Landmark XIII-E One joint off-collision prescription for the selected binary pair, or its obstruction in the frozen partial-diagonal class A common-frequency prescription attempt, localization independence, and the remaining E1 gate

We return to the joint-extension problem left open in XIII-C. The entries are the same switched metric–scalar vertex and the same admitted second-order composite. The carrier, action, off-shell lifts, and partial-diagonal ambiguity class are unchanged.

Exactly one route is attempted: a common-frequency regularization of the complete time-ordered graph factors, with their inherited tensor coefficients retained. We prove that the finite-regulator graph functionals are compact microcausal polynomials, that the regularization converges to the prescribed products on the microlocally admissible open sets, and that auxiliary compact localization changes have a vanishing remainder. We also prove that a partial-diagonal contact cannot change any prescription on the collision complement.

These results do not establish the remaining limit through the noncoincident characteristic configurations. The inherited wavefront cancellation prevents an application of the sufficient product theorem there, while neither divergence of the complete pair nor an unavoidable normalization incompatibility is proved. The earliest unfinished gate is therefore E1.

We formulate the subsequent transverse, overlap, and pushforward obligations without treating them as completed. A relative-coordinate lemma identifies a sufficient balanced endpoint once a joint extension exists; it does not provide existence. The coefficient linear in $\hslash$ retains the ending Unresolved. No Ward defect, interacting differential, or restriction quotient is constructed.

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

Figure fig:scope-XIII-E records that separation. The drawing adds no hypothesis.

and the three endings{ Constructed means that a joint distribution satisfies all four successor obligations and has an admitted proper pushforward.

Proved obstructed means that an explicit incompatibility excludes every extension in the previously frozen class ${\mathfrak{C}}_{\mathrm{part}}$.

Unresolved means that a required prescription, transverse estimate, contact compatibility, or output theorem is missing.

Results here: a named common-frequency attempt, admitted finite-regulator functionals, convergence on the admissible open sets, auxiliary-localization independence, and the off-collision rigidity of the permitted contacts.

Ending for the requested coefficient: Unresolved at E1. The characteristic-set limit and its prescription independence have not been established. E2–E4 are not released.

A finite regularization is not a completed prescription. A failed sufficient product criterion is not a nonexistence theorem. No universal obstruction in ${\mathfrak{C}}_{\mathrm{part}}$ is claimed. The free differential is not applied to the absent coefficient.}

The following are mathematical inputs with their predecessor scope. They are not new conclusions about the regularization introduced below.

The next extension attempt must supply:

1. a joint prescription across the noncoincident characteristic configurations, or a proved cancellation of those configurations in the complete selected pair;

2. uniform transverse scaling and wavefront estimates for the three-vertex and four-vertex collision strata and their intersections;

3. compatible extensions with the permitted partial contacts, rather than only isolated primitive extensions;

4. the full output wavefront estimate after proper pushforward.

If these obligations are met, the successor may release one ${R_1}^{(1)}$ in the unchanged carrier and only then open the binary gauge-consistency question at order $\lambda \hslash$ XIIIC.

Constructed: the incidence inventory, the widened class ${\mathfrak{C}}_{\mathrm{tot}}\subset {\mathfrak{C}}_{\mathrm{part}}$, the off-collision products on their microlocally admissible open sets, and the elementary parallel-line extensions.

The parallel kinetic primitive has scaling degree $8$ and codimension $4$; its partial-contact ambiguity has normal derivative order at most $4$. Sum-zero bounds and collision exclusion do not establish the joint off-collision product: distinct null-related vertices can carry a nonzero wavefront cancellation.

The coefficient linear in $\hslash$ is Unresolved. That is a failed sufficient product criterion, not a nonexistence theorem XIIIC.

The eleven branches and their labelled incidence matchings are used as the fixed graph presentation. They are not listed a second time.

The vertex

is admitted. The tree datum

is constructed, proper, and microcausal.

Ordered values of $R_1$ are $0$ or $\frac{i}{\hslash }[\mathcal{O},{\mathcal{V}}_g{]}_{{\star }_H}$. The literal normalization $R_1={\mathcal{V}}_g{\star }_H\mathcal{O}$ is proved obstructed XIIIB.

