Landmark IX-C Actual-functional factorization and coefficientwise coherence for one sharp ordered word

We formulate the binary coherence problem for the ordered word $B_f,W_g,B_{f^{\prime }}$ on the actual-functional carrier of IX-A. Both admitted first-stage products are fixed, including the reverse contraction and its sign. We identify the three one-edge outputs and give the labelled expressions for the three two-edge outputs, without assuming that a labelled expression defines a distributional product. We prove exact algebraic descent criteria for second-stage coefficient maps and for the calibrated scalar assignment ${u_{\mu }}^{\rho }+cr$. The scalar relation tests restrict the real parent parameter to an empty set, a singleton, or all of $\mathbb{R}$. We also prove the coefficientwise equations required for associativity through order ${\hslash }^3$, and distinguish observation compatibility, involution, presentation independence, and bulk-relative agreement.

The release outcome is an unresolved gate. Neither sharp second multiplication is constructed, and no incompatibility excluding all such multiplications is proved. The completed results are the admitted partial stages and the factorization and coherence reductions. The numerical triangle calibration remains conditional on the explicit VII-D analytical premise restated in IX-B-v2.

Figure fig:split-IX-C records one view of the same separation. Figure fig:chain-IX-C records one view of the same separation. Figure fig:pair-IX-C records one view of the same separation. Figure fig:gate-IX-C records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-IX-C records that separation. The drawing adds no hypothesis.

claim and proof boundary{ This manuscript proves reductions and necessary tests for one-word binary coherence. It does not claim a completed binary factorization.

A missing map is an unresolved gate, not a no-go theorem. The unresolved outcome is stated explicitly for both associations. No ternary numerical finite part is renamed as a binary product.}

The selected word is

The target is a bulk-relative binary factorization of this word through order ${\hslash }^3$, with fully contracted joint-test kernel ${u_{\mu }}^{\rho }+cr$, followed by associativity, involution, and presentation independence on the sector actually constructed.

There are three logically different release outcomes:

1. Construction: the named maps exist and satisfy the declared interfaces.

2. Proved obstruction: an explicit incompatibility excludes every map satisfying those interfaces.

3. Unresolved gate: a required map, descent proof, admission check, or coherence identity has not been established.

This manuscript has outcome (iii). Its proved results identify what is already available and what must still be established.

Failure of one candidate excludes that candidate only. Conversely, the absence of a construction in the supplied inputs does not prove that construction impossible.

A proposed construction must specify which of the following clauses it verifies:

1. Every polynomial output belongs to ${\mathcal{P}}_{\Sigma }$, including the metric derivatives and directional descendants required by IX-A.

2. The order-zero coefficient is classical pointwise multiplication.

3. On every specified overlap with the old bulk domain, the coefficient maps agree with ${\star }_H$.

4. The first-stage left coefficient is the locked ${\mathcal{Q}}_{f,g}$; the right coefficient is its reverse contraction for the same covariance.

5. The fully contracted joint-test kernel is ${u_{\mu }}^{\rho }+cr$, with $c\in \mathbb{R}$. Declared observation compatibility transports that same parameter.

6. No subtraction changes an already fixed old bulk product.

7. Unchanged regular approximants are not required to converge to the sharp insertions in ${\tau }_{\Sigma }$.

8. The maps respect every finite linear relation between actual presentations of their arguments.

9. Every continuity assertion has a named domain and a proof. Fixed-support test-family continuity is distinct from continuity or hypocontinuity on arbitrary carrier stages.

Clause C-I8 is the factorization condition. A test-labelled assignment that fails this clause is not a binary map on ${\mathcal{P}}_{\Sigma }$.

Clause C-I9 does not follow from C-I8. Algebraic descent does not automatically imply continuity in the subspace topology inherited from ${\tau }_{\Sigma }$.

No Einstein equation, backreaction, retarded product, switched realization, conical identification, or all-graph catalogue is constructed. No algebra on all of ${\mathcal{P}}_{\Sigma }$ is claimed. The VII-C obstruction is not repaired. IX-D and the separate G2 gate are not opened.

Let

Write $x=(t,\mathbf{x})$, $y=(s,\mathbf{y})$, and $x^{\prime }=(t^{\prime },{\mathbf{x}}^{\prime })$. Set

The operator ${\mathsf{E}}_0$ is the zero-value, zero-first-time-derivative Cauchy inverse on $\{t=0\}$.

