_μ(W),
h Bal(j_𝛇(W))=c𝓒_(I_s)(R(W))
                   +hΣ_μ(∂_μc)j_μ(W).              (U26)

Under the additional seven identities ∂μc=v, its last line is h Bal(j_𝛇(W))=c𝓒(I_s)(R(W)). Conversely every supplied c with those seven identities furnishes this class by the list ζ_(I_s)=c and ζ_K=v for K≠I_s. The stronger U8 states that precisely this class is forced by the all-input residual-only identity itself.

Let 𝛇^[s] and 𝛇^[0] be the recovered coefficient lists of Ω_s and e₀. Their representatives are Ω_s and e₀, their 𝓥 elements are v, and their tuples are respectively j and j^c. For every supplied central Z=a+bΩ, with no equation or n=u condition,

ZᶜΩ_s=aΩ_s+b e₀,       Zᶜe₀=a e₀−bΩ_s,
j(ZW)=a j(W)+b j^c(W),
j^c(ZW)=a j^c(W)−b j(W),
Bal(j(ZW))=a Bal(j(W))+b Bal(j^c(W))
             +Σ_μ[(∂_μa)j_μ(W)+(∂_μb)j^c_μ(W)],
Bal(j^c(ZW))=a Bal(j^c(W))−b Bal(j(W))
             +Σ_μ[(∂_μa)j^c_μ(W)−(∂_μb)j_μ(W)].     (U27)

All derivative terms of the same supplied a,b remain in these identities.

The forcing

1. Recover the full list, then derive the product transfers

V03-PB-0018 G1 gives the scalar character and complete coefficient formula of g. V01-PB-0004 C1–C3 applied to the complete supplied presentation in U0 gives 𝓒_K(𝓤_𝛇)=ζ_K. V03-PB-0018 G1 therefore gives, on exactly the same list,

g(𝓤_𝛇,Y)=Σ_K ζ_K𝓒_K(Y)=Λ_𝛇(Y).

This proves U3 before any derivative or source condition is imposed. The pairing's symmetry, addition, scalar and void laws are already the complete-input identities G2. Its separation G3 uses all blade pairings, not a square-sum positivity assertion.

Fix an actual generator label μ. V01-PB-0005 gives the complete blade products

e_μe_I=χ({μ},I)e_({μ}△I),
e_Ie_μ=χ(I,{μ})e_(I△{μ}).                           (U12)

Apply the same generator once more. Association and its square rule give

χ({μ},I)χ({μ},{μ}△I)e_I=e_μ(e_μe_I)=ε_μe_I,
χ(I,{μ})χ(I△{μ},{μ})e_I=(e_Ie_μ)e_μ=ε_μe_I.

PB-0005 (1) cancels the same final blade e_I from each equality. Each χ sign has self-product u, so multiplication by the first sign gives

χ({μ},{μ}△I)=ε_μχ({μ},I),
χ(I△{μ},{μ})=ε_μχ(I,{μ}).                           (U13)

By G1, g(e_I,e_J) is u when I=J and v otherwise. The two pairings g(e_I,e_μe_J) and ε_μg(e_μe_I,e_J) are both v unless I={μ}△J. At that index their values are χ({μ},J) and ε_μχ({μ},I), equal by U13. Hence

g(e_I,e_μe_J)=ε_μg(e_μe_I,e_J)

for every actual pair I,J. For right multiplication the two possible nonvoid values occur when I=J△{μ}; they are χ(J,{μ}) and ε_μχ(I,{μ}), again equal by U13. Both generator transfers are established on the full blade-pair list, including its empty indices.

To extend to full inputs, reconstruct X=Σ_Ix_I(X)e_I and Y=Σ_Jx_J(Y)e_J before multiplying. The actual prefix step for the left product is e_μ(V+x_J(Y)e_J)=e_μV+x_J(Y)e_μe_J, starting at e_μv=v; the right-product step is (V+x_J(Y)e_J)e_μ=Ve_μ+x_J(Y)e_Je_μ, starting at ve_μ=v. G2 distributes each complete paired sum into the occurrences x_I(X)x_J(Y) times the blade pairings just proved. Start the paired-list accumulator at v and append its next occurrence; replacing that occurrence with its established equal expression preserves the equality at that step. At the last prescribed pair the complete inputs therefore obey the first two U4 identities. This is full coefficient collection, not an inference from an unexamined selected support.

For the actual two factors in B, apply the right transfer in their displayed order:

g(X,YB)
  =g(X,(Ye₁)e₂)
  =ε₂g(Xe₂,Ye₁)
  =ε₂ε₁g((Xe₂)e₁,Y)
  =g(Xe₂e₁,Y)=−g(XB,Y).                              (U14)

The middle sign product is (−u)(−u)=u and e₂e₁=−B. This proves exactly the required right-B sign.

For Ω, apply the first transfer successively at its seven actual positions. The resulting coefficient is ε₀ε₁ε₂ε₃ε₄ε₅ε₆=u, since the six spatial negatives group into three pairs. The first input after the seven transfers is e₆e₅e₄e₃e₂e₁e₀X=Ω̃X. V03-PB-0017 C1 proves Ω̃=−Ω. Hence

g(X,ΩY)=g(−ΩX,Y).

The actual expression Zᶜ=a−bΩ is Z(a,−b), hence central by V03-PB-0017(C2). Apply G2 and the two supplied scalar coefficients to obtain

g(X,ZY)=a g(X,Y)+b g(X,ΩY)
         =g((a−bΩ)X,Y)=g(ZᶜX,Y).

This completes U4 directly from the whole blade products. Only the proved pairing and the seven displayed generator transfers are used.

2. Recover the whole remainder as one paired written element

Using U3, then the first and third transfers in U4, gives each tuple component:

j_(𝛇,μ)(W)
  =ε_μg(𝓤_𝛇,e_μWB)
  =ε_με_μg(e_μ𝓤_𝛇,WB)
  =−g(e_μ𝓤_𝛇B,W).                                   (U15)

V01-PB-0008 P2 gives coefficient–derivative compatibility. With V02-PB-0011's fixed blade derivatives, it gives

∂_μ𝓤_𝛇=Σ_K(∂_μζ_K)e_K.

All terms arise by the actual prefix application of 04's product rule

∂_μ(V+ζ_Ke_K)=∂_μV+(∂_μζ_K)e_K+ζ_K∂_μe_K
              =∂_μV+(∂_μζ_K)e_K,

starting at ∂_μv=v. The final equality removes only the proved blade derivative. Consequently the μ row of the remainder U0a is

Σ_K ε_μ(∂_μζ_K)𝓒_K(e_μWB)
   =ε_μg(∂_μ𝓤_𝛇,e_μWB)
   =−g(e_μ(∂_μ𝓤_𝛇)B,W).

G2 adds all seven rows, with their actual coefficients unchanged, to g(𝓥_𝛇,W). This proves the second equality in U5 on every full W.

For the last U5 equality, expand the actual D and use star's product rule, its derivative commutation, and the identity (X*)*=X:

[D(𝓤_𝛇*)]*
  =Σ_μ ε_μ e_μ* [∂_μ(𝓤_𝛇*)]*
  =Σ_μ ε_μ e_μ* ∂_μ𝓤_𝛇.

At μ=0, ε₀e₀*=−e₀; at each of the six spatial positions ε_ie_i*=−e_i. Thus this complete sum is −Σ_μe_μ∂_μ𝓤_𝛇. Right multiplication by B yields 𝓥_𝛇 exactly. The star identities are V01-PB-0007 and the derivative identities are V02-PB-0011; no coefficient of 𝓤_𝛇 is removed.

The scalar pairing product rule is also explicit. Differentiate its full G1 presentation, using V01-PB-0008 P2 to identify each coefficient derivative with the recovered derivative-input coefficient:

∂_μg(X,Y)
  =Σ_I[(∂_μx_I(X))x_I(Y)+x_I(X)(∂_μx_I(Y))]
  =g(∂_μX,Y)+g(X,∂_μY).                              (U16)

The two accumulations start at v. Denote their preceding prefixes by Acc_left and Acc_right, and the next actual summands by term_left and term_right. At that position the combined prefix changes by

(Acc_left+Acc_right)+(term_left+term_right)
  =(Acc_left+term_left)+(Acc_right+term_right).

This is the same finite additive regrouping as V01-PB-0004. Therefore the split retains every occurrence, with no assumption on either derivative list.

Apply U16 to the original component ε_μg(𝓤_𝛇,e_μWB). The fixed ε_μ,e_μ,B have void derivatives, so the two resulting rows are the full remainder row already identified and

ε_μg(𝓤_𝛇,e_μ(∂_μW)B).

Their seven-row sum is g(𝓤_𝛇,D(W)B) by G2 and the definition of D. This proves the first U6 identity. Finally R(W)=hD(W)B−F(W) and U3 give

Λ_𝛇(R(W))=h g(𝓤_𝛇,D(W)B)−Λ_𝛇(F(W)).

Substitution proves the second U6 identity. It gives both written forms of the full contraction balance U6 through one whole representative.

3. Classify the all-input remainder, keeping its coupled coefficients

The declared inversion gives hk=u and 04 rule 8 gives kh=hk=u. U6 and this kh=u show that the first U7 identity is equivalent to Drem_𝛇(W)=v on each supplied W. U5 shows that 𝓥_𝛇=v implies that equality on every W. Conversely, suppose the equality holds for every supplied W. It then holds at each member of the already complete blade list W=e_I, giving

g(𝓥_𝛇,e_I)=v  for every I in 𝓑.

V03-PB-0018 G3 reconstructs 𝓥_𝛇 from exactly those coefficients, so 𝓥_𝛇=v. This proves the equivalence without requiring that g(𝓥_𝛇,𝓥_𝛇)=v force any conclusion.

By U5, 𝓥_𝛇=v iff [D(𝓤_𝛇*)]B=v. Right multiplication by the actual −B, using PB-0006(B0)'s B(−B)=u, removes B in one direction; multiplication by B gives the reverse. Applying star to the resulting equality and using the identity (X)=X proves that this is equivalent to D(𝓤_𝛇)=v. No coefficient inverse has appeared.

For the complete scalar version, expand the whole sum before recovering a coefficient:

Σ_μe_μ∂_μ𝓤_𝛇
  =Σ_(μ,K)(∂_μζ_K)e_μe_K
  =Σ_(μ,K)χ({μ},K)(∂_μζ_K)e_({μ}△K).                 (U17)

The product expansion starts from the actual paired list μ first and K second. At output I, an occurrence from direction μ qualifies exactly when {μ}△K=I, equivalently K={μ}△I. There is one such K for each μ. V01-PB-0004 C2 recovers precisely p_I(𝛇) in U1. Its C4 therefore gives

Σ_μe_μ∂_μ𝓤_𝛇=v iff p_I(𝛇)=v for every I.

Negating the whole expression and multiplying or undoing B relates this equality to 𝓥_𝛇=v. This proves every direction in U7.