In particular, the tree datum is not the requested coefficient linear in $\hslash$.

Under $({\mathrm{UCP}}_W)$, ${\Delta }_{\xi }\mathcal{O}{|}_{{\mathcal{C}}_W}=0$. The coefficientwise kernel ${\mathcal{N}}_W$ is not a two-sided ideal. No reduced algebra was handed off. ${\mathcal{C}}_W$ remains an evaluation set XIIIA.

On the detecting data, every shell-descending action in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ induces the zero representation, and the pairing is nonzero on

The ending is Proved obstructed for that class only. The admitted cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ is not an integrated completion XIIID.

That classical cocycle calculation is not a product obstruction and supplies no numerical graph distribution.

The copied witness is retained:

Its hypotheses and evaluated sign are those of XII-F XIIF. It is not used to determine a regulator, a contact coefficient, or an extension normalization.

The new arguments use the distribution product criterion, submersion pullback, differential-operator wavefront bounds, and proper pushforward. Their microlocal continuity formulation is the one recalled in BF (Section 2) and developed in Hormander (Section 8.2).

Point extension and extension across a submanifold are distinct. Scaling and wavefront conditions for the latter are described in BF (Sections 5–6) and Dang. No existence theorem for local entries is imported as an existence theorem for the present nonlocal pair.

This freeze precedes the regularized kernel construction.

Use

The functions $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$ and ${\sigma }_{\epsilon }$ are fixed backgrounds. All fields are ambient smooth multiplets. No decay assumption is added.

The carrier is

There is no quotient by ${\mathcal{N}}_W$.

The action remains XII-C's cubic-only density weighting,

No Einstein–Hilbert interaction vertex is inserted into the selected pair XIIC.

The only interaction entry is ${\mathcal{V}}_g$ of Input statement input:B. The observable is

with

The insertion sign and both scalar-completion legs are unchanged XIIE.

The relevant inverse blocks are

The frozen translation-invariant product data satisfy

Metric–scalar cross blocks vanish for these frozen data. All numerical tensor factors and action signs are retained.

The selected expansion is

Only the coefficient of $\lambda \hslash$ is sought. There is no $R_2$, no ${\mathsf{T}}_3$, and no claim of completeness of a higher coupling-order relational coefficient.

The operation remains attached to these off-shell lifts. A representative-independent operation on all of ${\mathcal{E}}_H$ is not part of the target XIIIB.

For a graph $\Gamma$, let $y$ be the interaction vertex. For each connected subset $I$ containing $y$, with $|I|\ge 2$,

Include the intersection and contracted-graph loci specified in XIII-C. Observable-only subgraphs retain their inherited normalization.

The ambiguity class is exactly XIII-C's class of well-defined graphwise contacts

Here $r_I$ is transverse to ${\Delta }_I$, $S_{\Gamma /I}$ is the inherited spectator coefficient with the two remaining field slots, and $D_{\Gamma }$ is the fixed finite graph differential operator.

The external weights are those built from $g,\partial g,{\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$ and the already present smooth graph weights. Contacts are even, have ghost number zero and field degree two, and remain in the existing compact graph closure. They leave the tree datum and inherited observable-subgraph normalizations unchanged.

Only well-defined contact products and pushforwards belong to the class. The bound six is the copied conservative bound, not a new power-counting theorem. The parallel kinetic primitive has the smaller proved normal-order bound four.

This class includes $y=z$ with $x\ne z$. It is not replaced by ${\mathfrak{C}}_{\mathrm{tot}}$. A singular spectator is not declared smooth merely because its external weight is smooth.

The extension domain is the configuration space of the selected graphs, not ${\mathcal{C}}_W$. The four obligations are exactly Input statement input:four.

No tensor-cancellation theorem and no universal class obstruction are claimed. The limit attempted here is named explicitly, and its unproved part is not hidden in the word “renormalized.”