Configurations $(h,\varphi )$ are smooth and off shell. The selected functionals are

IX-A defines ${\mathcal{P}}_{\Sigma }$ as a space of actual finite polynomial functionals with compact symmetric distributional kernels. Its admission criterion is

Here

and ${\mathcal{M}}_1$ consists of compact linear metric kernels with no nonzero nonspacelike wavefront covectors.

The topology ${\tau }_{\Sigma }$ records the strong compact-distribution coefficients and every intrinsic boundary-trace coordinate

We inherit the carrier, intrinsic trace, classical closure, directional closure, and Hausdorff topology from IX-A IXA (def:Padm, lem:charge, thm:closure, def:topology). The truncated coefficient module is

not a constructed quantum algebra IXA (eq:module).

The Wick convention is

The fixed covariance blocks are

with

We inherit

and the mixed component

These are fixed input identities, not new calculations IXA (lem:tensor, eq:K) IXB (eq:tensor, eq:K).

IX-B fixes

Its existence and admission are inherited from IX-A; its sign and multiplicity are fixed in IX-B IXA (prop:knownoutputs, item 3) IXB (prop:partial, eq:Qpartial).

For the prescribed radial regulator placement,

There are distributions $r,{u_{\mu }}^{\rho }\in {\mathcal{D}}^{\prime }(X)$ such that

and, for every compact $K_0\subset X$,

The estimate includes compacts meeting $\mathrm{Rel}\cap {\Delta }_{13}$.

The residue is

where

and

Thus $\mathrm{Rel}=\mathrm{supp}r$.

For the prescribed regulator family,

The parent distributions satisfy

where dagger conjugates and exchanges $x,x^{\prime }$.

These statements are the analytical premise of VII-D, explicitly restated in IX-B IXB (ass:VIID, eq:adjoint) VIID (thm:finitepart, thm:conversion; Section “Adjoint symmetry”). No independent verification of VII-D is asserted here.

Fix $\rho ,\mu$. The permitted scalar calibration is

This is an imposed parent class. It is not a classification of all possible graph extensions or binary-product ambiguities.

These distinctions follow from the definitions in the supplied V manuscript V (Sections “The external-metric first retarded product”, “The one-oscillator mixed insertion”, and “Contrast: the Newton–wave cubic form”). No theorem from V enters a proof below. In particular, its subcritical extension uniqueness and switched bilinear identity are not premises for triangle factorization or G2.

Let $\mathcal{T}={C_c}^{\infty }(M_{+};\mathbb{C})$, and define

The maps $f\mapsto B_f$ and $g\mapsto W_g$ are injective.

Proof. Injectivity of $f\mapsto B_f$ is IX-A's result IXA (cor:notbulk).

Suppose $W_g=0$. Then $\int g{v}^2=0$ for every smooth scalar function $v$, by taking $h_{00}=v$. Polarization gives $\int guv=0$ for every smooth $u,v$. Taking $v=1$ proves that $g$ is the zero distribution, hence the zero test.

The reverse ordered coefficient is

It belongs to ${\mathcal{P}}_{\Sigma }$, and

Proof. The real tensor coefficients in $H_{hh}$, together with $\overline{{\Delta }_0^{+}(z)}={\Delta }_0^{+}(-z)$, give reversal of the mixed covariance by kernel conjugation. The real zero-Cauchy operator preserves this relation. Therefore the reverse mixed component is $\overline{K(x^{\prime },y)}$.

The two slots of ${h_{00}}^2$ supply multiplicity $2$. There is one insertion $B_{f^{\prime }}$, so its defining minus sign gives $-2$. This proves (eq:QR). The formula immediately gives (eq:Qadjoint). Admission follows from IX-A's selected reverse-output result, or from its conjugation closure applied to (eq:Qadjoint) IXA (prop:knownoutputs, item 3; thm:closure).

Define actual-functional subspaces

and

The sums are direct because the two summands are homogeneous of field degrees four and two. The underlying pointwise spaces ${\mathcal{L}}_4,{\mathcal{R}}_4$ coincide when the test ranges coincide. We retain the ordered notation to distinguish their roles. Likewise, their common elements must not be assigned inconsistent values by any future product.