The contribution with the distinguished label 0 can be written separately in this same collected relation. For the actual blade K, the ordered product e₀e_K has sign u: e₀ is already the first generator when absent, and the leading pair e₀e₀ removes to u when present. Thus U1 gives the equivalent written equality

∂₀ζ_({0}△I)
  =−Σ_(i=1,…,6)χ({i},{i}△I)∂_i ζ_({i}△I).            (U18)

This rearranges the full collected condition; it does not declare the six derivatives independently specifiable. U7 and U18 supply no replacement of these coupled equalities by termwise vanishings. No existence claim for a supplied scalar expression with nonvoid derivatives is needed or made.

4. Classify the stronger universal residual-only balance

Assume U8a for the same retained list 𝛇. By U6 it means

Λ_𝛇(F(W))+h g(𝓥_𝛇,W)=v                              (U19)

for every supplied W. The quadratic source is even and the second summand is odd in the supplied input. To establish those statements directly, selector additivity/negation gives Sel(−W)=−Sel(W), so

F(−W)=(−W)e₀[−Sel(W)]=F(W).

G2 gives g(𝓥_𝛇,−W)=−g(𝓥_𝛇,W). Therefore the same universal condition at the supplied input −W gives

Λ_𝛇(F(W))−h g(𝓥_𝛇,W)=v.                             (U20)

Add U19 and U20 in full. The two paired remainder terms cancel, leaving hΛ_𝛇(F(W))=v; multiplication by k gives Λ_𝛇(F(W))=v. Subtract U20 from U19 in full. The source terms cancel, leaving hh g(𝓥_𝛇,W)=v; multiplication by k and then k gives g(𝓥_𝛇,W)=v. This separation used two actual substitutions into a universal identity, not a polynomial sampling or an inference from separate terms in one equation.

The first conclusion holds for every supplied W. V04-PB-0023 S6 therefore gives

ζ_K=v for K≠I_s,
Λ_𝛇(Y)=c𝓒_(I_s)(Y),       c=ζ_(I_s),
𝓤_𝛇=cΩ_s.                                            (U21)

The second conclusion gives 𝓥_𝛇=v by U7. The word Ω_s contains no e₀, so Ω_s*=Ω_s, and all its derivatives are v. Hence, under the actual U21 identity,

D(𝓤_𝛇*)=D(cΩ_s)=(Dc)Ω_s.

By U7 this is v. The written undoing −Ω_s is available because Ω_s=e₀Ω and the parent's central ΩΩ=−u give Ω_sΩ_s=−u. Thus right multiplication by −Ω_s gives Dc=v. Its full seven-blade presentation is

Dc=Σ_μ ε_μ(∂_μc)e_μ.

These seven blade indices are distinct. Recovering its coefficient at {μ} gives ε_μ∂_μc=v; multiplication by the same sign gives ∂_μc=v at each actual μ. This proves all of U8b. Termwise derivatives are obtained here only after the source theorem has forced the single coefficient representative cΩ_s.

Conversely, retain U8b. V04-PB-0023 S6 gives Λ_𝛇(F(W))=v for every W. Differentiate each actual identity ζ_K=v at K≠I_s, using ∂_μv=v; the remaining derivative is ∂_μc=v by premise. Thus ∂μ𝓤_𝛇=v at every μ, so 𝓥_𝛇=v. U6 now yields U8a for every supplied W. The exported tuple recovery V04-PB-0024 J14, applied to Λ_𝛇(Y)=c𝓒(I_s)(Y), gives j_𝛇=cj, and the residual contraction is the displayed U8c. This establishes both directions without assuming derivative constancy at the outset.

5. Transport the whole list through central multiplication

Definition U2 and V01-PB-0004(C1)'s complete reconstruction give 𝓤_(𝛇^Z)=Zᶜ𝓤_𝛇. U3–U4 then give for every full Y

Λ_𝛇(ZY)=g(𝓤_𝛇,ZY)=g(Zᶜ𝓤_𝛇,Y)=Λ_(𝛇^Z)(Y).

Because Z commutes with each supplied generator and B, e_μZW B=Z(e_μWB). Substitution in the actual tuple definition therefore gives j_𝛇(ZW)=j_(𝛇^Z)(W) component by component. This proves U9 without an n=u or derivative premise.

Differentiate the complete representative equality before collecting terms:

∂_μ𝓤_(𝛇^Z)
  =(∂_μZᶜ)𝓤_𝛇+Zᶜ(∂_μ𝓤_𝛇).

Insert this in U0's definition of 𝓥. The second term's Zᶜ is central, so it passes the individual e_μ. Finite distribution gives

𝓥_(𝛇^Z)
  =−Σ_μ e_μ(∂_μZᶜ)𝓤_𝛇B
     −ZᶜΣ_μ e_μ(∂_μ𝓤_𝛇)B
  =−Σ_μ e_μ(∂_μZᶜ)𝓤_𝛇B+Zᶜ𝓥_𝛇.                   (U22)

No contracted derivative of Zᶜ is moved through 𝓤_𝛇. For each actual μ, V03-PB-0017(C1) gives ∂_μΩ=v and V02-PB-0011(D1) gives derivative compatibility with negation. Thus the full product-rule expansions are

∂_μZ=∂_μa+(∂_μb)Ω,
∂_μZᶜ=∂_μa−(∂_μb)Ω.

Both expressions are central by V03-PB-0017(C2), applied to these actual scalar derivatives. The defined c operation on a two-term central expression therefore gives (∂_μZ)ᶜ=∂_μZᶜ. With precisely these facts retained, U4 and U15 give

j_(𝛇,μ)((∂_μZ)W)
  =ε_μg(𝓤_𝛇,e_μ(∂_μZ)WB)
  =ε_μg((∂_μZᶜ)𝓤_𝛇,e_μWB)
  =−g(e_μ(∂_μZᶜ)𝓤_𝛇B,W).                           (U23)

Also U4 gives g(Zᶜ𝓥_𝛇,W)=g(𝓥_𝛇,ZW)=Drem_𝛇(ZW). Pair U22 with W, use U23 on all seven terms, and apply U5. This proves the second U10 identity. By the original tuple definition, its added sum is

Σ_μ ε_μΛ_𝛇(e_μ(∂_μZ)WB)
  =Λ_𝛇((Σ_μ ε_μe_μ∂_μZ)WB)
  =Λ_𝛇((D(Z))WB).

Only V04-PB-0023(S2)'s finite contraction laws and distribution are used. Combining U6 for 𝛇^Z with U9–U10 gives U11.

For comparison with the full source and residual laws under central multiplication, V03-PB-0017 C5–C6 give

F(ZW)=nF(W),
R(ZW)=ZR(W)+(Z−nu)F(W)+h(D(Z))WB.

Adding their complete contractions yields

Λ_𝛇(F(ZW))+Λ_𝛇(R(ZW))
  =Λ_(𝛇^Z)(F(W))+Λ_(𝛇^Z)(R(W))
       +hΛ_𝛇((D(Z))WB).

The n source occurrence cancels only against the displayed −n occurrence. Substitution in U6 at ZW is exactly U11. Central multiplication of the complete input and transport of the coefficient list are thus two recoveries of the same written identity.

6. Recover the two distinguished lists and the source-annihilating class

For U25, append each actual coefficient term in the definitions of 𝓤, Λ and j. Scalar distribution at that term gives ζ_(mix,K)e_K=a_mix ζ_K e_K+b_mix ζ′K e_K and ζ(mix,K)𝓒_K(Y)=a_mix ζ_K𝓒_K(Y)+b_mix ζ′_K𝓒_K(Y). At an actual paired prefix, the same additive regrouping used in §2 carries the combined accumulation to its two complete accumulations. Both start at v. This proves the first two identities at the end of the fixed list; substituting Y=e_μWB and multiplying by ε_μ proves the tuple identity at each original μ. No derivative was taken in this calculation.

By V01-PB-0004 recovery, the lists 𝛇^[s] and 𝛇^[0] have representatives Ω_s and e₀. Every derivative of those fixed words is v by V02-PB-0011, so their 𝓥 elements are v. V04-PB-0024's exported J14 and J14c identify their tuples, component by component, as j and j^c.

Since Ω_s=e₀Ω and ΩΩ=−u, centrality gives

ZᶜΩ_s=(a−bΩ)Ω_s=aΩ_s+b e₀,
Zᶜe₀=(a−bΩ)e₀=a e₀−bΩ_s.                           (U24)

Coefficient recovery applied to U24 gives the actual transformed lists

(𝛇^[s])^Z=a𝛇^[s]+b𝛇^[0],
(𝛇^[0])^Z=a𝛇^[0]−b𝛇^[s],

where each list equality is entrywise on all of 𝓑. Insert them into U9 and U25 to obtain the two tuple identities U27. Thus the list-linearity step and the recovered list are both explicit.

At each actual μ, apply 04's product rule to those component identities:

∂_μj_μ(ZW)=(∂_μa)j_μ(W)+a∂_μj_μ(W)
             +(∂_μb)j^c_μ(W)+b∂_μj^c_μ(W),
∂_μj^c_μ(ZW)=(∂_μa)j^c_μ(W)+a∂_μj^c_μ(W)
               −(∂_μb)j_μ(W)−b∂_μj_μ(W).

Accumulate all seven rows in the defined order. Finite scalar distribution and additive regrouping give both complete Bal identities in U27.

Finally retain S6 of V04-PB-0023, without a derivative condition. It gives ζ_K=v at every K≠I_s and ζ_(I_s)=c. Thus 𝓤_𝛇=cΩ_s and Λ_𝛇(Y)=c𝓒_(I_s)(Y). Applying the latter at e_μWB, F(W) and R(W), and using J14, gives respectively j_𝛇=cj, Λ_𝛇(F(W))=v and Λ_𝛇(R(W))=c𝓒_(I_s)(R(W)). Differentiate the actual identities ζ_K=v at all K≠I_s; the corresponding entries in Drem vanish by ∂_μv=v. Its sole remaining K occurrence at each μ is

ε_μ(∂_μc)𝓒_(I_s)(e_μWB)=(∂_μc)j_μ(W).

Its complete sum, together with U5, therefore establishes the exact recovery

g(𝓥_𝛇,W)=Drem_𝛇(W)=Σ_μ(∂_μc)j_μ(W).

Substitute all three contractions into U6 to obtain U26. The further premise ∂_μc=v removes every displayed remainder term. Conversely the supplied list concentrated at I_s satisfies S6 and, under those seven identities, has void coefficient derivatives; the same calculation returns precisely the stated tuple and balance. This is both directions of the constant-coefficient class recovery, contained in the larger all-input equivalence U8.

What is not forced

U7 is the exact criterion for this fixed-list construction, allowing varying entries. Its scalar version is the complete coupled list p_I(𝛇)=v. This record does not assert that arbitrary derivative data satisfying those displayed sums are simultaneously realizable by supplied coefficient expressions, nor infer all ∂_μζ_K=v from U7. It does not assert the converse strict nonimplication by introducing an unprovided scalar with nonvoid derivatives. The stronger all-input residual-only identity U8 does force coefficient constancy after the full source class has first been determined.

All universal statements keep the same supplied list while W changes. They do not classify lists chosen afresh as functions of W, nonlinear contractions, every possible finite derivative-sum identity, or identities valid only on actual solutions of R=v. The two substitutions W and −W in the proof of U8 use its all-input premise; a solution-only premise is a different scope.