The triangle identified in XIII-C has numerical structure

with $F_{\mu }$ the corresponding time-ordered boundary-value line. The configuration is

Figure fig:xiii-e-null-line is the selected configuration. It is not a cancellation theorem.

XIII-C supplies a nonzero line-covector selection whose sums vanish at all three vertices. This incidence occurs in the selected pair. It does not prove that summing the complete tensor coefficients cannot cancel a singular contribution.

The present attempt retains those coefficients. It neither deletes the incidence nor adds a subtraction supported on a noncoincident null locus.

Let $X_{\Gamma }=M^{n_{\Gamma }}$ be the numerical vertex space of one graph in the copied inventory. Let

Observable-only singular subgraphs keep their inherited definition.

Write $k_e(v_a-v_b)$ for each numerical line, including its specified derivatives and constant fiber maps. The lines are tempered distributions in relative coordinates. The two new cross-lines use the full $H_T$ kernels.

Let ${\mathcal{U}}_{\Gamma }$ be the open set on which the complete line product satisfies the Hörmander sum condition. We use

The inherited pinch configuration shows why this set cannot be identified with the entire collision complement by the supplied product argument.

Fix the Euclidean quadratic form $|p{|}_{\mathrm{E}}^2={{\sum }_{\mu =0}}^3{p_{\mu }}^2$ in the given Minkowski coordinate chart. It is auxiliary regulator data, not a change of spacetime metric.

For $\tau >0$ define

The parameter $\tau$ is distinct from the fixed ramp width $\epsilon$. Only $\tau$ is sent to zero.

Every line uses the same $\tau$. Smooth external weights are not regularized. Differentiation commutes with $S_{\tau }$, so the inherited derivative placement is preserved.

The complete time-ordered graph factor is

with finite tensor sums taken exactly as in the selected pair. The grouping uses the full time-ordered lines. It does not expand the desired difference into separately undefined mixed products of $H$ and ${G_{\mathrm{act}}}^{\mathrm{A}}$.

For fixed $\tau$, (eq:A-reg) is a smooth function. The attempted off-collision prescription is its weighted distributional limit on ${X_{\Gamma }}^{\circ }$, followed by the collision extensions demanded at E2 and E3. Existence on all of ${X_{\Gamma }}^{\circ }$ is an obligation, not a definition by assertion.

Gaussian smoothing creates tails, so finite-$\tau$ support is not the original retarded support. This must be handled explicitly.

Let $K_{\Gamma }$ be the copied compact closed support set determined by the graph weights and inherited internal retarded lines. Choose

It is an auxiliary localization, not a contact normalization.

Let $z_1,z_2$ be the two residual field variables, attached to vertices $v_{a_1},v_{a_2}$. Before pushforward define

The fixed operator $D_{\Gamma }$ includes the prescribed residual field derivatives. The attachments are separately labelled even when they occur at the same vertex.

With the inherited Wick coefficients $c_{\Gamma }$, define the regularized time-ordered coefficient kernel

No new combinatorial multiplier is introduced.

For every $\tau >0$, the kernel (eq:t-reg) is compact and defines an even, ghost-number-zero quadratic polynomial. Its wavefront set is contained in $Z_2$.

Proof. The numerical coefficient in (eq:Pi) is smooth and compactly supported in the vertex variables. The two delta attachments are well defined: each has its own external variable, so a nonzero covector cancellation between them would require a nonzero covector in an external variable where no other factor contributes.

Their wavefront covectors are balanced between an external variable and its attached vertex. Multiplication by the smooth coefficient and the finite differentiations preserve that bound.

The support is compact because the vertex variables are compact and the delta attachments place each external variable at a vertex. The projection is therefore proper. Its wavefront condition sets all integrated vertex covectors to zero, leaving $p_1+p_2=0$ in the two external slots.

The sum-zero bound excludes both all-causal cones. Finite tensor sums and species symmetrization preserve the bound. The remaining species are even and have ghost number zero Hormander.