The formulas

define bilinear maps on $\mathcal{B}\times \mathcal{W}$ and $\mathcal{W}\times \mathcal{B}$, respectively, with values in ${{\mathcal{V}}_{\Sigma }}^{[3]}$. They satisfy classical normalization, the locked first-stage normalization, and reversal under complex conjugation.

Proof. Lemma lem:injective identifies each element of $\mathcal{B}$ and $\mathcal{W}$ with its unique test. The formulas are bilinear in those tests and therefore are well defined on the named actual-functional spaces.

Classical closure places $B_f{W}_g$ and $W_g{B}_{f^{\prime }}$ in ${\mathcal{P}}_{\Sigma }$. The selected contractions are admitted by the inherited result and Proposition prop:reverse. Thus both formal outputs belong to ${{\mathcal{V}}_{\Sigma }}^{[3]}$. Their order-zero term is the pointwise product. Equation (eq:Qadjoint) proves reversal. No additional binary contraction is possible for these pairs: each sharp insertion has only one metric slot and no middle scalar slot with which its scalar slot could contract.

For fixed compact observation supports, both maps in (eq:first) depend jointly continuously on their tests, with target ${{\mathcal{V}}_{\Sigma }}^{[3]}$ carrying the inherited topology. This is a test-family statement, not a quantum hypocontinuity theorem on ${\mathcal{P}}_{\Sigma }$.

Proof. The pointwise terms have the fixed-support test-family continuity proved in IX-A IXA (thm:admission).

For the contraction term, its compact coefficient kernel is obtained from $f(x)g(y)K(x,y)$ by a fixed scalar-slot derivative and bundle placement. On fixed compact supports, multiplication of a fixed distribution by varying smooth coefficients and fixed finite differentiation are continuous into the strong compact-distribution topology. This follows directly by applying the distribution to bounded smooth tests and using its finite-order estimate on the common compact set.

The proof of IX-A's selected-output admission shows that every first metric derivative of this contraction, at a smooth configuration and after directional differentiation, is a smooth compact kernel IXA (prop:knownoutputs, item 3). Its boundary coordinates therefore vanish. The coefficient continuity consequently gives continuity in ${\tau }_{\Sigma }$. Conjugation gives the same result for the reverse contraction.

Theorem thm:first constructs only the two first stages. It does not define multiplication of their outputs by another sharp insertion.

Put

IX-A admits the scalar-contracted sharp self-product

This is inherited, not a second-stage closure theorem IXA (prop:knownoutputs, item 2).

Define

All three functionals in (eq:oneedges) exist and belong to ${\mathcal{P}}_{\Sigma }$. They are homogeneous of field degree four.

Proof. Each factor in (eq:oneedges) is already admitted. Classical pointwise closure in IX-A gives the products and their full admission criterion, including directional descendants IXA (thm:closure). The degree is the sum of the two residual degrees. No new singular-variable multiplication is used in this argument.

The following are labelled candidate expressions:

The superscript “lab” means that the expressions identify the desired contraction placements. It does not assert the existence of the indicated distributional products.

At positive cutoff, (eq:g1223)–(eq:g1323), with the correspondingly regulated kernels and residual fields, are the three two-edge placements. Each has sign $+1$ from the two insertion signs and assignment multiplicity $2$. Their residual field degree is two. The fully contracted placement has coefficient

Proof. The allowed nonzero edges are the sharp–middle metric edge $12$, the scalar edge $13$, and the middle–sharp metric edge $23$. Each choice of two leaves exactly two field slots.

For $12+23$, both middle slots are used, leaving the two scalar slots. There are two assignments of the middle slots. For $12+13$, either middle slot can be used on edge $12$; the remaining middle slot and third-vertex metric slot survive. The case $13+23$ is the exchanged case. Both sharp insertions contribute one minus sign, so all three coefficients have positive sign.

The inherited tensor cancellation selects the zero vector component on every sharp–middle metric edge. This gives $S_{0\nu }$ and $S_{\mu 0}$ in the latter two placements. Full contraction uses both metric edges and the scalar edge, giving the inherited triangle and multiplicity IXB (prop:unique, eq:T).

For (eq:g1223), a compact scalar-only polynomial kernel, if established, would pass the metric admission test because all its metric derivatives are zero. The existence of that kernel must nevertheless be proved.

For (eq:g1213) and (eq:g1323), establishing a compact distributional kernel is not sufficient: their metric derivatives must satisfy (eq:admission). The supplied calibration theorem does not provide these graph-specific results.

