Landmark XV-C The Full Joint Characteristic Target: Verified Reconstruction Tests and the Instantiation Boundary

XV-C addresses the proposed instantiation of XV-B's relative-lifting framework for the frozen six-branch joint characteristic target. The supplied data retain the grouped $K_{47}$ contribution, both placements of $K_5$, and the supported sector $K_6+K_8+K_9$. They do not supply the complete cross-line operators, reconstruction space, correction class, or output definitions needed to construct the full graph operation.

We therefore take the explicitly permitted unresolved outcome, rather than inventing the missing graph data. The new mathematical results are local tests needed by the proposed application. We give a complete reconstruction-null criterion in a specified finite normal-jet module, collect the supplied supported-sector coefficients through normal order three, and prove an exact distributional extension criterion expressed by test-function estimates. This criterion yields a necessary and sufficient test for unrestricted supported realization of the supplied normal form. It also provides explicit failure witnesses and a support-sensitive description of the remaining ambiguity.

These results do not evaluate the unsupplied cross-line coefficients or establish membership in the frozen correction class. The full joint map remains unresolved before graph-level instantiation. No exhaustive obstruction, complete-graph wavefront theorem, or admitted quantum coefficient is asserted.

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

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

The intended target is a map

for the actual frozen joint graph. Its value on the retained sum must use the prescribed grouped cross-lines and mixed vertex Hessian, agree with canonical composition on its proved domain, descend under reconstruction, and satisfy the required complete-graph analytic bounds.

The supplied XV-C plan expressly requires a prerequisite stop if the defining graph data cannot be recovered. That stop applies here. A construction or a nonexistence theorem for the actual full graph cannot be justified from operators and classes that have not been specified.

Accordingly, this manuscript has two distinct conclusions:

1. the local mathematical tests stated below are proved completely;

2. their application to the full frozen graph remains unresolved at the source-completeness gate.

The second conclusion is not a mathematical obstruction to existence.

We retain the technical-continuation convention adopted in the supplied XV-B manuscript XVB. Thus XV-B and XV-C may study the unresolved construction before admission. The admission success criterion remains

It is a prerequisite for the intended downstream quantum use, not a prerequisite for writing this technical continuation.

The supplied XV-A text uses the earlier convention under which “XV-B remains closed” XVA. That naming convention is not reimposed here. The retained mathematical boundary is that the required quantum coefficient has not been admitted in the supplied releases.

The direct inputs are:

1. the supplied full XV-A and XV-B texts;

2. the supplied XIV-H, XIV-I, and XIV-J excerpts;

3. the proposed XV-C plan, used as a statement of objectives, not as a source of construction theorems.

The broad retrieval account supplies documentary context only. A passage having been retrieved from a manuscript does not supply every definition in that manuscript or transfer its theorems to the present problem.

No result from the relative classical branch or the unfinished XIII-E route is imported. No claim about the current state of an external manuscript registry is made.

All distributional arguments below are local in Euclidean charts. They extend componentwise to finite-dimensional residual fibers. Statements attributed to predecessors retain their supplied domains and are not independently verified here.

The retained kernel is

The first sum contains six original branches $B_4,\dots ,B_9$: $K_{47}$ groups $B_4+B_7$. The $K_5$ branch has two retained placements.

Write

These superscripts label placements, not additional original branches.

The operational list

has six entries, but it is not a list of six independently labelled original branches: the first entry groups two originals and the next two entries belong to the same original.

This ledger checks the supplied decomposition. It does not verify unsupplied tensor coefficients in a proposed joint operation.

The two supplied $K_5$ placements are

The supplied XIV-H proposition expresses them as finite differential operations on the tensor-product pullbacks associated with

Hkernel. XV-B independently checks the existence mechanism for these internal products XVB.

The other displayed source expressions are

and

Their complete inherited ingredients are not defined by these formulas alone Iamplitude.

In particular, the existence of internal products in independent variables does not establish their product with additional singular cross-lines in the attachment variables.