Let $t^{H,[2]}$ be the already defined kernel of the coefficient of ${\hslash }^2$ in ${\mathcal{V}}_g{\star }_H\mathcal{O}$. Then

is an admitted finite-regulator functional. It is not denoted ${R_1}^{(1)}$. Admission at each positive $\tau$ gives no convergence or uniform wavefront theorem as $\tau \downarrow 0$.

For $k\in {\mathcal{S}}^{\prime }({\mathbb{R}}^4)$, a compact set $L$, and any multi-index $\alpha$, there are constants $C,N$ such that

If $L\cap \mathrm{supp}k=\varnothing$, then for every $M$,

Proof. A tempered distribution is bounded by finitely many Schwartz seminorms. Apply that bound to ${\partial }^{\alpha }{G}_{\tau }(r-\cdot )$. For $r$ in a fixed compact set, its seminorms grow at most as a fixed inverse power of $\tau$, proving the first assertion.

For the second, insert a smooth cutoff equal to one near $L$ and supported away from $\mathrm{supp}k$. The pairing with its near-$L$ part is zero. In the remaining pairing the Gaussian argument stays a positive distance from zero. Gaussian decay, including all derivatives and polynomial weights required by the tempered-distribution seminorm, dominates every power of $\tau$. This proves the uniform remainder estimate.

Replace ${\chi }_{\Gamma }$ by another compact smooth function ${\tilde{\chi }}_{\Gamma }$ equal to one near $K_{\Gamma }$. For every compactly supported graph test and every $M$, the difference of the resulting pairings in (eq:Pi) is $O({\tau }^M)$.

Thus auxiliary compact localization cannot change a prospective distributional limit. This statement does not prove that the limit exists.

Proof. On the compact support of $({\chi }_{\Gamma }-{\tilde{\chi }}_{\Gamma })w_{\Gamma }$, the original closed graph support conditions cannot all hold. Locally, either the weight is zero or at least one numerical line is evaluated on a compact set disjoint from its support.

Use a finite cover and a smooth partition of unity. On each nonzero patch, Lemma lem:smoothing makes that line smaller than every power of $\tau$, while every other line and its derivatives has at most polynomial growth in ${\tau }^{-1}$. The finite product, including the specified differentiations, therefore has the same arbitrarily small remainder.

Pairing the delta attachments and their derivatives with a fixed smooth graph test preserves this estimate. Summing the patches proves the assertion.

This eliminates an artificial dependence introduced to restore compactness at finite regulator. It does not prove independence of the quadratic form in (eq:smoothing), relative damping rates, or another prescription at the characteristic set.

On every relatively compact open subset of ${U_{\Gamma }}^{\mathrm{good}}$, the regularization (eq:A-reg) converges distributionally to the prescribed complete line product.

The limit there is independent of the smoothing choice among standard approximate identities satisfying the same local microlocal approximation estimates.

Proof. Localize every line near the compact set under consideration. Approximate-identity regularization converges weakly and in the wavefront-restricted sense outside any slightly enlarged cone containing the localized wavefront set. The latter assertion follows from the rapid Fourier decay in those directions; the separated tails are controlled by Lemma lem:smoothing.

The sum condition permits such enlarged cones to be chosen without introducing a zero covector sum on this compact set. The sequentially continuous product, equivalently the admissible diagonal pullback of the tensor product, therefore takes the regularized factors to their canonical product. This is the microlocal continuity statement recalled in BF (Section 2, properties M1–M3).

Smooth weights and finite differential operators preserve the result. The product is canonical on this domain, so two such regularizations give the same limit there.

There is also a useful ordered subdomain. If the interaction attachment $y$ is strictly on the same chronological side of both cross-line endpoints, their possible nonzero wavefront covectors at $y$ lie in the same pointed causal cone. They cannot sum to zero. Removing the cross-lines leaves the inherited observable tree, where the forest argument rules out a remaining circulation. Such strictly ordered attachment domains are therefore covered by the admissible product calculation.

At the copied configuration $u,y,x$, the two cross-line orientations at $y$ instead permit cancellation. The preceding proof supplies no uniform estimate there.