The pairing transfers and coefficient separation apply to complete supplied inputs. Neither representative restricts their coefficient support. The pairing is used through its recovered coefficient identities and separation on all blades.

U9–U11 move the central multiplier between a complete input and its complete coefficient list. Their algebraic validity does not assert that Z preserves sources or solutions. Those additional questions retain the full native central residual condition. In particular no n=u condition has been silently inserted or cancelled.

What it means

The complete coefficient list can be treated as one written element using the already derived scalar pairing. Exact generator-product signs then move every fixed multiplication to the other input, so the entire varying-list remainder becomes one paired derivative element. The complete blade evaluations determine exactly when that element is void. This supplies the exact condition beneath the sufficient constant-coefficient case.

If the requested universal identity removes both the source and the varying-list remainder, the even quadratic source and the odd derivative remainder separate by two exact substitutions. The existing source-generation theorem and full coefficient recovery then recover precisely the constant multiples of the canonical tuple. Central multiplication uses the same whole representative and the same terms from 04's product rule, recovering both distinguished tuples with their signs and retaining the full source and residual contributions. These specializations are explicit claims of the same general interface.

The complementary family has a complete derivative criterion before aa+bb=u (V05-PB-0039)

LAYER: 10 FORCED BY: V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V01-PB-0008, V02-PB-0011, V02-PB-0014, V03-PB-0019, V04-PB-0029, V07-PB-0043, V04-PB-0031, V05-PB-0038, V06-PB-0040, V07-PB-0042

Declares

Retain 04's finite written operations, h=u+u, k=h⁻¹ with h≠v, and B=e₁e₂. Coefficient recovery is V01-PB-0004's; D,L,F,R are V01-PB-0006's, T,A_X,K_A are V01-PB-0007's, and Gr is V07-PB-0043's operation. Retain the fixed (3,4) expressions I_{34},P_{34,±},W₄,M_t,N_t,K_t,Δ_t of V04-PB-0029 and V04-PB-0031. I,P_± abbreviate those actual pair objects.

Supply actual scalar expressions t,a,b without an additional guard, an aa+bb=u condition or a derivative condition. Define the complete supplied expressions:

V_{ab}=ae₅+be₆,
n_{ab}=aa+bb,
S_σ=k(u+σe₀V_{ab}),
Ξ_σ=S_σΔ_t,       σ∈{u,−u},
n_σ=e₀−σV_{ab}.                                    (UF0)

Both signs are retained. V_{ab} denotes the displayed two-generator expression; the subscript distinguishes it from the factor witness V. n_{ab} abbreviates the displayed scalar expression; it is not assumed equal to u. Every derivative a_μ=∂_μa,b_μ=∂_μb,t_μ=∂_μt belongs to these same supplied expressions.

Θ_t and c_J retain V05-PB-0038's notation, κ_t retains V06-PB-0040(N0)'s definition, Mix retains V02-PB-0014's, and Cmp_C retains V03-PB-0019's. The coordinates below describe the complete expressions UF0; they impose no restriction on Ψ.

What is forced

Without imposing n_{ab}=u the complete factor products are

V_{ab}V_{ab}=−n_{ab},      (e₀V_{ab})(e₀V_{ab})=n_{ab},
S_σS_σ−S_σ=kk(n_{ab}−u),
S_σS_{−σ}=S_{−σ}S_σ=kk(u−n_{ab}),
S_σ+S_{−σ}=u,
T(S_σ)=Gr(S_σ)=S_σ.                                (UF1)

Both S factors commute with e₁,I,P_±,M_t,N_t,K_t and Δ_t. Thus the following entire statements need no hypothesis n_{ab}=u:

T(Ξ_σ)=Ξ_σ,       Ξ_σK_t=v,       Ξ_σ≠v,
Ξ_σ+Ξ_{−σ}=Δ_t.                                   (UF2)

For these actual S factors, idempotence and the void product between opposite signs are each equivalent to n_{ab}=u, by the available h,k alone. The equality n_{ab}=u therefore has a precise algebraic role; it is not needed to obtain UF2.

In V05-PB-0038 CN1–CN7's complete twelve-coefficient presentation, the same Ξ_σ has exactly the recovered coordinates

c_empty=k,       c_05=σka,       c_06=σkb,
every other c_J=v.                                 (UF3)

This identity follows by recovering Θ_t(Ξ_σ), not by selecting a trial support. In particular the parameter recoveries hold without the condition n_{ab}=u:

t=−h𝓒_1(Ξ_σ),
a=σh[𝓒_05(Ξ_σ)−𝓒_012345(Ξ_σ)],
b=σh[𝓒_06(Ξ_σ)−𝓒_012346(Ξ_σ)].                    (UF4)

For every actual μ,

∂_μΞ_σ=v iff t_μ=a_μ=b_μ=v.               (UF5)

The entire contracted derivative has the exact first-stage equivalence

D(Ξ_σ)=v
  iff D(S_σ)=v and D(t)S_σ=v.                     (UF6)

Its complete coefficient criterion, without imposing n_{ab}=u, is the simultaneous collection

a_μ=b_μ=v                  for μ=0,1,2,3,4;
a₅+b₆=v,      b₅−a₆=v;
t₁=t₂=t₃=t₄=v;
t₅=σa t₀,     t₆=σb t₀;
(u−n_{ab})t₀=v.                                (UF7)

Every condition in UF7 is necessary, and together they are sufficient. The final product is retained as written. Neither its scalar factors nor a,b,t or their derivatives are cancelled.

The associated products, with their full scalar remainders, are

n_σn_σ=u−n_{ab},
n_σS_σ=k(u−n_{ab})e₀,
𝓒_{\{0\}}(n_σ)=u≠v.                                (UF8)

Under the last three lines of UF7,

D(t)=t₀n_σ,
D(t)D(t)=v.                                         (UF9)

The second equality follows from the retained product (u−n_{ab})t₀=v, not from an unproved n_σn_σ=v. At an actual instance satisfying UF7, the additional condition t₀≠v makes ∂₀Ξ_σ nonvoid by UF4–UF5, while UF7 gives D(Ξ_σ)=v. Existence of such a scalar instance is not asserted.

The original nonlinear equation and the separate source comparison are exactly

F(Ξ_σ)=v,
R(Ξ_σ)=v iff UF7;

K_A(X)=K_t  ⇒
  A_{X+Ξ_σ}=A_X,
  F(X+Ξ_σ)=F(X),
  R(X+Ξ_σ)−R(X)=hD(Ξ_σ)B.                         (UF10)

The comparison is valid at the displayed actual-factor premise, and under that same premise

R(X+Ξ_σ)=R(X) iff UF7.

A solution replacement additionally needs R(X)=v. Independently, A_(Ξ_σ)=v for every supplied t,a,b; Ξ_σ is a standalone solution only when UF7 holds, and does not provide that base.

For the same supplied t,a,b, the two signs both solve their original equations exactly when

t_μ=v for all seven μ,
a_μ=b_μ=v for μ=0,1,2,3,4,
a₅+b₆=v,      b₅−a₆=v.              (UF11)

Their unique center and half-difference remain

C=kΔ_t,
E=kσe₀V_{ab}Δ_t,
T(C)=C, T(E)=E.                                      (UF12)

and the full center-and-half-difference criterion recovers UF11 as L(C)=L(E)=v. The criterion for their sum alone remains only t_μ=v for all μ; it imposes no a,b derivative condition.

Finally, when the additional actual identity n_{ab}=u is retained, UF7 is equivalent to the following criterion at n_{ab}=u:

a_μ=b_μ=v for all seven μ,
t₁=t₂=t₃=t₄=v,
t₅=σa t₀,     t₆=σb t₀.              (UF13)

Under that same n_{ab}=u condition UF1 gives the complementary idempotent factors, UF8 gives n_σn_σ=v and n_σS_σ=v, and UF11 becomes derivative constancy of all three parameters. The parameter recoveries UF4, individual-derivative criterion UF5, nonvoid coefficients UF2/UF8 and center identities UF12 retain the same expressions at n_{ab}=u; UF13 states the entire stronger derivative criterion.

For the actual same family define H_σ=S_σP_− and d_ab=u−n_{ab}. Its recovered corner and opposite-sign product are

P_−Ξ_σP_−=H_σ,
H_σH_{−σ}=kk d_abP_−.                                     (UF18a)

At an actual supplied scalar r_ab with d_ab r_ab=u, the complete constructed expression

U_(c,σ)=hh r_ab H_{−σ}

satisfies H_σU_(c,σ)=P_−. Under exactly this supplied relation, the whole original equation for Ξ_σ is equivalent to

t_μ=v for all seven μ,
a_μ=b_μ=v for μ=0,1,2,3,4,
a₅+b₆=v,     b₅−a₆=v.                    (UF18b)

Its sufficiency does not use r_ab. Without that supplied undoing, UF7 retains the full product d_ab t₀=v. Independently of r_ab and independently of n_{ab}=u,

UF7 ⇒ κ_t=v,                              (UF19)

with exactly the κ_t of V06-PB-0040 N0/N12.

The forcing

1. Separate the products that require n_{ab}=u from those that do not

Distribute the full two-generator square and use e₅e₆+e₆e₅=v:

V_{ab}V_{ab}=aa e₅e₅+ab(e₅e₆+e₆e₅)+bb e₆e₆
         =−(aa+bb)=−n_{ab}.

Each e₀e_j is the negative of e_je₀, so e₀V_{ab}=−V_{ab}e₀. Hence (e₀V_{ab})(e₀V_{ab})=−e₀e₀V_{ab}V_{ab}=n_{ab}. The full square and opposite-sign product are

S_σS_σ=kk[u+hσe₀V_{ab}+n_{ab}],
S_σS_{−σ}=kk[u−σe₀V_{ab}+σe₀V_{ab}−(e₀V_{ab})(e₀V_{ab})]=kk(u−n_{ab}),
S_{−σ}S_σ=kk[u+σe₀V_{ab}−σe₀V_{ab}−(e₀V_{ab})(e₀V_{ab})]=kk(u−n_{ab}).

The guarded inverse gives hk=u, and 04's scalar commutation gives kh=u. Since S_σ=kkh(u+σe₀V_{ab}) and h=u+u, subtraction yields S_σS_σ−S_σ=kk(n_{ab}−u). Addition gives S_σ+S_{−σ}=kh=u. Multiplying either scalar difference by hh proves both directions of the n_{ab}=u equivalences stated after UF2. No inverse for n_{ab} is used.

For j=5,6, V01-PB-0007's product reversal gives T(e₀e_j)=T(e_j)T(e₀)=−e_je₀=e₀e_j. V07-PB-0043(G1,G2) gives Gr(e₀e_j)=(−e₀)(−e_j)=e₀e_j; both operations fix scalars. This proves their S signs without using the square. The actual exchanges

e₀e_je₁=−e₀e₁e_j=e₁e₀e_j

give commutation with e₁. For each of the two letters 0,j, V04-PB-0031(BC5a) gives commutation past the complete four-letter I because 0,j are outside its index list. Thus e₀e_j commutes with I, then with P_± and W₄=e₁P_+. Distribution through the actual formulas gives commutation with M_t,N_t,K_t,Δ_t and their T images. None of these statements used n_{ab}=u.