The supplied internal pole chart is

with $s\in \{+1,-1\}$ and

It represents the existing retarded line Ipole. Its delta part is not adjusted in this manuscript. No graph regulator, finite-part prescription, or restoring limit is introduced.

A source-complete instantiation requires:

1. the actual presentation space ${\mathcal{P}}_{\mathrm{int}}$, reconstruction map $\mathsf{Rec}$, carrier ${\mathcal{K}}_{\mathrm{int}}=\mathrm{im}\mathsf{Rec}$, and the topology required of the map;

2. the attachment domain $W$, characteristic loci, canonical open domain $O=U_{\mathrm{can}}$, and the relevant incidence equations;

3. the canonical contraction $m^{\circ }$, the grouped cross-line operators, mixed vertex Hessian, tensor coefficients, weights, and normalization;

4. the permitted candidate and correction classes, with their contact, partial-diagonal, continuity, tensor, and microlocal restrictions;

5. the complete-graph target bounds and the definitions of the output operations, $Q_H$, and ${\mathcal{E}}_H$.

The supplied material does not satisfy the instantiation-data requirement above. In particular, it does not define an evaluable preliminary full characteristic lift for the actual graph.

Proof. The supplied full XV-B text explicitly records that the source definitions of the cross-line factors, frozen map class, $Q_H$, and ${\mathcal{E}}_H$ were not reproduced. Its reconstruction interface is introduced as an abstract interface, separate from the unsupplied physical definitions.

The additional XIV-H–XIV-J excerpts reproduce internal formulas, open-gate statements, a supported reference, and a conditional extension interface. They do not reproduce the missing grouped cross-line operators or the definition of the frozen map class. The supported coefficients are displayed using $v_{\alpha }$ and $c_{D,\beta }$ without explicit formulas for those ingredients.

Consequently F1–F5 are not all supplied. This is a statement about the provided text, not a claim that the missing definitions do not exist elsewhere.

The stop in Proposition prop:prerequisite is documentary:

It neither proves failure of an analytic construction nor changes the inherited analytic stop at the full joint map.

The remaining sections prove local tests that identify exactly what must be checked once the missing definitions are supplied.

Let

where $X\subset {\mathbb{R}}^d$ and $V_q\subset {\mathbb{R}}^m$ are open, and $0\in V_q$.

For a multi-index $\gamma \in {\mathbb{N}}^m$ and $b\in {\mathcal{D}}^{\prime }(X)$, define

Fix $T\in {\mathcal{D}}^{\prime }(X)$ and an integer $r\ge 0$. Define the specific presentation space

and its reconstruction

Multiplication here is by smooth functions and is defined.

This is a local test module. It is not declared to equal the inherited ${\mathcal{P}}_{\mathrm{int}}$.

For distributions $b_{\gamma }\in {\mathcal{D}}^{\prime }(X)$,

Proof. Choose $\chi \in \mathcal{D}(V_q)$ equal to one near zero and define

For all multi-indices $\beta$,

The fact that $\chi$ is constant near zero makes all its positive order derivatives at zero vanish.

Testing the finite insertion sum against ${\mathsf{J}}_{\gamma }\psi$ therefore gives $⟨b_{\gamma },\psi ⟩$. This proves the forward implication; the reverse is immediate.

For $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}(T)$, set

Then

and

Thus (eq:completekernel) is a complete null-relation test for this module, not merely a list of necessary identities.

Proof. For one summand and any test $\varphi$,

The Leibniz rule gives

Using (eq:insert), this becomes

Collecting the coefficient of ${\mathsf{E}}_{\gamma }$ yields (eq:B) and (eq:jetnormal). Lemma lem:extraction then proves (eq:completekernel).

The equations in (eq:completekernel) are distributional equations. They cannot be replaced by equations saying that the smooth scalar coefficients vanish pointwise. A nonzero distribution $T$ can be annihilated by a nonzero smooth function. For example, $x\delta (x)=0$. The criterion therefore retains, rather than suppresses, tangential annihilator relations.