For the fixed weighted graph family, the remaining obligation is:

1. establish a distributional limit through every noncoincident characteristic configuration in ${X_{\Gamma }}^{\circ }$;

2. retain the full tensor and branch summation when deciding whether singular contributions cancel;

3. show that admissible changes of auxiliary regularization give the same off-collision prescription, or differ only by the permitted collision-supported ambiguities after extension;

4. preserve the prescribed values on the strictly ordered attachment domains.

No estimate proving (i)–(iii) at the characteristic configuration has been established here. No test proving divergence of every prescription has been established either. The selected move is therefore an attempt, not a Constructed E1 prescription.

Every well-defined contact in ${\mathfrak{C}}_{\mathrm{part}}$ restricts to zero on ${X_{\Gamma }}^{\circ }$. Consequently it cannot alter convergence, divergence, or prescription dependence tested by graph tests supported in ${X_{\Gamma }}^{\circ }$.

Proof. Each contact is supported on a collision locus ${\Delta }_I$. Differentiation does not enlarge support. Its pairing with a test supported away from the union of those loci is zero. This remains true for a family of contacts whose coefficients depend on an auxiliary regulator.

This lemma separates two questions. Contacts may normalize a collision extension once the off-collision distribution exists. They cannot manufacture that distribution at a distinct-vertex characteristic configuration.

It also gives a precise obstruction test for a future proof: if a compulsory off-collision normalization were incompatible with every joint prescription on such an open set, then no member of ${\mathfrak{C}}_{\mathrm{part}}$ could repair it. No such universal incompatibility is proved in this manuscript.

Suppose two candidate distributions agree on ${U_{\Gamma }}^{\mathrm{good}}$. Their difference is not thereby proved to be a member of ${\mathfrak{C}}_{\mathrm{part}}$.

To identify it as a permitted ambiguity, agreement on the full collision complement and the remaining contact-class conditions must also be established.

Proof. Agreement on an open set only places the support of the difference in its complement. That complement can contain noncoincident characteristic configurations as well as collisions. A distribution supported on such a characteristic locus need not be collision-supported.

Membership in ${\mathfrak{C}}_{\mathrm{part}}$ additionally restricts normal derivative order, spectator structure, external weights, normalization, and well-defined composition. Support alone does not verify those requirements.

In particular, the inherited sum-zero bound does not turn every difference between two regularization limits into a permitted partial contact.

E1 has not been crossed. No scaling degree is assigned here to an undefined full triangle or four-vertex distribution.

For a collision subset $I$,

Choose local coordinates $(q,r)$, with $q$ tangential to ${\Delta }_I$ and $r\in {\mathbb{R}}^{q_I}$ normal to it. A prospective off-diagonal distribution $a(q,r)$ would be rescaled by

In distributional notation this includes the usual ${\rho }^{-q_I}$ test-function Jacobian.

The required estimate is not merely a bound for one fixed smooth value of $q$. It must hold uniformly on compact tangential sets, with the relevant distributional and wavefront seminorms controlled, including the approach to the other collision strata.

After a joint E1 prescription has been obtained, E2 requires finite transverse degrees for the three-vertex and four-vertex strata, together with compatible estimates at their intersections.

In each chart, an asserted degree ${\omega }_I$ must control the rescaled distribution on bounded families of tests, with the necessary microlocal control in the tangential variables. The singular spectators must remain distributions in those variables; they are not replaced by pointwise smooth coefficients.

For example, a partial collision $y=z$ has codimension four while an observation vertex $x$ remains tangential and may satisfy $x\ne z$. A degree-four point-contact classification at the total diagonal says nothing by itself about the singular dependence on this uncollapsed variable.

The inherited primitive statement is retained:

Its finite contact ambiguity has normal derivative order at most four XIIIC. That primitive extension is not repeated.

The graph expression $2E_I+D_I$ remains a superficial checklist. It is not substituted for the uniform estimates of Definition def:transverse.

The available submanifold extension theorems assume an already defined off-submanifold distribution and appropriate scaling and microlocal bounds BF,Dang (Sections 5–6). Here the prerequisite distribution at the characteristic configuration is missing. Those theorems cannot discharge E1.