No claim is made here that these products fail to exist. Their existence and admission remain unproved in this manuscript.

All tensor products in this section are algebraic tensor products over $\mathbb{C}$. No completion or quotient topology is silently introduced.

Let

Define presentation maps

Their images are the corresponding actual-functional spaces. Their kernels contain all finite linear relations between the displayed presentations.

Let $v:E\to V$ be a surjective linear presentation map, let $Z,Y$ be vector spaces, and let $\lambda :E\otimes Z\to Y$ be linear. There is a unique bilinear map

such that

if and only if

Proof. Necessity follows by evaluating at $v(e)=0$. For sufficiency, define the value using any presentation $e$ of the first argument. Two presentations differ by an element of $\ker v$, so (eq:descentcondition) makes the value independent of the choice. Linearity of $\lambda$ gives bilinearity. Surjectivity of $v$ gives uniqueness.

For presentation maps on both arguments, the analogous condition is vanishing on

This follows by applying Theorem thm:descent twice.

Since $f^{\prime }\mapsto B_{f^{\prime }}$ is injective, define

Define similarly

A calibrated bilinear map

exists algebraically if and only if

The analogous right map

exists if and only if

When they exist, both maps are unique on these domains.

Proof. Apply Theorem thm:descent to $q_L$ and $q_R$, using injectivity of the sharp test map on the other argument. Constants belong to ${\mathcal{P}}_{\Sigma }$, so the scalar target has no additional admission issue.

In particular,

requires

Theorem thm:scalarfactor is an exact equivalence for algebraic scalar descent. It does not verify the conditions for the actual kernels $q_L,q_R$, nor does it construct the nonconstant maps.

Let $\mathcal{Z}$ be the direct sum of the two relation-test spaces in (eq:leftscalarcondition) and (eq:rightscalarcondition). Define linear functionals $\alpha ,\beta$ on $\mathcal{Z}$ by evaluating the corresponding joint tests against ${u_{\mu }}^{\rho }$ and $r$, respectively. Then both scalar relation tests are equivalent to

The set

is empty, a singleton, or all of $\mathbb{R}$. More precisely:

1. if $\beta =0$, it is all of $\mathbb{R}$ when $\alpha =0$ and empty otherwise;

2. if $\beta \ne 0$, choose $z_0$ with $\beta (z_0)\ne 0$. The only possible parameter is

   $$c_0=-\frac{\alpha (z_0)}{\beta (z_0)}.$$

   It is allowed exactly when $c_0\in \mathbb{R}$ and $\alpha +c_0\beta =0$ on every relation test.

Proof. If two distinct real parameters solve (eq:affine), their difference times $\beta$ is zero, hence $\beta =0$; then $\alpha =0$. Otherwise there is at most one solution. The displayed alternatives follow directly.

An empty set here would prove an incompatibility with the imposed parent calibration class, before any continuity claim. This manuscript does not establish that the set is empty, nonempty, or all of $\mathbb{R}$.

Let $D_1,D_2$ be subspaces of a vector space and $a_i:D_i\to Y$ linear. A linear map on $D_1+D_2$ extending both $a_i$ exists if and only if

When it exists, it is unique.

Proof. Necessity is immediate. For sufficiency, define $a(d_1+d_2)=a_1(d_1)+a_2(d_2)$. Any two decompositions differ by an element of the intersection, on which the maps agree.

For a proposed coefficient of one partial binary product, relevant domains inside ${\mathcal{P}}_{\Sigma }\otimes {\mathcal{P}}_{\Sigma }$ include

The last domain carries the prescribed old Wick coefficient. Lemma lem:gluing shows exactly where overlap checks are required. Calling the left and right maps by different names does not permit them to disagree on a common actual pair.

The lemma is algebraic. Continuity of a glued map remains a separate requirement.

A candidate construction must supply coefficient maps

on the orders needed below, with $L_0,R_0$ pointwise multiplication.

For the selected contraction-degree prescription, no third binary contraction against a degree-two sharp factor is present. This is a declared prescription for the candidate sector, not a consequence excluding arbitrary additional contact terms. Any departure must be separately formulated.

Write

Under the selected contraction-degree prescription, the two candidate compositions expand as

Their selected field degrees are $6,4,2,0$, respectively.