Use V04-PB-0029(E1)'s exact T(Δ_t)=Δ_t and Δ_tK_t=v:

T(Ξ_σ)=T(Δ_t)T(S_σ)=Δ_tS_σ=S_σΔ_t,
Ξ_σK_t=S_σ(Δ_tK_t)=v,
Ξ_σ+Ξ_{−σ}=(S_σ+S_{−σ})Δ_t=Δ_t.

This proves the algebraic parts of UF2 independently of idempotence.

2. Recover the full constructed expression inside the complete null interface

Transform the actual expression, using the established commutations and undoings:

Θ_t(Ξ_σ)=T(M_t)S_σT(N_t)P_−N_tM_t
          =S_σP_−
          =[ku+σka e₀e₅+σkb e₀e₆]P_−.

V05-PB-0038(CN2,CN3)'s unique twelve-coefficient recovery now gives UF3. The calculation identifies the complete supplied Ξ_σ in the full corner domain; these coordinates are not asserted for other right-null inputs.

Substitute UF3 into the already proved full coefficient formulas V05-PB-0038 CN6. The complete remaining list is

Index	Coefficient of Ξ_σ
empty	kk(u−tt)
1234	−kk(u+tt)
1	−kt
05	σkk a(u−tt)
06	σkk b(u−tt)
012345	−σkk a(u+tt)
012346	−σkk b(u+tt)
015	σk at
016	σk bt

Every unlisted coefficient is v because its fully recovered c_J is v or it is outside the proved thirty-six-position presentation. The index 1 coefficient gives t after multiplication by −h. At 05 and 012345 their difference is σkk a[(u−tt)+(u+tt)]=σka; multiplication by σh gives a. The same complete difference at 06 and 012346 gives b. This proves UF4 on the full domain before n_{ab}=u.

If Ξ_σ=v, its index 1 recovery makes t=v. Its empty coefficient is then kk and would have to be v. Multiplication by hh gives u=v, which would make h=v contrary to the original guard. Thus Ξ_σ≠v without a condition on n_{ab} or u±tt.

For the individual derivatives, V06-PB-0040 CN8 requires all c_{J,μ}=v and all t_μc_J=v. The actual c_empty=k is derivative-constant and has undoing h, so t_μc_empty=v is exactly t_μ=v. The two further nonzero-form entries have c_{05,μ}=σka_μ and c_{06,μ}=σkb_μ; their voidness is exactly a_μ=b_μ=v by multiplication by σh. Every other entry and its derivative is identically v. These calculations give both directions of UF5 and preserve all actual parameter dependencies.

3. Separate the whole collected derivative by its derived grade parts

V04-PB-0029(E1)'s exact expression Δ_t=P_−−te₁−ttP_+ gives

Ξ_σ=S_σP_−−ttS_σP_+−tS_σe₁.

V04-PB-0031(BC5) gives the void derivatives of P_±; the generator derivatives are supplied void in 04. Applying 04's product rule in every term of the seven-label contraction yields

D(Ξ_σ)=O_d−N_de₁,
O_d=D(S_σ)P_−−D(ttS_σ)P_+,
N_d=D(tS_σ).                                       (UF14)

Every derivative of t,a,b is retained in these three terms. UF1 and V07-PB-0043(G1,G2) give Gr(S_σ)=S_σ and Gr(P_±)=P_±: I has four generator factors, so its four negative images multiply with sign u; Gr fixes scalars. V07-PB-0043 G3's exact derivative law makes D(S_σ),D(ttS_σ),N_d odd. Thus O_d is odd while N_de₁ is even. If the full UF14 is void, apply Gr to obtain −O_d−N_de₁=v. Adding and subtracting these two equations, then multiplying by k, gives O_d=v and N_de₁=v. Right multiplication by −e₁ makes the latter N_d=v. Conversely those two conditions give the original whole derivative void.

Right-multiply O_d=v by P_−. The exact ordered products P_−P_−=P_− and P_+P_−=v recover

D(S_σ)P_−=v.                                      (UF15)

One cannot cancel a general P_−. The complete derivative of this particular S_σ and the recovery needed for UF15 are derived next.

4. Derive every coefficient of D(S_σ) and the permitted recovery

Distribute the two scalar terms of S_σ through all seven original derivative labels. V02-PB-0011(D1) gives void derivatives of k,u,σ, and 04 gives the generator derivatives void. The complete expression is

D(S_σ)=σk[−(a₅+b₆)e₀
   +a₀e₅+b₀e₆
   +(b₅−a₆)e₀e₅e₆
   +a₁e₀e₁e₅+b₁e₀e₁e₆
   +a₂e₀e₂e₅+b₂e₀e₂e₆
   +a₃e₀e₃e₅+b₃e₀e₃e₆
   +a₄e₀e₄e₅+b₄e₀e₄e₆].                (UF16)

The label 0 contributions use e₀e₀=u. For the repeated outside labels the actual products are −e₅e₀e₅=−e₀ and −e₆e₀e₆=−e₀. The two crossed products are −e₅e₀e₆=e₀e₅e₆ and −e₆e₀e₅=−e₀e₅e₆. At m=1,2,3,4 and j=5,6 the distinctness gives −e_me₀e_j=e₀e_me_j. These are all fourteen differentiated contributions, collected at the twelve listed indices; none relies on n_{ab}=u.

Their complete routing under right multiplication by I is

Actual index in UF16	Index after multiplying by I
0	01234
5	12345
6	12346
056	0123456
015	02345
025	01345
035	01245
045	01235
016	02346
026	01346
036	01246
046	01236

V01-PB-0008(P1), applied to the two single-blade inputs e_J and I, has only the output index J△{1,2,3,4}; all other coefficients are void. V01-PB-0004(C1,C4) then gives each actual routed index. Every associated product sign is a scalar sign, but its value cannot affect this disjointness: none of the right-column indices occurs in the left column. Consequently the coefficient of D(S_σ)P_−=k[D(S_σ)−D(S_σ)I] at each left-column position is exactly k times its coefficient in D(S_σ). If UF15 holds, multiply those recovered coefficients by h. All UF16 coefficients are then void, and V01-PB-0004(C1,C4) reconstructs D(S_σ)=v. This is a derived support recovery, not a projection cancelled by assumption.

For any supplied scalar c and expression S, each original derivative obeys

e_μ∂_μ(cS)=e_μ[(∂_μc)S+c(∂_μS)].

The scalar c alone commutes past e_μ; appending all seven signed terms gives D(cS)=D(c)S+cD(S). At c=t and c=tt this is

N_d=D(t)S_σ+tD(S_σ),
D(ttS_σ)=htD(t)S_σ+ttD(S_σ).

We have proved D(Ξ_σ)=v implies D(S_σ)=v and N_d=v, hence D(t)S_σ=v. Conversely those two conditions make N_d and D(ttS_σ) void, hence make O_d and all UF14 void. This proves UF6 in both directions without n_{ab}=u.

Finally, the twelve distinct UF16 indices and the known undoing σh of σk give exactly

D(S_σ)=v
  iff a_μ=b_μ=v for μ=0,1,2,3,4,
      a₅+b₆=v, b₅−a₆=v.             (UF17)

The last two collected coefficients stay as coupled equations. These equations retain the supplied a,b without differentiating an unsupplied identity n_{ab}=u.

5. Resolve the entire scalar-derivative product without cancelling its remainder

Retain t_μ=∂_μt. The entire product is

D(t)S_σ=k[D(t)+σD(t)e₀(ae₅+be₆)].

After removing only the known common factor k, its singleton coefficients are

Index	Complete coefficient after the k factor
0	t₀−σ(at₅+bt₆)
1	−t₁
2	−t₂
3	−t₃
4	−t₄
5	−t₅+σat₀
6	−t₆+σbt₀

The remaining coefficients after that same k factor are σ(bt₅−at₆) at 056, σat_m at 0m5, and σbt_m at 0m6 for m=1,2,3,4. There are no others: these are the same full ordered generator products used in UF16, with the original scalar derivative singleton terms also retained.

If the product is void, singleton recovery first gives

t₁=t₂=t₃=t₄=v,
t₅=σat₀,       t₆=σbt₀.

Its coefficient at 0 then gives

t₀−σ[a(σat₀)+b(σbt₀)]
  =(u−aa−bb)t₀=(u−n_{ab})t₀=v.

This is a remaining condition, not a factor that can be discarded because the conditional equality n_{ab}=u would make it void. Conversely retain exactly these displayed equations. Every 0m5 and 0m6 coefficient is void because t_m=v. The 056 coefficient is

σ[b(σat₀)−a(σbt₀)]=(ba−ab)t₀=v

by scalar commutation. The singleton coefficients are void by the equations themselves, including the retained final product. Thus reconstruction gives D(t)S_σ=v. These are both directions of the last three lines of UF7.

The same conclusion can be kept as a whole product. The recovered t_μ values give D(t)=t₀n_σ. Direct expansion, retaining n_{ab}, gives

n_σn_σ=e₀e₀−σ(e₀V_{ab}+V_{ab}e₀)+V_{ab}V_{ab}=u−n_{ab},
n_σS_σ=k[e₀+σV_{ab}−σV_{ab}−V_{ab}e₀V_{ab}]
         =k(e₀−n_{ab}e₀)=k(u−n_{ab})e₀.

Here V_{ab}e₀V_{ab}=−e₀V_{ab}V_{ab}=n_{ab}e₀. Multiplication by t₀ gives the void product exactly under (u−n_{ab})t₀=v. Also

D(t)D(t)=t₀t₀(u−n_{ab})
         =t₀[(u−n_{ab})t₀]=v.

The full expression n_σ=e₀−σae₅−σbe₆ has coefficient u at the distinct index {0}. V01-PB-0004(C1) recovers it exactly; u is nonvoid because u=v would imply h=v. This proves UF8–UF9 with the full remainder present. Combining this product criterion with UF6 and UF17 proves every part of UF7.

6. Apply aa+bb=u with its exact derivative consequences

Now, and only in this section, retain the additional actual identity n_{ab}=aa+bb=u. UF1 immediately gives both complementary idempotent factors. UF8 gives n_σn_σ=v, n_σS_σ=v, and its same nonvoid e₀ coefficient. The final product in UF7 becomes v at this actual hypothesis.

It remains to determine precisely how n_{ab}=u strengthens UF17. Its first five-label conditions already give a_μ=b_μ=v for μ=0,1,2,3,4. The two coupled UF17 equations give b₆=−a₅ and a₆=b₅. Differentiate the actual equality aa+bb=u at labels 5 and 6, retaining both terms of each scalar product:

v=∂₅(aa+bb)=h(a a₅+b b₅),
v=∂₆(aa+bb)=h(a b₅−b a₅).

Multiply each by k. The two complete combinations are

a(a a₅+b b₅)−b(a b₅−b a₅)
  =aa a₅+ab b₅−ba b₅+bb a₅=a₅,
b(a a₅+b b₅)+a(a b₅−b a₅)
  =ba a₅+bb b₅+aa b₅−ab a₅=b₅.