Let $e_j$ denote a coordinate multi-index. Directly from (eq:insert),

For $\alpha =e_j$ this gives

The supplied XIV-J chart uses the corresponding one-variable reconstruction convention

Consequently its reported null presentation is

Theorem thm:jetnull includes this identity in its local model Jnull.

For a candidate $F$ defined on the two supplied summands, descent requires the concrete equality

Canonical agreement establishes only its restriction to the proved canonical domain.

Theorem thm:jetnull does not prove that:

1. the full inherited presentation space is a finite normal-jet module;

2. all its null relations are generated by (eq:inheritednull) or by (eq:qidentity);

3. both complete $K_5$ placements, $K_{47}$, and every residual term are covered by this local module;

4. the actual joint operation acts continuously on this module.

The full descent test remains

A verified local null test cannot replace a complete description or other complete treatment of the actual reconstruction kernel.

For a vertex component $D$, the supplied supported reference is

It is a predecessor input for $K_{\mathrm{supp}}$ only Jreference.

The products in (eq:reference) are understood on their proved source domain. This manuscript does not define them at the omitted characteristic points.

The unique coefficients in the supplied normal chart are

In particular:

Proof. In (eq:reference), put $\gamma =\alpha -\beta$. Then $\alpha =\gamma +\beta$ and

This gives (eq:collected). The displayed degree formulas are the finite sums according to $|\beta |\le 3-|\gamma |$.

Uniqueness is Lemma lem:extraction, applied locally on the source domain.

The formulas collect the actual supplied reference expression. They are not a numerical or analytic evaluation of $v_{\alpha }$ or $c_{D,\beta }$, whose defining formulas are absent.

In a realization supported on the response incidence and preserving normal order at most three, lower insertion orders cannot replace a missing coefficient at a different insertion order.

Proof. Every coefficient is independently extracted by the corresponding ${\mathsf{J}}_{\gamma }$ test. Thus a finite sum of changes at other insertion orders has zero extracted coefficient at order $\gamma$.

This does not rule out cancellations inside a collected coefficient in (eq:collected). Such cancellations would have to be proved using its actual ingredients.

This section supplies the estimate that must be checked for the coefficients in Section sec:coefficients.

For an open $X\subset {\mathbb{R}}^d$ and $\varphi \in \mathcal{D}(X)$, extend $\varphi$ by zero outside $X$ and set

For a compact $K\subset X$, tests with support in $K$ are understood to have the usual test-function topology.

Let $Z\subset X$ be relatively closed and $b^{\circ }\in {\mathcal{D}}^{\prime }(X\setminus Z)$. The following are equivalent:

1. There exists $b\in {\mathcal{D}}^{\prime }(X)$ with

   $$b{|}_{X\setminus Z}=b^{\circ }.$$

2. For every compact $K\subset X$, there are an integer $k_K\ge 0$ and a constant $C_K<\infty$ such that

   $$|⟨b^{\circ },\varphi ⟩|\le C_K{p}_{k_K}(\varphi )$$

   for every $\varphi \in \mathcal{D}(X\setminus Z)$ with $\mathrm{supp}\varphi \subset K$.

When extensions exist, their differences are exactly the distributions in ${\mathcal{D}}^{\prime }(X)$ supported in $Z$.

Proof. Suppose (i) holds. Distributional continuity on tests supported in a fixed compact $K$ gives a finite-order estimate

On the tests in (ii), the pairing with $b$ is the pairing with $b^{\circ }$. This proves necessity.

Assume (ii). Choose a relatively compact open set $V\subset X$ and a compact $K\subset X$ containing $\overline{V}$. On the linear subspace $\mathcal{D}(V\setminus Z)\subset \mathcal{D}(V)$, the functional defined by $b^{\circ }$ is bounded by $C_K{p}_{k_K}$.

The real or complex Hahn–Banach theorem extends this functional to a linear functional $B_V$ on $\mathcal{D}(V)$ satisfying the same bound. Since $p_{k_K}$ is continuous on every fixed-compact test space, $B_V$ is a distribution on $V$. It agrees with $b^{\circ }$ on $V\setminus Z$.