Proof. Expand the coefficientwise formal product of $P_L+\hslash Q_L$ with $C$, and of $A$ with $P_R+\hslash Q_R$. Collect equal powers of $\hslash$. The degree-two sharp factor permits at most two cross-contractions. Every contraction removes two field slots, giving the degree ledger.

Assuming the maps in (eq:AL)–(eq:AR) are well defined, the equality ${\mathcal{A}}_L={\mathcal{A}}_R\mathrm{mod}{\hslash }^4$ holds if and only if the following three identities hold for every permitted $f,g,f^{\prime }$:

The order-zero identity holds by classical associativity.

Proof. Equality in the truncated module is equality of its four coefficient coordinates. The zero-order terms coincide because $P_{L}C=(AW)C=A(WC)=A{P}_R$. The other three coordinates are exactly (eq:assoc1)–(eq:assoc3).

If the two-edge functionals have been established, the intended placement prescriptions are

These are labelled prescription equations until descent through $p_L,q_L,p_R,q_R$, existence, and admission have been proved.

If the prescriptions (eq:firstprescription)–(eq:mixedprescription) define actual coefficient maps satisfying admission and descent, and both scalar maps use the same allowed $c$, then (eq:assoc1)–(eq:assoc3) hold.

Proof. At order one, both sides sum ${\mathsf{G}}_{12}+{\mathsf{G}}_{13}+{\mathsf{G}}_{23}$. At order two, both sides sum ${\mathsf{G}}_{12,23}+{\mathsf{G}}_{12,13}+{\mathsf{G}}_{13,23}$. At order three, both equal ${\mathcal{N}}_c(f,g,f^{\prime })1$.

The corollary proves coherence of maps if they exist with the stated prescriptions. It does not establish their existence by defining both associations to be the same graph sum. That shortcut would omit actual-functional factorization.

A nonconstant discrepancy in (eq:assoc1) or (eq:assoc2) cannot be repaired by adding $cr$ to the constant coefficient. The direct-sum coefficient structure separates these equations.

For the fixed scalar calibration (eq:Cal), and a fixed smooth-coefficient differential operator $D$ on $X$, the corresponding numerical descendant is

For example,

Under Assumption ass:calibration, IX-B supplies the associated strong numerical finite-part estimate with test-order loss $64\to 64+\mathrm{ord}D$.

Proof. The displayed equality is the definition of distributional differentiation. Smooth multiplication and differential operators act linearly, so the parent shift $cr$ becomes $caDr$. The quantified transfer is inherited from IX-B IXB (thm:descendants, cor:conversion).

This proposition does not prove that a descended carrier-level binary map satisfies any additional configuration-derivative identity. The three operations remain distinct:

1. differentiation of observation tests or their joint distribution;

2. fixed smooth configuration-direction differentiation in IX-A;

3. exposed functional derivative slots in a new contraction graph.

For fixed compact supports of $f,g,f^{\prime }$, the map $(f,g,f^{\prime })\mapsto {\mathcal{N}}_c(f,g,f^{\prime })$ is jointly continuous. The same holds for each fixed observation descendant.

Proof. A distribution has a finite-order estimate on every fixed compact support. Apply that estimate to $f\otimes g\otimes f^{\prime }$. Leibniz's rule bounds its test seminorms by products of finitely many smooth seminorms of the three factors. Fixed differential operators add only finitely many required test derivatives.

Even if Theorem thm:scalarfactor yields an algebraic map, Proposition prop:scalarcontinuity proves its continuity along the named generating test family, not automatically in the subspace topology of ${\mathcal{L}}_2\times \mathcal{B}$. No quotient-topology identification has been proved. No hypocontinuity theorem in ${\tau }_{\Sigma }$ is asserted.

A general smooth joint multiplier $a(x,y,x^{\prime })$ defines a numerical operation on the joint distribution. It need not encode a binary operation on the original carrier stages.

For polynomial functionals, star denotes complex conjugation. For a joint distribution, dagger is

For real $c$,

If scalar descent holds on the left, the prescription

defines the corresponding right scalar map on the conjugate reverse stage, with the required calibration.

Proof. Assumption ass:calibration gives $({u_{\mu }}^{\rho }+cr{)}^{†}={u_{\mu }}^{\rho }+cr$. Pairing this identity with product tests proves the first claim. Conjugation sends ${\mathcal{L}}_2$ onto ${\mathcal{R}}_2$ by (eq:Qadjoint). Thus the displayed prescription is well defined whenever the left map is, and its value is the reversed calibration.