2 status No uniform transverse theorem for the selected triangles, four-vertex cycles, or their intersections is released. E2 is Unresolved and has not been reached by the present E1 attempt.

The isolated primitive extensions are inherited constructions. The remaining problem is compatible extension of the complete graphs, not another extension of the same primitive.

Suppose, conditionally, that local extensions $u_i$ have been constructed on an open cover $\{U_i\}$ of a graph domain. A permitted adjustment $c_i$ must be a well-defined contact of the frozen class. Agreement on overlaps requires

or equivalently

If the local extensions and contacts are defined and satisfy (eq:overlap), they determine a unique distribution on their union. This statement does not establish existence of contacts solving (eq:overlap) in ${\mathfrak{C}}_{\mathrm{part}}$.

Proof. A smooth locally finite partition of unity subordinate to the cover defines a distributional pairing by summing the local pairings. The overlap identities make this pairing independent of the partition and agree with each local distribution. Uniqueness follows from localization of every compact test.

The automatic identity

on a triple overlap is only a consistency condition. It does not prove solvability within the restricted spectator, jet, and normal-order class.

Moreover, before E1 is completed, local prescriptions may differ on noncoincident characteristic sets. Such a difference is not made contact-supported by the gluing notation.

The ordered values that must survive are

These are the inherited values for full functional supports, not a statement using $\mathrm{supp}f$ alone.

Every allowed interaction–observable contact vanishes on a strictly ordered, disjoint attachment region, by Lemma lem:rigidity. Thus contacts cannot change its prescribed off-collision value. This establishes the support part of the ordered-value audit, not the complete overlap construction.

For the common-frequency attempt, agreement with the canonical product on the admissible ordered attachment domains follows from Proposition prop:good-limit. No exact causal factorization is asserted at finite $\tau$: Gaussian damping has noncausal tails.

3 status The overlap equations and their contact-support restrictions are explicit. No family of compatible complete-graph extensions satisfying them has been constructed. E3 is Unresolved.

No joint pushforward is performed at this gate because E1–E3 are unfinished. The following lemma identifies one useful sufficient wavefront input for a future endpoint proof.

Let $a\in {\mathcal{D}}^{\prime }({\mathbb{R}}^{4(n-1)})$ and define a numerical vertex distribution by submersion pullback,

Then

Multiplication by smooth external weights and finite differential operators preserves this inclusion.

Furthermore, a distributional limit of unweighted amplitudes (eq:A-reg), when such a limit exists, is invariant under simultaneous translation of all vertices and has this relative-coordinate form.

Proof. In coordinates $s=v_1$ and $r_j=v_j-v_1$, the distribution is $1_s\otimes a(r)$. Its wavefront covectors have zero $s$ component. Changing back to vertex coordinates makes that component the sum of all vertex covectors. This gives the asserted bound by the submersion pullback theorem Hormander.

Smooth multiplication and differentiation do not enlarge the wavefront set.

Each unweighted regularized amplitude depends only on differences, so its derivative in every simultaneous-translation direction is zero. Distributional convergence preserves those equations. In the coordinates $(s,r)$, a distribution whose $s$ derivatives all vanish is constant in $s$, hence is $1_s\otimes a(r)$.

The lemma is not an existence theorem for $a$. Nor is the relative-coordinate ansatz imposed as an additional restriction on every extension in ${\mathfrak{C}}_{\mathrm{part}}$. It is one sufficient endpoint structure that the selected attempt would preserve if the needed limits and compatible extensions were established.

Conditionally, if the complete extended numerical graphs have the balance of Lemma lem:relative, their residual delta attachments are compatible with them: a cancellation in the product would have to vanish at each separate external slot, forcing the corresponding attachment covectors to vanish. The compact graph closure then makes the projection proper.

The proper-pushforward wavefront condition sets internal covectors to zero and leaves a sum-zero bound on the two residual covectors. This is the conditional endpoint argument, not an application to an existing joint graph distribution.

The required output would be an even, ghost-number-zero polynomial of degree two with compact kernel $t^{(1)}$ satisfying

Only after that proof could the selected coefficient $Q_H{R}_1^{(1)}\in {\mathcal{E}}_H$ be released.