Their left sides are void, hence a₅=b₅=v. Therefore a₅,b₅,a₆,b₆ are all void. Conversely if all fourteen derivatives are void, UF17 holds directly. This proves the strengthened derivative constancy only at the supplied equality n_{ab}=u, without dividing a or b. Together with the remaining t equations it establishes UF13 exactly.

The original individual-derivative, source, parameter-recovery and nonvoid claims have already been proved in §§1–5 on the larger domain. The additional uses of n_{ab}=u are exactly complementary idempotence, the two bare n_σ null products, elimination of (u−n_{ab})t₀, and the last four a,b derivative identities. All other displayed claims retain their already proved larger scope.

7. Rejoin the original equation, both signs and the center

UF2 and its actual corner coordinates UF3 put Ξ_σ under V06-PB-0040 N18. That claim gives F(Ξ_σ)=v and R(Ξ_σ)=v iff D(Ξ_σ)=v; UF7 is the exact latter criterion. For a separate supplied X with K_A(X)=K_t, apply V07-PB-0042 FC4 to T(Ξ_σ)=Ξ_σ and Ξ_σK_A(X)=v for the active-part and source equalities. Its FC3 at σ=u gives the residual difference. FC4 gives residual equality iff D(Ξ_σ)=v; UF7 supplies precisely the displayed criterion. Its solution corollary FC4s keeps R(X)=v when this comparison is used to produce another solution. Independently, the definition in V01-PB-0007 and UF2 give

A_(Ξ_σ)=k[Ξ_σ−T(Ξ_σ)]=k(Ξ_σ−Ξ_σ)=v.

This proves the active-part statement in UF10 without a base premise.

The additional nonvoid-derivative implication follows by differentiating the actual recovery t=−h𝓒_1(Ξ_σ): if ∂₀Ξ_σ were v, then t₀ would be v. This implication does not construct such a nonconstant instance.

If both signs solve their equations, both derivatives D(Ξ_σ),D(Ξ_{−σ}) are void. Their sum is D(Δ_t), because UF2 is an identity without imposing n_{ab}=u. V04-PB-0029(E2) then makes all seven t derivatives void. Each sign's UF6 also requires D(S_σ)=v, whose exact criterion is UF17. Thus the conditions in UF11 are necessary. Conversely all t derivatives void give D(t)S_σ=v for both signs, and UF17 gives D(S_σ)=v for both signs. UF6 and the full residual criterion prove both solutions. This proves UF11 without cancelling any a,b or n_{ab}.

The two actual sum and difference identities are

Ξ_σ+Ξ_{−σ}=Δ_t,
Ξ_σ−Ξ_{−σ}=σe₀V_{ab}Δ_t.

V03-PB-0019(C2,C6)'s unique center and half-difference give UF12. Each is T-fixed because e₀V_{ab} and Δ_t are T-fixed and commute. V01-PB-0007's T-fixed selector criterion therefore gives Sel(C)=Sel(E)=v. Their sources are void by V01-PB-0006's definition, and V02-PB-0014's definition gives Mix(C,E)=Ce₀Sel(E)+Ee₀Sel(C)=v. The complete pair criterion V03-PB-0019(C11), Cmp_C(E)=v and R(C)=F(E), is therefore exactly L(E)=L(C)=v at these actual expressions. This recovers UF11 through the unrestricted center-and-half-difference interface, without asserting that their separate derivatives vanish.

Finally the sum equation is D(Δ_t)=v iff all t derivatives are void, independently of a,b because the actual sum has already removed their entire contributions. The individual equations retain UF17 in addition. Under n_{ab}=u, §6 makes UF17 exactly complete derivative constancy of a,b, giving the complete paired criterion at n_{ab}=u.

8. Retain the supplied scalar undoing only where it is used

The recovered corner in UF3 is H_σ=S_σP_−. Since S_σ commutes with P_−, UF1 and V04-PB-0031(BC5) give

H_σH_{−σ}=S_σS_{−σ}P_−P_−=kk d_abP_−.

This proves UF18a. If the actual d_ab r_ab=u is additionally supplied, then

H_σ(hh r_ab H_{−σ})=hh r_ab kk d_ab P_−=(hk)(hk)r_ab d_ab P_−=P_−.

The statement supplies the displayed witness only under that relation. To obtain UF18b directly, UF7 has d_ab t₀=v; multiplying by r_ab gives t₀=v. Its two equations t₅=σa t₀ and t₆=σb t₀ then give the two remaining outside t derivatives void, while the four inside ones were already void. The a,b equations are unchanged. Conversely, UF18b makes each line of UF7 true without using r_ab. UF10 therefore proves the exact original-equation equivalence with precisely its stated premise.

Finally substitute UF7's t₁=t₂=t₃=t₄=v, t₅=σat₀ and t₆=σbt₀ directly into V06-PB-0040(N0)'s definition of κ_t. Scalar commutation and σσ=u give

κ_t=t₀t₀−(σat₀)(σat₀)−(σbt₀)(σbt₀)
   =[(u−aa−bb)t₀]t₀=(d_ab t₀)t₀=v.

The last equality is UF7's retained product d_ab t₀=v. This proves UF19 without another scalar undoing or an n_{ab}=u premise; UF9 already states the corresponding whole-product equality D(t)D(t)=v.

What is not forced

UF7 is the exact derivative criterion for the complete supplied expressions Ξ_σ=S_σΔ_t before imposing n_{ab}=u. Its complete corner coordinates were recovered from the full class before the family-specific derivative calculation. No support of Ψ is imposed by these parameterized expressions.

The scalar product (u−n_{ab})t₀=v remains one collected condition; its factors are not split in UF7. The two a,b conditions at labels 5,6 are retained as collected equations; the calculation at aa+bb=u in §6 derives the four individual derivative identities under its explicit additional premise. These retained equations include every algebraic case; they do not select one by a numerical or symbolic trial.

The derived criteria do not supply actual nonconstant scalar realizations. Such expressions would have to obey the original arithmetic, all their derivative relations and derivative commutations together. Nor does this selector-null construction provide a solution with nonvoid active part and factor K_t. That base remains an explicit condition of the comparison.

UF9 is a whole-product consequence at its retained conditions. Its voidness is not replaced by a factor-by-factor alternative.

What it means

The equality aa+bb=u makes the two S factors complementary and strengthens the two coupled a,b derivative equations into four separate ones. Before that condition, the exact criterion retains the full product (u−aa−bb)t₀ together with the coupled a,b equations. The parameter recoveries are identical at both scopes; each extra conclusion keeps the condition that supplies it.

The conditional product H_σU_(c,σ)=P_− in UF18b is an actual corner undoing. Its identification with V07-PB-0045 U0 at H=H_σ is a non-use reference: the present criterion is proved directly from UF7 by the supplied scalar product d_ab r_ab=u.

The local calculation of κ_t in UF19 is also the instance of V06-PB-0040(N12) with c_empty=k: its κ_tk=v is undone by h. This identifies the general consequence without using it in the proof.

The complete corner expression has three recovered second-derivative conditions (V07-PB-0044)

LAYER: 10 FORCED BY: V01-PB-0001, V01-PB-0004, V02-PB-0011, V03-PB-0022, V04-PB-0031, V05-PB-0038, V06-PB-0040

Declares

All expressions use the supplied operations and their retained hypotheses. P_{34,−},P_{34,+} are V04-PB-0031's fixed (3,4) pair; D retains V01-PB-0006's definition. The supplied scalar t has its actual derivatives t_μ=∂_μt. The expressions q_t=D(t) and κ_t=Σ_μ ε_μt_μt_μ are V06-PB-0040(N0), with ε₀=u and εᵢ=−u at the other six labels.

Sq is exactly V03-PB-0022's signed seven-term second derivative. For an actual full corner H=P_{34,−}HP_{34,−}, retain V05-PB-0038's expressions

Y=(u−te₁)H,
Z=Y−tYe₁.                                          (SD1)

No T condition is imposed on H. Put H_μ=∂_μH and define

C_t(H):=(Sq(t))H+hΣ_(μ=0,…,6) ε_μt_μH_μ.            (SD2)

The coefficient f_{r,J}=𝓒_J(F_(out,r)) refers to the full encoder H=Enc(F_out) of V05-PB-0038(N1,N8,N16), with r in (0,2,3,4) and J in its complete eight-entry outside list 𝒪. These are actual recovered scalar coefficients. When T(H)=H is added, c_J instead denotes that parent's twelve-coordinate sublist. No coefficient is declared invertible.

What is forced

The complete expression SD1 expands to

Z=H−te₁H−tHe₁+tt e₁He₁.                              (SD1e)

For the full corner family SD1, the four constant corners give

P_{34,−}ZP_{34,−}=H,
P_{34,+}ZP_{34,−}=−te₁H,
e₁P_{34,+}ZP_{34,−}=tH,
P_{34,−}ZP_{34,+}=−tHe₁,
P_{34,−}ZP_{34,+}e₁=tH,
P_{34,+}ZP_{34,+}=tt e₁He₁,
e₁P_{34,+}ZP_{34,+}e₁=ttH.                          (SD4)

The complete second derivative is

Sq(Z)=Sq(H)−e₁Sq(tH)−Sq(tH)e₁+e₁Sq(ttH)e₁.          (SD5)

At the same full corner premise, each output is recovered by

P_{34,−}Sq(Z)P_{34,−}=Sq(H),
e₁P_{34,+}Sq(Z)P_{34,−}=Sq(tH),
P_{34,−}Sq(Z)P_{34,+}e₁=Sq(tH),
e₁P_{34,+}Sq(Z)P_{34,+}e₁=Sq(ttH).                 (SD5a)

These recoveries give the exact criterion

Sq(Z)=v iff Sq(H)=Sq(tH)=Sq(ttH)=v.                   (SD6)

For arbitrary supplied H, without a corner premise, the product identities are

Sq(tH)=tSq(H)+C_t(H),
Sq(ttH)=ttSq(H)+ht C_t(H)+hκ_tH,                     (SD7)
D(q_tH)+q_tD(H)=C_t(H).                              (SD10)

Consequently, on the full corner family SD1,

Sq(Z)=v iff Sq(H)=v, C_t(H)=v, κ_tH=v.                (SD8)

For the full encoder, its last condition has precisely the coordinate form

κ_tH=v iff κ_t f_{r,J}=v for every r∈(0,2,3,4), J∈𝒪. (SD9)

Under the additional T-fixed condition it is equivalently κ_tc_J=v at each of the twelve inherited positions. If κ_tH=v is retained and one actual recovered coefficient f_{r,J} has an actually supplied scalar s with f_{r,J}s=u, then κ_t=v. Under T(H)=H the same conclusion requires κ_tH=v and the actual c_Js=u at one of its twelve positions. Both the void-product premise and the particular undoing remain hypotheses of these implications.