Take a locally finite cover $\{V_i\}$ of $X$ by such relatively compact open sets and a smooth partition of unity $\{{\chi }_i\}$ subordinate to it. Choose the extensions $B_{V_i}$ above and define

For each compactly supported test, only finitely many terms contribute. On each fixed compact test space this is a finite sum of continuous functionals, hence a distribution.

If $\varphi$ is supported in $X\setminus Z$, every term is the pairing of $b^{\circ }$ with ${\chi }_i\varphi$. The partition of unity gives

This proves sufficiency.

Finally, two extensions have zero difference after restriction to $X\setminus Z$, which is equivalent to support in $Z$. Conversely, adding a distribution supported in $Z$ changes no such restriction.

The proof constructs unrestricted distributional extensions. It does not select a normalized characteristic prescription, ensure a prescribed wavefront cone, or prove membership in the inherited contact class. Its arbitrary extension choices are not adopted as a prescription for the frozen joint graph.

If (eq:extensionbound) fails for a compact $K$, then there exist tests

such that

Any such sequence excludes a distributional extension of $b^{\circ }$ to $X$.

Proof. Failure of the estimate means that for each $n$ the bound with order $n$ and constant $n$ fails. Choose a nonzero test ${\psi }_n$ with

The seminorm $p_n({\psi }_n)$ is positive because it includes the zeroth derivative. Dividing by it gives (eq:failuresequence).

If an extension existed, on $K$ it would satisfy $|⟨b,\varphi ⟩|\le C{p}_k(\varphi )$ for some $C,k$. For $n\ge k$, the sequence obeys $p_k({\varphi }_n)\le p_n({\varphi }_n)\le 1$, whereas its pairings exceed $n$. This contradicts the estimate for all sufficiently large $n$.

This is an exact failure test. No sequence of the form (eq:failuresequence) is asserted for an inherited coefficient in this release.

Continue in a product normal chart

Let $A\subset U$ be relatively closed and set

Suppose $t^{\circ }\in {\mathcal{D}}^{\prime }(U\setminus A)$ is supported on $S\setminus A$ and has the local finite normal form

The equality is understood in product neighborhoods contained in $U\setminus A$ that meet $S$; away from $S$, the distribution is zero.

A supported realization of normal order at most $r$ means a distribution supported in $S$ that annihilates tests whose normal jets through order $r$ vanish. This is the support-explicit convention of XV-B.

Under these hypotheses, the following are equivalent:

1. There exists a supported realization $t\in {\mathcal{D}}^{\prime }(U)$ of normal order at most $r$ with

   $$t{|}_{U\setminus A}=t^{\circ }.$$

2. For every compact $K\subset X$ and every $|\gamma |\le r$, there are constants $C_{K,\gamma }$ and integers $k_{K,\gamma }$ such that

   $$|⟨{b_{\gamma }}^{\circ },\psi ⟩|\le C_{K,\gamma }{p}_{k_{K,\gamma }}(\psi )$$

   whenever $\psi \in \mathcal{D}(X\setminus Z)$ and $\mathrm{supp}\psi \subset K$.

When these conditions hold, every such realization has a unique representation

with $b_{\gamma }$ extending ${b_{\gamma }}^{\circ }$. The unrestricted supported ambiguity is

Proof. A distribution of the stated supported normal order has a unique finite insertion representation. For completeness, define its coefficients by

For a test $\varphi$, the difference between $\varphi$ and the finite sum of its lifted signed normal jets has zero normal jets through order $r$. The defining annihilation condition gives (eq:realization). Lemma lem:extraction proves uniqueness.

If (i) holds, near each point of $S$ outside $A$ choose a product neighborhood disjoint from $A$. The realization and the prescribed reference agree there. Extracting coefficients shows that each $b_{\gamma }$ restricts to ${b_{\gamma }}^{\circ }$. Theorem thm:extension gives (eq:jetbound).