The tree coefficient would remain unchanged because the new datum enters at order ${\hslash }^2$ in ${\mathsf{T}}_2$ and at order $\hslash$ after the retarded division. The inverse signs and the inherited branch overlap cannot be changed at that order.

Properness is available as a support theorem from XIII-B. It cannot define a missing distribution. Likewise, Proposition prop:finite cannot be passed to an arbitrary distributional limit without a uniform or structural wavefront argument.

4 status{ There is no admitted joint output ${R_1}^{(1)}$ in this release. The endpoint implication is conditional on the preceding gates. E4 is Unresolved.}

Four predecessor boundaries remain separate:

1. XIII-D excludes integrated classical completions in its parameter-local/one-retarded, shell-descending class on the detecting data. Its admitted cochain is not an integrated completion and is not an extension coefficient.

2. XII-F's nonzero witness remains a boundary of unqualified full-shell invariance. It is not a normalization condition for the present product.

3. XIII-A's additional vanishing space is not a quantum ideal. No quotient by ${\mathcal{N}}_W$ is introduced.

4. XIII-B's literal retarded-factorization equality is already obstructed. Its corrected ordered values are retained.

None of these is a theorem that the selected two-line pair has no extension.

No action of the free differential on the absent quantum coefficient is printed. No binary Ward defect is formed. A Constructed release of the selected pair would remove one necessary prerequisite for that later question; it would not automatically construct the binary data with differentiated entries or prove their compatibility.

No antifields, master equation, interacting differential, state, or physical cohomology class are added. No reflected-cone subtraction or triangle residue is used. The ramp width remains fixed, and no limit $\epsilon \to 0$ is taken.

Evaluation on ${\mathcal{C}}_W$ is subsequent to admission. It neither defines the graph product nor permits a restriction of its configuration-space domain. The tree zero-evaluation subclasses do not imply zeros of a degree-two quantum coefficient.

For the fixed selected pair and the copied class ${\mathfrak{C}}_{\mathrm{part}}$:

1. The common-frequency regularization defines compact admitted quadratic graph functionals at every positive regulator.

2. Its numerical graph products converge to the prescribed products on the microlocally admissible open sets. Auxiliary compact-localization changes have a remainder smaller than every regulator power.

3. Contacts in ${\mathfrak{C}}_{\mathrm{part}}$ cannot alter a prescription on the collision complement. Agreement only on the admissible subset does not prove that two prescriptions differ by permitted contacts.

4. The selected characteristic-set limit, the uniform transverse estimates, the contact overlaps, and the complete output theorem have not all been established.

The requested coefficient therefore retains the ending Unresolved, with E1 the earliest unfinished gate. No assertion excludes every extension in ${\mathfrak{C}}_{\mathrm{part}}$.

Proof. Proposition prop:finite, Lemma lem:smoothing, Propositions prop:localization and prop:good-limit, Lemma lem:rigidity, and Proposition prop:agreement prove the affirmative statements.

Definition def:E1-missing identifies the missing characteristic-set limit and its prescription independence. Sections sec:E2–sec:E4 state the subsequent obligations without asserting that they have been discharged. Under the declared release criteria, the absence of those proofs permits neither Constructed nor Proved obstructed for the full coefficient.

The conclusion is deliberately narrower than a joint-extension theorem. Finite regularization, canonical products on proper subdomains, and compact support control do not complete that theorem.

A successor receives:

1. the unchanged pair and contact class;

2. the explicit common-frequency family (eq:smoothing)–(eq:t-reg);

3. the finite-regulator admission and compact-localization remainder estimate;

4. the canonical limits on the admissible open sets;

5. the theorem that collision contacts cannot repair an off-collision prescription failure;

6. the precise unfinished characteristic-set obligation;

7. the transverse, overlap, and proper-pushforward interfaces.

It does not receive a named admitted ${R_1}^{(1)}$. It receives no universal class obstruction and no permission to form the binary Ward defect.