Parent symmetry does not establish reversal for the nonconstant second-stage coefficient maps. Those maps would have to satisfy

on their conjugate reverse domains, together with the overlap conditions of Lemma lem:gluing.

Observation differentiation exchanges first and third observations under dagger; a descendant is not generally individually self-adjoint.

Suppose labelled coefficient prescriptions descend to maps on the named actual-functional spaces and are consistent on their specified overlaps. Then their values are unchanged by any rewriting that preserves their arguments as actual off-shell polynomial functionals. This includes finite linear presentation relations, kernel permutations, tensor symmetries, polarization identities, and integration by parts with all boundary terms retained.

Proof. A descended map assigns a value to the actual arguments, not to their labels. Any equality-preserving rewriting leaves those arguments unchanged. Equivalently, the difference of two presentations lies in the relevant kernel of a presentation map and is annihilated by the descent condition. Overlap consistency ensures the same statement if the rewriting moves between two named prescriptions.

The IX-A identity

is equality of actual kernels IXA (prop:adjoint, thm:presentation). Dropping the delta term does not preserve the argument and is not an admissible presentation change.

Theorem thm:presentation is conditional on descent. It cannot be used to supply the missing descent proof.

IX-B's separated witness proves that observation-diagonal counterterms cannot remove the selected logarithm IXB (thm:diagonal). The unchanged regularized coefficient remains divergent on that witness. The normal-topology extension obstruction of VII-C, under its declared hypotheses including (E1)–(E4), remains valid VIIC (thm:coefficient, ass:extension, cor:noalgebra).

IX-A explicitly proves

for nonzero real $f$ IXA (thm:IX1, cor:compatibility). No opposite continuity requirement is introduced here.

A numerical subtraction from ${T_{\epsilon }}^{\rho }$ is not a change inside the unchanged old product. Subtracting old regular coefficients would change those products; assigning only a sharp value would not restore the forbidden normal continuity IXB (rem:subtraction, Section “Continuity reconciliation”).

The separate G2 condition is untouched. Where IX-A's stated switch-gap hypotheses apply, its first-order incompatibility continues to apply IXA (prop:G2). No unconditional gauge theorem outside those hypotheses is inferred.

Relative to the inherited IX-A and IX-B results and the explicit analytical premise in Assumption ass:calibration, this manuscript establishes:

1. both admitted first-stage products, with locked signs and reverse contraction;

2. actual-functional existence and admission of the three one-edge outputs;

3. the labelled signs, multiplicities, and residual degrees of the three two-edge placements;

4. necessary and sufficient algebraic descent tests for a prescribed labelled coefficient map;

5. necessary and sufficient scalar relation tests for the calibrated assignment on the named partial-output spaces;

6. the empty/singleton/all-real classification of parameters passing those scalar relation tests;

7. the three nontrivial coefficient equations for associativity through order ${\hslash }^3$;

8. conditional placement coherence, scalar reversal, and presentation independence after actual descent.

Neither sharp second multiplication is constructed. No incompatibility excluding every map satisfying Requirement req:interface is proved. The release outcome is therefore an unresolved gate.

Proof. The assertions follow from Theorem thm:first, Proposition prop:oneedges, Proposition prop:placementcount, Theorem thm:descent, Theorem thm:scalarfactor, Proposition prop:parameters, Theorem thm:coherence, Corollary cor:placementcoherence, Proposition prop:scalarreverse, and Theorem thm:presentation.

The two-edge existence and admission requirements in Remark rem:treeboundary have not been discharged. The scalar relation tests have not been evaluated for the actual presentation kernels. Consequently the hypotheses needed to construct the second-stage maps and invoke the conditional coherence results have not been established. No violated relation or other universal incompatibility has been exhibited. The unresolved classification follows.

{}1.12

A construction must now do the following:

1. establish the actual two-edge distributions and their compact polynomial kernels;

2. check (eq:admission) for both metric-quadratic outputs, at every configuration and after the required descendants;

3. prove descent of all second-stage prescriptions through $p_L,q_L,p_R,q_R$;

4. evaluate the scalar relation tests and determine whether ${\mathcal{C}}_{\mathrm{desc}}$ is nonempty;

5. verify overlap agreement with the old bulk coefficients and between any two prescriptions for the same actual pair;

6. prove the named continuity claims without importing normal-stage sharp-admission convergence;

7. establish reversal of every nonconstant coefficient map.