Conversely, (ii) and Theorem thm:extension extend each ${b_{\gamma }}^{\circ }$ to a distribution on $X$. The finite insertion sum is defined on $U$, supported in $S$, and has normal order at most $r$. On each product neighborhood outside $A$ meeting $S$, its coefficients agree with those in (eq:puncturedform); hence the distributions agree by coefficient uniqueness. Off $S$, both vanish. These local equalities prove the required restriction.

The extension differences of the coefficients are exactly the distributions supported in $Z$, by Theorem thm:extension. Uniqueness of the insertion representation gives (eq:ambiguity).

For the supplied supported reference, take

Its unrestricted supported normal-order-preserving extension exists locally if and only if the collected distributions (eq:degree3)–(eq:degree0) satisfy (eq:jetbound) on the tangential characteristic complement.

Proof. The source provides the supported normal form on its stated domain. Proposition prop:collected gives its unique coefficients. Apply Theorem thm:supported.

Corollary cor:sourceestimate replaces the bare assertion “the coefficients extend” by a precise test-function estimate. It does not prove that estimate for the actual coefficients.

In particular, the top-order requirement is a bound for

and the order-two requirement is a bound for the full difference in (eq:degree2), not merely a bound for each internal line before the cross-line traces are taken.

Suppose, on a relatively compact tangent neighborhood, coefficient extensions can be chosen with a common estimate

Then the corresponding realization satisfies a local estimate of total derivative order at most $k+r$.

Proof. For a test $\varphi$ in a fixed product neighborhood, put

Tangential derivatives of ${\psi }_{\gamma }$ through order $k$ are bounded by derivatives of $\varphi$ through total order $k+|\gamma |\le k+r$. Thus

with the last seminorm taken in the product coordinates.

For the supplied sector, the normal-order contribution is at most three. The estimate is ordinary local distributional continuity. It is not a target transverse scaling degree, a complete-graph wavefront estimate, or a normalized admission theorem.

For

one has

Consequently, if $C\subset U$ is relatively closed, a supported normal-order-preserving change $\Delta t=\sum {\mathsf{E}}_{\gamma }{d}_{\gamma }$ is supported in $C$ if and only if every $d_{\gamma }$ is supported in

Proof. The inclusion from right to left as an upper bound on the support follows from the insertion definition.

For the reverse inclusion, suppose $t$ vanishes in a product neighborhood of $(x,0)$. Choose the cutoff in ${\mathsf{J}}_{\gamma }$ inside that normal neighborhood. Testing there extracts every coefficient, so all $b_{\gamma }$ vanish near $x$. This proves the support equality and then the stated equivalence.

Thus identifying (eq:ambiguity) with a contact class requires both a support comparison and all further frozen tensor, normalization, continuity, and microlocal restrictions. The support comparison alone is not enough.

Once the actual reconstruction and canonical contraction are specified, write

and

No equality between $W\setminus O$ and the characteristic locus is assumed.

A preliminary lift must first be defined:

Only then is its null defect

an evaluable object.

If the actual permitted candidate family is proved to be $F_0+\mathcal{A}$, with $\mathcal{A}\subset \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$ a specified vector space, XV-B gives the exact descent criterion

It uses membership in the actual correction image, not merely its closure XVB.

The supplied data do not provide an evaluation of $d_0$ or of (eq:relative) for the full frozen graph.

Proof. Proposition prop:prerequisite identifies the missing full cross-line operation, full presentation space, and actual correction class. No characteristic preliminary lift $F_0$ for these data is defined in this release.

Without such a lift, $F_0{|}_H$ has no computed value. Without the actual correction class, its restriction image and relative quotient cannot be evaluated for the inherited instance. The local equations in Sections sec:relations and sec:supported do not supply either missing object.

If all permitted corrections have support in a closed contact locus $C$, a computed null presentation $A\in H$ and an open set $G\subset W\setminus C$ satisfying

would exclude descent in that correction family. Indeed, for every permitted $a$,

This is the contact-support test of XV-B.

No such separated defect is computed here. Nor is the actual correction class proved to be exhausted by a contact-valued affine family. Therefore no obstruction outcome is asserted.