To continue the same attempt, the successor must prove convergence and prescription independence on compact tests meeting the noncoincident characteristic set, with complete tensor sums retained. Only then can the three- and four-vertex transverse estimates and compatible partial contacts be addressed as extensions of an existing distribution.

A different prescription must be named as a different E1 attempt, while preserving the copied pair and ambiguity class. The auxiliary-cutoff theorem does not establish equivalence of every possible prescription.

The tree datum is not the quantum coefficient. The sum-zero test does not cross the noncoincident characteristic set. Exactly one prescription route has been attempted, and its earliest missing theorem is explicit. Neither the partial results nor the unresolved interface constitute an interacting BRST algebra.

scope. XIII-C supplies the graph inventory, contact class, primitive extensions, characteristic cancellation, and four successor obligations. XIII-B supplies the tree datum, convention, properness, and ordered values. XIII-A and XIII-D supply boundaries only. XII-F's witness is copied and not recomputed.

proof scope. The new results concern the common-frequency family, its finite-regulator kernels, convergence on the admissible sets, auxiliary compact-localization independence, off-collision contact rigidity, and a conditional relative-coordinate endpoint.

scope. No convergence theorem through the full noncoincident characteristic set is asserted. No superficial degree is promoted to a scaling theorem for an undefined distribution. No isolated primitive extension is promoted to a compatible joint extension.

scope. The selected regulator does not change the frozen action inverse, the insertion sign, the Wick combinatorics, or the external backgrounds. Its positive-regulator tails are not called exact causal factorization. Ordered values are obligations of the limit.

. No successful pdflatex execution, proof-assistant check, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The manuscript releases the stated partial proofs and an explicit unresolved joint gate, not a completed quantum construction.

XIIIC

N. Cheung, Landmark XIII-C: The joint two-line extension of one selected binary pair, on the unchanged carrier. Supplied manuscript XIII-C.tex. Inventory, partial-diagonal class, primitive extensions, characteristic cancellation, and four successor obligations.

XIIIB

N. Cheung, Landmark XIII-B: One binary causal datum for the displayed coefficient, on the unchanged carrier. Supplied manuscript XIII-B.tex. Tree datum, properness, retarded convention, ordered values, and unresolved two-line gate.

XIIIA

N. Cheung, Landmark XIII-A: Classical cancellation on a fixed subclass, and the obstruction to a coefficientwise restriction quotient. Supplied manuscript XIII-A.tex. Evaluation-only boundary and non-ideal theorem.

XIIID

N. Cheung, Landmark XIII-D: One bounded classical completion of the auxiliary pairing, or its obstruction on the unchanged sheet. Supplied manuscript XIII-D.tex. Class-qualified classical obstruction and distinction between an admitted cochain and an integrated completion.

XIIF

N. Cheung, {Landmark XII-F: The gauge-fixed remainder of ${{\Phi }^{(2)}}_{\mathrm{D}}$: companion support, the auxiliary pairing, and what is not a Ward identity}. Supplied manuscript XII-F.tex. Retained full-shell witness and its classical scope.

XIIE

N. Cheung, Landmark XII-E: Assembly of the second-order relational coefficient after admission of the density-weighted cubic insertion. Supplied manuscript XII-E.tex. Displayed coefficient, insertion sign, and scalar-sourced slot.

XIIC

N. Cheung, Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question. Supplied manuscript XII-C.tex. Unchanged selected action.

BF

R. Brunetti and K. Fredenhagen, “Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds,” Communications in Mathematical Physics 208 (2000), 623–661, arXiv:math-ph/9903028. Microlocal product continuity, scaling degree, and extension across points and submanifolds.

Dang

N. V. Dang, “The extension of distributions on manifolds, a microlocal approach,” Annales Henri Poincaré 17 (2016), 819–859, arXiv:1412.2808. Scaling and wavefront conditions for submanifold extensions.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428. Functional domains, time ordering, and renormalization; not an imported joint-extension theorem for the selected pair.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, second edition, Springer, 1990. Submersion pullback, distribution products, differential-operator wavefront bounds, and proper pushforward.