For this same family, D(Z)=v implies each of the three conditions in SD8. They follow directly from the full first-derivative pair D(Y)=v, q_tY=v in V06-PB-0040(N5), as well as from Sq(Z)=D(D(Z)) and SD8. The first-derivative criterion remains that complete pair. (SD15)

The forcing

1. Apply the recovered constant corners to the whole expression

Distribute the two products in Z=Y−tYe₁ with Y=(u−te₁)H:

Z=(H−te₁H)−t(H−te₁H)e₁
 =H−te₁H−tHe₁+tt e₁He₁.

This proves SD1e. Retain H=P_{34,−}HP_{34,−}. Applying V04-PB-0031(BC5)'s P_{34,−}P_{34,−}=P_{34,−} and the two void opposite-sign products on the two sides of this equality gives

P_{34,−}H=HP_{34,−}=H,
P_{34,+}H=HP_{34,+}=v.

V04-PB-0031(BC5a)'s e₁ exchange and e₁e₁=−u give

e₁H=P_{34,+}e₁H,
He₁=He₁P_{34,+},
e₁He₁=P_{34,+}e₁He₁P_{34,+}.

Thus the four terms of Z in SD1e occupy its four indicated corners, respectively. Multiplying the complete expression by each pair of actual P_{34,±} factors retains that term and removes only the products containing P_{34,+}P_{34,−} or its reverse. These are void by BC5. This gives all four corner values in SD4. Left multiplication of −te₁H by e₁ gives tH; right multiplication of −tHe₁ gives the same tH. Multiplication of tt e₁He₁ on both sides by e₁ gives ttH because its two adjacent e₁e₁ factors both equal −u. This proves every recovery in SD4 without undoing t or H.

2. Pass the actual second derivative through the constant products

For each of the seven supplied labels μ, two applications of 04's product rule and V04-PB-0031(BC5)'s ∂μP{34,±}=v give

∂_μ(P_{34,−}WP_{34,−})=P_{34,−}(∂_μW)P_{34,−},
∂_μ∂_μ(P_{34,−}WP_{34,−})=P_{34,−}(∂_μ∂_μW)P_{34,−}.

The same two displayed product-rule applications, with the actual left and right factors replaced, give the identities for the other three corners and for an exterior e₁ on either side. Adding all seven signed expressions defining Sq therefore commutes Sq with those constant left and right factors. Applying this to all four terms of SD1e gives SD5.

The actual identities H=P_{34,−}HP_{34,−}, tH=P_{34,−}(tH)P_{34,−}, and ttH=P_{34,−}(ttH)P_{34,−} show that their Sq outputs are again corners. Hence SD5's four products recover

P_{34,−}Sq(Z)P_{34,−}=Sq(H),
e₁P_{34,+}Sq(Z)P_{34,−}=Sq(tH),
P_{34,−}Sq(Z)P_{34,+}e₁=Sq(tH),
e₁P_{34,+}Sq(Z)P_{34,+}e₁=Sq(ttH).

If Sq(Z)=v all three outputs are void. Conversely if all three outputs are void, substitution into SD5 gives Sq(Z)=v. This proves SD6 in both directions. No equation for an individual second derivative is inferred from Sq.

3. Keep every term in the two product differentiations

For each actual μ, with t_{μμ}=∂_μ∂μt and H{μμ}=∂_μ∂_μH, the complete expressions are

∂_μ(tH)=t_μH+tH_μ,
∂_μ∂_μ(tH)=t_{μμ}H+t_μH_μ+t_μH_μ+tH_{μμ}
             =t_{μμ}H+h t_μH_μ+tH_{μμ}.

Multiply by each actual ε_μ and add the seven equalities. Move only scalar factors. The result is the first SD7 equality, by the definition of C_t. The actual scalar product tt satisfies

∂_μ(tt)=t_μt+tt_μ=ht t_μ,
∂_μ∂_μ(tt)=h t_μt_μ+ht t_{μμ}.

Use these in the just-written two-factor formula with tt replacing t:

∂_μ∂_μ(ttH)=ttH_{μμ}+hh t t_μH_μ
                        +(h t_μt_μ+ht t_{μμ})H.

Add its seven signed instances. The first term is ttSq(H); the terms with ht collect to ht[(Sq(t))H+hΣ_μ ε_μt_μH_μ]; the remaining sum is hκ_tH. This is the second SD7 equality. No derivative of t or H has been deleted.

If the three conditions of SD6 hold, SD7's first equality gives C_t(H)=v. Its second then gives hκ_tH=v. The guarded inverse gives hk=u and scalar commutation gives kh=u, so multiplication by k gives κ_tH=v. Conversely Sq(H)=C_t(H)=κ_tH=v makes both product outputs of SD7 void. This proves SD8 through SD6 without cancelling any of their factors.

4. Recover every coefficient of the remaining whole product

V05-PB-0038(N8,N16) gives the unique complete encoder and the recovery f_{r,J}=h𝓒_{J∪R_r}(H). Scalar coefficient recovery in V01-PB-0004(C3) gives

h𝓒_{J∪R_r}(κ_tH)=κ_t f_{r,J}.

Thus κ_tH=v implies every displayed product void. Conversely, if all of them are void, the complete encoder reconstruction of κ_tH has only void coefficients, so κ_tH=v. These are all thirty-two coordinates, proving SD9. Its T-fixed form follows by substituting the entire twelve-position list from V05-PB-0038(N2), whose remaining encoder coefficients are proved void there. If f_{r,J}s=u is an actual supplied relation, then

(κ_t f_{r,J})s=κ_t(f_{r,J}s)=κ_t.

The left side is void under its product condition, so κ_t=v. The same calculation applies to c_J. Neither nonvoidness alone nor a further scalar-domain assumption enters this argument.

5. Obtain the mixed derivative identity from the existing singleton products

V03-PB-0022(S8–S9) gives the equal-label and distinct-label products for the signed singleton factors b_{{μ}}=ε_μe_μ. Since q_t=Σ_ν b_{{ν}}t_ν, retain all seven terms to obtain, for each actual μ,

b_{{μ}}q_t+q_tb_{{μ}}
  =Σ_ν (b_{{μ}}b_{{ν}}+b_{{ν}}b_{{μ}})t_ν
  =hε_μt_μu.

For ν≠μ the parent establishes that particular paired product as void; for ν=μ its two terms give hε_μu. V02-PB-0011(D7p) gives the complete product derivative, and V03-PB-0022(S1) gives D(q_t)=Sq(t). Therefore

D(q_tH)+q_tD(H)
  =(Sq(t))H+Σ_μ(b_{{μ}}q_t+q_tb_{{μ}})H_μ
  =(Sq(t))H+hΣ_μ ε_μt_μH_μ=C_t(H).

This proves SD10.

6. Recover the same consequences from the first-derivative pair

Retain D(Z)=v and apply V06-PB-0040(N5): D(Y)=v and q_tY=v. V02-PB-0011(D1) makes every derivative of v void, hence D(v)=v. Applying D to the first equality and using V03-PB-0022(S1) gives Sq(Y)=v. V05-PB-0038(CA1) recovers H=P_{34,−}Y, so the constant-product calculation of section 2 gives Sq(H)=v.

V06-PB-0040(N12) directly gives κ_tH=v for this recovered corner under the same D(Z)=v premise. Finally SD10 at Y gives

C_t(Y)=D(q_tY)+q_tD(Y)=D(v)+q_tv=v.

The definition of C_t uses only scalar coefficients and the individual derivatives, so P_{34,−}C_t(Y)=C_t(P_{34,−}Y)=C_t(H). It is therefore void as well. This proves all three implications in SD15 directly. The alternative Sq route is D(D(Z))=D(v)=v followed by SD8; both routes concern the same actual t,H.

What is not forced

SD6 and SD8 are equivalent criteria for Sq(Z), with the full corner hypothesis retained. SD15 is an implication from the first-derivative criterion to those conditions. No converse to D(Z)=v is asserted here. The scalar products κ_tf_{r,J}=v remain complete products unless their particular actual undoing relations are supplied. There is no corner-undoing hypothesis in this record. All derivative expressions belong to the same supplied t,H; no independent realization of their derivative data is asserted.

What it means

The complete expression contains H, tH and ttH in recoverable constant corners. Its second derivative therefore has three recovered conditions, with every product derivative retained. The mixed derivative contribution and the scalar κ_t acting on the whole corner also follow directly from the full first-derivative pair. The identities relate these exact consequences without replacing the first-derivative equation by them.

The direct corner recovery SD4 is the instance of V04-PB-0031(BC5a)'s unrestricted four-corner identity at the complete expression SD1e; its proof here multiplies that expression by the four actual pairs.

With an actual corner undoing, the derivative equation has seven scalar conditions and two coupled corner equations (V07-PB-0045)

LAYER: 10 FORCED BY: V01-PB-0001, V01-PB-0004, V04-PB-0029, V04-PB-0031, V05-PB-0038, V06-PB-0040

Declares

Retain the actual fixed pair, source factor K_t, and whole corner construction of V05-PB-0038. For a supplied scalar t and a supplied H satisfying the actual corner condition H=P_{34,−}HP_{34,−}, define

Y=(u−te₁)H,
Z=Y−tYe₁.

No T condition is imposed. T retains V01-PB-0007's definition; D and R retain V01-PB-0006's. The expressions D_out, D_+, D_−, q_out, b_± and t_μ=∂_μt are exactly V06-PB-0040(N0). In particular every derivative refers to the same actual supplied t,H.

Where explicitly retained below, U_c is an actual supplied expression satisfying

HU_c=P_{34,−}.                                          (U0)

There is no scalar inverse or derivative rule for U_c in this premise; U_c is not required to be a corner. The statement does not supply U_c for an arbitrary H.

What is forced

The complete expression has the exact product form

Z=(u−te₁)H(u−te₁).                                    (U0e)

For every supplied corner H and t satisfying U0, the complete first-derivative criterion is

D(Z)=v iff
   t_μ=v for all seven original labels μ,
   D_out(H)=tD_+(H),
   D_−(H)=tD_out(H).                                   (U1)

The right-to-left implication holds without supplying U_c. The left-to-right implication is stated under U0. The two remaining equations concern the entire same H and keep their collected terms.

At H=P_{34,−}, V05-PB-0038 CA4 gives the actual identity Z=Δ_t, and V04-PB-0031 BC5 supplies U0 with U_c=P_{34,−}. The derivative criterion at this same expression is V04-PB-0029 E2:

D(Δ_t)=v iff every t_μ=v.                            (U2)

Both coupled equations of U1 have void sides at this derivative-constant corner, so U2 is exactly the already proved E2 specialization.

At the additional condition T(H)=H, V05-PB-0038 CN2 gives T(Z)=Z and its twelve-coordinate expression; V06-PB-0040 N18 then gives

R(Z)=v iff D(Z)=v.                                      (U3)

Consequently U1 is also the complete original-equation criterion at both its retained corner-undoing premise U0 and that retained T sign. No second-derivative condition is used in U1–U3.

The forcing

1. Apply the actual undoing to the two complete product equations

Distribution, with the scalar t commuted through the supplied factors, gives

Z=Y−tYe₁=Y(u−te₁)=(u−te₁)H(u−te₁).