Failure of one selected extension, one correction, or one ordinary product criterion would not exclude all permitted constructions. Conversely, arbitrary Hahn–Banach extensions or algebraic corrections would not establish a construction in the frozen class.

The full decision requires either:

1. an actual admissible full map, with its defining operations and all required bounds proved; or

2. an incompatibility theorem in a class proved to exhaust the permitted maps.

Neither is supplied by the present local tests.

The predecessor data distinguish three operations:

1. reconstruction of defined internal distributions;

2. canonical cross-line composition on its proved open domain;

3. the full characteristic graph operation on tests meeting $N$.

The XIV-I internal amplitude result supplies the first operation in its named module, not the third Iamplitude,Ijoint.

The supported reference provides a wavefront bound only for $K_{\mathrm{supp}}$ on $U\setminus N_U$ Jreference. The estimates in this paper concern unrestricted coefficient extendability and finite local distributional order. They do not prove:

1. a complete-graph wavefront bound across the characteristic locus;

2. uniform transverse estimates for the actual joint section;

3. compatibility on intersections of characteristic and collision strata;

4. compatibility with the frozen contact and partial-diagonal ambiguity;

5. proper output pushforward or restriction estimates;

6. the normalized output admission (eq:admission).

A response-normal insertion order is not a transverse degree of the quantum graph. Likewise, internal frequency homogeneity is not a transverse estimate for the later composition. The supplied null presentation is an additional reason presentation degree must not be treated as an intrinsic reconstructed filtration without a descent proof.

The ambient characteristic locus, its attachment-variable lift, the response incidence, and collision loci are not identified with one another. Their equations and intersections must be supplied before stratum-specific estimates can be formulated.

No interacting differential is applied to an unconstructed coefficient, and no quantum Ward defect is formed.

1. Outcome selection. The manuscript takes the plan's unresolved alternative. It does not turn a proposed title or target into a proved construction.

2. Numbering. XV-B denotes the supplied technical-continuation manuscript. Older downstream exit language is retained only as an admission boundary.

3. Evidence. A broad retrieval inventory is not treated as the complete text of every retrieved manuscript.

4. Branch counting. $K_{47}$ groups two original branches; the two $K_5$ placements belong to one original branch. The supported sector is not the full retained sum.

5. Local versus full relations. The normal-jet null criterion is complete for its explicitly defined module only. No complete generating claim about the inherited $\ker \mathsf{Rec}$ is made.

6. Tangential annihilators. Coefficient equations remain equations in ${\mathcal{D}}^{\prime }(X)$. Smooth multiplication of a fixed nonzero distribution is not assumed injective.

7. Extendability versus admissibility. The test-function estimate is equivalent to unrestricted distributional extension. It is not equivalent to membership in the frozen graph prescription class.

8. Simultaneous maps. Pointwise extension of every displayed coefficient does not itself construct a continuous linear operation on the entire inherited presentation space.

9. Support. The supported coefficient theorem explicitly requires support on the response incidence. Additional characteristic terms outside that incidence are not silently discarded.

10. Domain. The canonical open domain is not identified with an entire characteristic complement.

11. Estimates. Ordinary finite distributional order, response-normal order, internal frequency degree, and target transverse degree remain distinct.

12. Prescription. The retarded pole, propagators, carrier, ambiguity class, and normalization are not changed. Hahn–Banach choices are not selected as a graph regulator or finite part.

13. Obstruction. A separated-contact witness is stated as a conditional test. No witness or exhaustive no-go class for the actual graph is invented.

14. Admission. The coefficient is not asserted to be nonexistent. It is not constructed or admitted in this release.

For the local objects explicitly defined in this manuscript:

1. the reconstruction-null space of ${{\mathcal{P}}_r}^{\mathrm{jet}}(T)$ is exactly the simultaneous kernel of the collected coefficient maps (eq:B);

2. the supplied supported-sector normal form has the unique collected coefficients (eq:degree3)–(eq:degree0);

3. a distribution on a closed-set complement extends distributionally exactly when it satisfies the compact-uniform finite-order test estimate (eq:extensionbound);

4. a supported normal-order-preserving realization of the supplied local normal form exists in the unrestricted distribution class exactly when its collected coefficients satisfy (eq:jetbound);

5. its unrestricted supported ambiguity is (eq:ambiguity), with contact support characterized by Proposition prop:support;

6. a sequence (eq:failuresequence) for an actual coefficient would exclude its unrestricted distributional extension.