An explicit relation with nonzero calibrated value would give a genuine factorization obstruction for the imposed parent class. A demonstrated failure of one proposed graph extension would exclude that candidate only, unless all permitted alternatives were separately excluded.

conclusion{ The admitted partial coefficient has a definite place in the IX-A carrier. The scalar calibration has an exact relation test for factoring through that place. Associativity has three separate nontrivial coefficient equations.

None of those statements supplies the missing second multiplication. Both sharp associations are explicitly unresolved. The ternary numerical calibration is retained without being advertised as a binary product or as an associative algebra.}

For the selected word,

under the declared contraction-degree prescription. The fully contracted order-three term is a constant functional, but the order-two terms are not.

Observation differentiation changes neither the number of insertions nor the contraction order. It does not supply an extension theorem for independently opened functional derivative slots.

Each first-stage contraction contains one insertion sign:

The complete word contains two insertion signs:

The two middle-slot assignments give multiplicity $2$, not $4$. The partial factor $2$ is already present when its two remaining slots are contracted with the last sharp insertion.

All presentation maps act on actual off-shell functionals. No replacement of a kernel by an equation-of-motion-equivalent kernel is included. Boundary terms in the sharp adjoint remain part of the argument.

The new proofs are algebraic descent, parameter consistency, formal coefficient comparison, first-stage reversal, and the specified inherited-closure consequences.

The parent ultraviolet analysis is the VII-D analytical premise. The sharp two-edge products, their metric admission, and scalar descent for the actual presentation kernels have not been verified.

The source is designed for two-pass pdfLaTeX compilation. Compilation has not been executed in this interface. No successful build, page count, proof-assistant check, or independent review is claimed.

of the result. This is a proof of partial-stage and coherence reductions with an explicit unresolved release outcome. It is not a completed quantum factorization theorem.

boundary. IX-A and IX-B-v2 are supplied sources. VII-D is used only through the explicit analytical premise restated in IX-B-v2; its full proof is not supplied or independently certified here. VII-C's stated obstruction is retained through the source-locked boundary recorded in IX-B-v2. V is used only to identify different objects.

release boundary. No sharp second multiplication, global quantum algebra, interacting realization, or gauge-compatible switched construction is claimed.

IXA

N. Cheung. Landmark IX-A: An off-shell admission domain and boundary-sensitive topology for sharp Cauchy relational insertions. Supplied manuscript, IX-A.tex. Relevant labels: def:Padm, lem:charge, thm:closure, def:topology, thm:admission, prop:adjoint, lem:tensor, prop:knownoutputs, cor:notbulk, thm:IX1, cor:compatibility, thm:presentation, eq:module, and prop:G2.

IXB

Landmark IX-B: Derivative-stable triangle calibration and counterterm constraints for sharp Cauchy insertions. Supplied manuscript, IX-B-v2.tex. Relevant labels: eq:tensor, eq:K, eq:T, ass:VIID, prop:unique, thm:diagonal, thm:descendants, cor:conversion, thm:independence, prop:weights, thm:regularassociation, rem:subtraction, prop:partial, eq:Qpartial, prop:missing, and eq:adjoint.

VIID

Finite part of the regulated Cauchy-reflection triangle: A strong numerical distributional limit for the VII-C family. Companion manuscript, VII-D.tex, identified in IX-B-v2. The statements used here are explicitly restated in Assumption ass:calibration. Relevant source labels reported by IX-B-v2: thm:finitepart, thm:conversion; also the section “Adjoint symmetry”. No independent verification of the companion proof is claimed.

VIIC

Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions. Companion manuscript, VII-C.tex, identified in IX-B-v2. Relevant source labels reported by IX-B-v2: thm:coefficient, ass:extension, and cor:noalgebra. The obstruction boundary is retained as recorded in the supplied IX-B-v2 manuscript.

V

N. Cheung. Admissible currents and first retarded corrections for a fixed-time scalar response. Supplied manuscript, V.tex. Used only for the separation of its external retarded coefficient, one-oscillator coefficient, and Newton–wave cubic form from the selected sharp triangle. No theorem from V enters a proof.