This proves U0e. Under the corner premise, V06-PB-0040(N7) gives the exact equivalence between D(Z)=v and its four whole corner equations. Retain D(Z)=v and the supplied HU_c=P_{34,−}. Right-multiply N7's two product equations by that same U_c, preserving order:

(q_out−tb_+)HU_c=(q_out−tb_+)P_{34,−}=v,
(b_−−tq_out)HU_c=(b_−−tq_out)P_{34,−}=v.               (U4)

This is the only use of U_c. No derivative of it occurs.

2. Recover the seven scalar derivatives through the native corner encoder

The first expression in U4 is exactly V05-PB-0038's full encoder at the following complete outside tuple F_out:

F_(out,0)=q_out−t t₁,
F_(out,2)=−t t₂,
F_(out,3)=−t t₃,
F_(out,4)=−t t₄,
(q_out−tb_+)P_{34,−}=Enc(F_out).

These are written products of the actual t and its actual derivatives. N8/N16 of that record therefore recovers F_(out,0)=v, including all its outside coefficients. The full F_(out,0) is

−t t₁+t₀e₀−t₅e₅−t₆e₆.

Its four indices empty,0,5,6 are distinct; V01-PB-0004(C1)'s coefficient recovery gives t₀=v, t₅=v and t₆=v. The additional product t t₁=v may also be recovered, but is not divided or used in place of a derivative equality. Thus q_out=v.

Substitute that proved whole equality into the second U4 expression. It becomes

b_−P_{34,−}
  =[t₁−B t₂−e₁e₃ t₃−e₁e₄ t₄]P_{34,−}
  =Enc((t₁,−t₂,−t₃,−t₄))=v.

The same complete encoder recovery now yields t₁=t₂=t₃=t₄=v. Together the two actual recoveries give every t_μ=v. No scalar t or coordinate of H was cancelled, and no extra operation separating grades was needed.

3. Retain the two remaining coupled equations and prove the converse

With all seven t_μ=v proved in §2, V06-PB-0040 N13 applies to this actual corner H and its full Z. It equates N7, and hence D(Z)=v, with the two collected equations

D_out(H)=tD_+(H), D_−(H)=tD_out(H).

This proves U1's remaining necessity. Conversely, with those two equations and every t_μ=v retained, the same N13 equivalence gives D(Z)=v without a U_c premise. U_c was used only in §1's two right products.

For U2, V04-PB-0031(BC5) supplies U0 at H=U_c=P_{34,−}, and V05-PB-0038(CA4) identifies Z=Δ_t. BC5 also gives ∂μP{34,−}=v at all seven labels. The displayed definitions therefore give D_out(P_{34,−})=D_+(P_{34,−})=D_−(P_{34,−})=v, so both coupled equations have void sides. V04-PB-0029(E2) at this identical Δ_t is exactly the criterion U2.

Finally, under T(H)=H the complete Z has the retained T-fixed sign by V05-PB-0038 CN2. V06-PB-0040 N18 supplies U3 directly, so the same first-derivative criterion applies to the original residual at that additional sign. No independent source comparison is re-derived here.

What is not forced

U0 remains an actual supplied relation except in specializations, such as U2, that construct it explicitly. U1’s necessity retains the actual relation HU_c=P_{34,−}; its sufficiency has no U_c premise. Its use fixes the derivatives of t; the equations on H remain the two coupled expressions of U1.

The proof treats identities of actual expressions and their original derivatives. It constructs neither nonconstant scalar realizations nor arbitrary derivative data. The full residual criterion U3 retains T(H)=H; it is not a statement about a different sign or an unrestricted right-null input.

What it means

For a supplied corner H with an actual right undoing HU_c=P_{34,−}, the condition D(Z)=v gives the two retained product equations. Their complete encoder recovery makes all seven derivatives of t void. The same coordinate recovery therefore provides the sharper criterion without an additional grade separation. What remains is exactly two whole coupled derivative equations on H.

Twelve internal relabellings carry the complete fixed-pair chain with every blade sign (V07-PB-0046)

LAYER: 11 FORCED BY: V01-PB-0000, V01-PB-0001, V01-PB-0003, V01-PB-0004, V01-PB-0006, V01-PB-0007, V03-PB-0022, V04-PB-0026, V04-PB-0029, V04-PB-0031, V05-PB-0038, V05-PB-0039, V06-PB-0040, V07-PB-0043, V07-PB-0044, V07-PB-0045

Declares

Use 04's whole-statement relabelling rule. Each map π is defined with π(0)=0, π(1)=1 and π(2)=2, and with its other four values in the table below. Both signs ω∈{u,−u} and all six pairs (i,j) from V04-PB-0031 are retained. Let (ℓ,m) be the increasing pair in {3,4,5,6} outside {i,j}. The last column defines the finite pair list ℐ_π; 34 there means (3,4), not a product.

Label	(π(3),π(4),π(5),π(6))	Pair list ℐ_π
34+	(3,4,5,6)	none
34−	(4,3,5,6)	34
35+	(3,5,4,6)	45
35−	(5,3,4,6)	34,35
36+	(3,6,4,5)	45,46
36−	(6,3,4,5)	34,35,36
45+	(4,5,3,6)	35,45
45−	(5,4,3,6)	34,35,45
46+	(4,6,3,5)	35,45,46
46−	(6,4,3,5)	34,35,36,45
56+	(5,6,3,4)	35,36,45,46
56−	(6,5,3,4)	34,35,36,45,46

The label ij+ uses ω=u and ij− uses ω=−u. For a blade index J, πJ means the increasing image index set. Define s_π(J) as the product of −u over the displayed list ℐ_π restricted to pairs whose two entries belong to J; the empty product is u.

ρ_π denotes the corresponding whole written-expression relabelling. It fixes u,v and the arithmetic operations, sends e_μ to e_(π(μ)), and sends each derivative label μ to π(μ). Every supplied scalar expression, derivative rule, guard and identity in a conditional application is transported with it. Use t^π=ρ_π(t), a^π=ρ_π(a), b^π=ρ_π(b), and X^π=ρ_π(X). No invariance of supplied scalar data under π is presumed.

D,F,R retain V01-PB-0006's definitions; T,A,K_A retain V01-PB-0007's; Sq retains V03-PB-0022's; Γ retains V04-PB-0026's; Gr retains V07-PB-0043's; and 𝓒_J retains V01-PB-0004's. The pair expressions I_{ij},P_{ij,±} are V04-PB-0031's; the fixed-pair M_t,N_t,W₄,K_t,Δ_t are V04-PB-0029's. The index list 𝒥 and s_J,λ_J,ξ_J,M_J are V05-PB-0038's declared table.

The superscript π on an existing object denotes its whole relabelled image. The additional label ord,π used below names an increasing-blade rewriting with its explicitly retained sign. Neither denotes an independently selected factor or solution.

What is forced

Each displayed π is a permutation fixing 0,1,2, hence fixes B=e₁e₂. Its list ℐ_π is exactly the pairs (a,b), a<b, whose images satisfy π(a)>π(b). The positive label gives π(3)=i,π(4)=j, and the negative label gives π(3)=j,π(4)=i; both give π(5)=ℓ,π(6)=m. (RT0p)

The whole images of the fixed-pair expressions are

I^π=ω I_{ij},
P_+^π=P_{ij,ω},     P_−^π=P_{ij,−ω},
W₄^π=e₁P_{ij,ω},
M_t^π=u+t^πW₄^π,   N_t^π=u−t^πW₄^π,
K_t^π=P_{ij,ω}+ωt^π e₁I_{ij}+(t^πt^π)P_{ij,−ω},
Δ_t^π=P_{ij,−ω}−t^πe₁−(t^πt^π)P_{ij,ω}.             (RT0)

The complete increasing-blade sign table is the following; +/− means u/−u.

J∩{3,4,5,6}	34+	34−	35+	35−	36+	36−	45+	45−	46+	46−	56+	56−
empty	+	+	+	+	+	+	+	+	+	+	+	+
3	+	+	+	+	+	+	+	+	+	+	+	+
4	+	+	+	+	+	+	+	+	+	+	+	+
5	+	+	+	+	+	+	+	+	+	+	+	+
6	+	+	+	+	+	+	+	+	+	+	+	+
34	+	−	+	−	+	−	+	−	+	−	+	−
35	+	+	+	−	+	−	−	−	−	−	−	−
36	+	+	+	+	+	−	+	+	+	−	−	−
45	+	+	−	+	−	+	−	−	−	−	−	−
46	+	+	+	+	−	+	+	+	−	+	−	−
56	+	+	+	+	+	+	+	+	+	+	+	+
345	+	−	−	+	−	+	+	−	+	−	+	−
346	+	−	+	−	−	+	+	−	−	+	+	−
356	+	+	+	−	+	+	−	−	−	+	+	+
456	+	+	−	+	+	+	−	−	+	−	+	+
3456	+	−	−	+	+	−	+	−	−	+	+	−

For every one of the full 128 blade indices, that table gives

ρ_π(e_J)=s_π(J)e_(πJ),
𝓒_(πJ)(X^π)=s_π(J)ρ_π(𝓒_J(X)).                     (RT1)

In RT1, ρ_π(𝓒_J(X)) is the whole transported expression whose scalar value is the recovered coefficient. It is not identified by notation alone with the increasing-blade extractor 𝓒_(πJ): the displayed s_π(J) is required. Every other transported coefficient assertion uses this same conversion. The relabelling has the inverse obtained by the inverse permutation, and carries each written product in its original order. It commutes with reversion, star, T, Γ and Gr. Its derivative and source identities are

ρ_π(∂_μX)=∂_(π(μ))X^π,
ρ_π(D(X))=D(X^π),       ρ_π(Sq(X))=Sq(X^π),
ρ_π(A_X)=A_(X^π),       ρ_π(K_A(X))=K_A(X^π),
ρ_π(F(X))=F(X^π),       ρ_π(R(X))=R(X^π).              (RT2)

Thus R(X)=v and all its supplied conditions carry to R(X^π)=v and precisely those transported conditions. No scalar derivative is identified with an independently chosen one.

The entire algebra and derivative chain of V05-PB-0038, V06-PB-0040, V05-PB-0039, V07-PB-0044 and V07-PB-0045 is carried by each of these twelve actual substitutions. In particular, its unrestricted right-null class is

ZK_t^π=v iff
   Z=[Enc_π(F_out)+e₁Enc_π(G_out)](u−t^πe₁),                  (RT3)

where Enc_π is the same encoder with outside generators e₀,e_ℓ,e_m, ordered inside factors u,B,e₁e_(π(3)),e₁e_(π(4)), and final P_{ij,−ω}. Both tuples are full four-by-eight outside tuples. Its recovered corners are

H=P_{ij,−ω}ZP_{ij,−ω},
G_c=−e₁P_{ij,ω}ZP_{ij,−ω}.

The T-fixed class is precisely

Z=(u−t^πe₁)H(u−t^πe₁),
T(H)=H,   H=P_{ij,−ω}HP_{ij,−ω}.                      (RT4)

Its complete twelve-position list is π𝒥, with no omitted image. For J∈𝒥 define

E_(πJ)^{ord,π}=T(N_t^π)e_(πJ)P_{ij,−ω}N_t^π.

Then ρ_π(E_J(t))=s_π(J)E_(πJ)^{ord,π}. The full transported three-coefficient identity is

𝓒_(πJ)(E_(πJ)^{ord,π})=k(u−λ_J t^πt^π),
𝓒_(πJ′)(E_(πJ)^{ord,π})=−s_Js_π(J)s_π(J′)k(u+λ_J t^πt^π),
𝓒_(πM_J)(E_(πJ)^{ord,π})=−ξ_Js_π(J)s_π(M_J)t^π,               (RT5)

with every other coefficient void. All thirty-six image positions remain distinct. If Z=ρ_π(Z_c), its coefficient on E_(πJ)^{ord,π} is c_(πJ)^{ord,π}=s_π(J)ρ_π(c_J), and it recovers from the whole Z by

c_(πJ)^{ord,π}=𝓒_(πJ)(Z)−s_Js_π(J)s_π(J′)𝓒_(πJ′)(Z).         (RT6)

The complete derivative collections are

D_out^π=e₀∂₀−e_ℓ∂_ℓ−e_m∂_m,
D_±^π=∂₁±[B∂₂+e₁e_i∂_i+e₁e_j∂_j].                (RT7)

Here each operator acts on the complete following expression. The four-unit multiplication and tuple rows carry literally with their ordered inside factors. Every condition in RD2 and N7/N11 of V06-PB-0040 is transported in that order, including every derivative of t^π times a corner. For the corner-built expression Z=(u−t^πe₁)H(u−t^πe₁), with H=P_{ij,−ω}HP_{ij,−ω}, retain the actual supplied relation HU_c=P_{ij,−ω}. Under precisely these conditions the full derivative criterion is

D(Z)=v iff
   ∂_νt^π=v for all seven ν,
   D_out^π(H)=t^πD_+^π(H),
   D_−^π(H)=t^πD_out^π(H).                            (RT8)

The right-to-left implication of RT8 holds without supplying U_c, exactly as in V07-PB-0045 U1. At T(H)=H this is the original-residual criterion by RT2 and V06-PB-0040 N18. The transported Sq criterion keeps its separate scope: it is the criterion for Sq(Z), and is used as a consequence of D(Z)=v wherever that implication was proved.

For the complete complementary family, both σ∈{u,−u} are retained independently of the core sign ω:

S_σ^π=k[u+σe₀(a^πe_ℓ+b^πe_m)],
Ξ_σ^π=S_σ^πΔ_t^π.

Its original equation is exactly the transported full criterion

∂_νa^π=∂_νb^π=v       for ν in {0,1,2,i,j},
∂_ℓa^π+∂_mb^π=v,     ∂_ℓb^π−∂_ma^π=v,
∂_νt^π=v             for ν in {1,2,i,j},
∂_ℓt^π=σa^π∂₀t^π,    ∂_mt^π=σb^π∂₀t^π,
[u−a^πa^π−b^πb^π]∂₀t^π=v.                            (RT9)

Its aa+bb=u and supplied-undoing refinements retain respectively a^πa^π+b^πb^π=u and the actual [u−a^πa^π−b^πb^π]r_ab^π=u. Neither is inserted into RT9 without that supplied condition.

The forcing

1. Write every allowed permutation and its complete signs

The six increasing pair choices are exactly V04-PB-0031's index. For each, the complement has two increasing entries ℓ,m. Assigning (π(3),π(4))=(i,j) or (j,i), and (π(5),π(6))=(ℓ,m), uses each of 3,4,5,6 exactly once. Hence every row is a permutation and has an actual inverse. Each fixes 0,1,2, so fixes the ordered B declaration exactly. These twelve cases need no relabelling of that declaration.

The pair list ℐ_π is verified by comparing the six actual ordered pairs of positions 34,35,36,45,46,56 in each displayed four-entry image. For example, the 46− image (6,4,3,5) puts precisely 34,35,36,45 in decreasing order; its remaining pairs 46 and 56 stay increasing. Every such result is displayed in the same column, including the empty list for 34+. This proves RT0p.

For the whole increasing word e_J, relabel its actual generators. Labels 0,1,2 remain below all images of 3,4,5,6 and keep their order, so they cause no new exchange. Use V01-PB-0003(3)'s actual sorting route on the remaining image letters: when the next letter is inserted, it moves left across precisely the preceding letters with larger images. Each listed reversed pair contained in J is crossed once at its later letter, and no other pair is crossed. Thus this written route uses one exchange for exactly each listed reversed pair whose two original entries lie in J. Each exchange contributes −u by V01-PB-0003; every pair of such signs multiplies to u by V01-PB-0001. The total word sign is therefore exactly the defined s_π(J):

s_π(J)=the product of −u over the displayed reversed pairs
       whose two entries belong to J∩{3,4,5,6}.

The sixteen subsets exhaust the four-label part of every blade. For each empty or singleton subset no pair qualifies, so its sign is u. For each of the six two-label subsets, its own reversed-pair-column membership determines its displayed sign. For the four triple subsets, the complete products are respectively the signs at (34,35,45), (34,36,46), (35,36,56), and (45,46,56). For the four-label subset the product uses all six pairs. Applying precisely those finite products to each of the twelve written columns gives every entry of RT1's table. No spatial square or repeated generator occurs in this sorting.

Apply that actual blade identity to X=Σ_J𝓒_J(X)e_J. Since the image-index map J↦πJ is a bijection of the full finite index list, exactly one original coefficient reaches each output position. V01-PB-0004(C1,C4)'s full coefficient recovery proves RT1's second identity. The inverse permutation gives the two-sided expression recovery.

2. Preserve every native operation and its conditions

04's relabelling clause carries each written rule instance, including arithmetic and the ordered B declaration. The displayed permutations preserve generator squares, exchanges, and the distinction between e₀ and the six spatial star images. Therefore the complete reversal and star of every relabelled written product are exactly the relabelled reversal and star. Their composite T, and the definitions Γ(X)=e₀T(X)e₀ and Gr(X)=e₀X*e₀, commute with ρ_π because e₀ is fixed.

The derivative-label identity in RT2 is the whole relabelling of the written derivative operation, with its rules and premises carried together. Consequently

ρ_π(D(X))
  =e₀∂₀X^π−Σ_(μ=1,…,6)e_(π(μ))∂_(π(μ))X^π
  =D(X^π).

Every spatial derivative label occurs exactly once, so the second equality is only the finite regrouping permitted by V01-PB-0001. The same seven-term regrouping for the repeated derivatives gives ρ_π(Sq(X))=Sq(X^π). It retains each derivative's actual argument and order; the existing derivative-commutation rules also transport with the whole statement.

The definitions A_X=k(X−T(X)), K_A(X)=e₀Ã_X, F(X)=Xe₀(X̃−X*), and R(X)=hD(X)B−F(X) now give the remaining RT2 identities by direct substitution. The nonvoid guard on h is unchanged; any other supplied scalar guard, equation or undoing is its actual relabelled premise. This proves RT2 before using a conditional class.

3. Carry the complete corner and its coefficients

Since π fixes 1,2,

ρ_π(I_{34})=e₁e₂e_(π(3))e_(π(4))=ωI_{ij}.

At the negative core sign the two final letters exchange once, and at the positive sign they are already increasing. Applying the actual P definitions gives P_+^π=P_{ij,ω}, P_−^π=P_{ij,−ω}. The factor and null-expression identities in RT0 then follow by applying ρ_π to every term of the corresponding complete identities of V04-PB-0029(E1).

Apply the same map to V05-PB-0038's unrestricted CA1–CA2. Its two actual corner products become precisely RT3's recoveries, and its full Enc becomes the stated Enc_π. Completeness and uniqueness transport in both directions because the inverse relabelling reverses the entire expressions and their conditions. In particular no new support is selected. Applying it to CN2 and N6 gives RT4, while applying it to the four-unit multiplication table keeps the order of the two inside image factors and all outside factors.

For each of the twelve actual J, preservation of multiplication and commutation with T give

ρ_π(E_J(t))
  =T(N_t^π)[s_π(J)e_(πJ)]P_{ij,−ω}N_t^π
  =s_π(J)E_(πJ)^{ord,π}.

Apply RT1's coefficient law at the three CN5 output positions J,J′,M_J, and multiply by the same sign s_π(J), which undoes itself. This yields the three lines of RT5 exactly. Their only additional factors are the complete signs already listed in RT1. A permutation preserves the distinctness of the thirty-six original positions and maps every remaining void coefficient to a void coefficient. The full Z_c sum therefore has coefficients c_(πJ)^{ord,π}=s_π(J)ρ_π(c_J). Applying the same coefficient identity to CN7 and using s_π(J)s_π(J)=u gives RT6 in both directions.

4. Carry the simultaneous derivative and family criteria

The outside list (0,5,6) becomes (0,ℓ,m). The inside list (1,2,3,4) becomes (1,2,π(3),π(4)), so the complete definitions of D_out,D_± give RT7. When ω=−u, the final two summands exchange order; their finite sum is unchanged, giving the displayed increasing i,j form. This does not exchange the corresponding ordered factors in Enc_π's multiplication table.

Apply the proved identities RT2 to every complete equality in RD1,RD2,N7,N11 of V06-PB-0040. Each derivative of tF_out carries both actual product-rule terms. The corner recovery maps remain the same left and right products with P_{ij,±ω}, so necessity and sufficiency are preserved in both directions. The actual HU_c premise of V07-PB-0045 becomes exactly the one in RT8; its criterion then gives RT8 without a new witness. The same transport carries V07-PB-0044's complete Sq identities and their stated one-way consequences. It supplies no converse that those claims did not state.

Finally, the outside generators of S_σ and its actual supplied scalars map exactly to the displayed S_σ^π. Apply RT2 to V05-PB-0039 UF7/UF10. Each label 0,1,2,3,4 maps to the set {0,1,2,i,j}; labels 5,6 become ℓ,m. Every scalar product retains its two factors and its order. These are precisely all the RT9 equations, including the uncancelled final product. Applying the same substitution to UF13 and UF18a–UF18b carries their additional aa+bb=u or actual undoing premise and nothing weaker. This completes the whole named chain at all twelve relabellings.

What is not forced

The relabellings above fix 0,1,2 and carry the chosen inside generator e₁ to itself. They therefore cover all six pairs and both core signs while preserving the declared ordered B. They do not identify an e₂-, e₃- or e₄-based construction with this e₁ chain. A general retained-core change remains a separate scope.

Every conditional scalar identity and derivative rule must be transported together. No equality of the original and relabelled scalar data is asserted unless that additional invariance is supplied. The coordinate completeness is algebraic; scalar realization and an active-source base solution remain separate obligations.

The full right-null coordinate class and the corner derivative identities transport before a T condition. The original-residual criteria of the named chain retain their T-fixed premise. This transport does not settle the T-negative original equation.

What it means

One complete signed relabelling table carries the fixed-pair algebra, its two-corner derivative criterion, its scalar family and its conditional undoing theorem to every remaining pair and the partner core sign. All blade signs and all hypotheses travel with the same full expression. The twelve instances share one derivation interface instead of requiring twelve separate proof chains.