Proof. Assertion (i) is Theorem thm:jetnull. Assertion (ii) is Proposition prop:collected. Assertion (iii) is Theorem thm:extension. Assertion (iv) is Theorem thm:supported and Corollary cor:sourceestimate. Assertion (v) is the ambiguity part of Theorem thm:supported, together with Proposition prop:support. Assertion (vi) is Proposition prop:failure.

For the inherited physical instance, the outcome is:

This conclusion is justified by Proposition prop:prerequisite and Proposition prop:noevaluation. The supplied XIV-J release already leaves the full map unresolved, and XV-B supplies interfaces rather than an instantiation Jverdict,Jrelease,XVB.

The local release theorem does not supply a value for the full sum on a test meeting the characteristic locus. It does not establish the admissibility of an extension, evaluate a full null defect, or prove complete-graph bounds. Therefore it neither closes the full-map gate nor establishes (eq:admission).

No full joint section or controlled remainder is constructed. The subsequent transverse, contact, output, and admission obligations remain undischarged for the required coefficient. The no-admission statement is not a universal nonexistence theorem Jadmission.

The next substantive calculation requires an explicit package:

1. the complete frozen cross-line and vertex contraction, including all six retained operational blocks;

2. the actual reconstruction space and a complete method of handling its null relations;

3. the incidence equations and the actual coefficient distributions $v_{\alpha }$, $c_{D,\beta }$;

4. the exhaustive permitted construction and correction class;

5. the full graph microlocal target and the output definitions.

Once these are supplied, the supported-sector extension subtask can be attacked by proving (eq:jetbound) for (eq:degree3)–(eq:degree0), or by producing an actual failure sequence. A preliminary full lift can then be tested on (eq:actualnulltest) and on the remaining reconstruction relations.

Successful supported extension would still leave the characteristic compositions of $K_{47}$ and both $K_5$ placements to be joined into one coherent map. A construction of that full map would still leave the subsequent output and admission obligations.

{{0.91} XV-C proves the local reconstruction and coefficient-extension tests stated in this manuscript and records the complete retained branch ledger. It does not replace the missing frozen graph data. The full joint characteristic map remains unresolved before instantiation; no exhaustive obstruction or admitted quantum coefficient is asserted. }

XVA

Manuscript XV-A, supplied XV-A-va.tex. A Model-and-Domain Audit of the Ramp, Graviton, and Joint-Extension Tracks. Full text supplied in the present record. Its earlier downstream numbering convention is distinguished from the technical-continuation convention used here.

XVB

Manuscript XV-B, supplied XV-B-va.tex. Relative Lifting and Reconstruction Descent for the Full Joint Characteristic Target. Full text supplied in the present record. Used for the abstract relative-lifting theorem, supported normal-jet interface, and explicit application boundary.

Hkernel

Manuscript XIV-H, version 1. XIV-H.tex, identifier 05d90ea1. Proposition prop:K5exact, “Exact two-placement kernel formula,” lines 1135–1202. Direct excerpt supplied.

Hverdict

Manuscript XIV-H, version 1. Theorem thm:H2, “Exact H2 verdict,” lines 1539–1582. Direct excerpt supplied. Historical internal-amplitude stop subsequently addressed by XIV-I.

Ipole

Manuscript XIV-I, version 1. XIV-I.tex, identifier 13365486. Proposition prop:pole, “Exact pole decomposition and chart matching,” lines 726–826. Direct excerpt supplied.

Iamplitude

Manuscript XIV-I, version 1. Theorem thm:I2, “The internal amplitude gate is Constructed,” and adjacent precomposition formulas, lines 1288–1378. Direct excerpt supplied. The adjacent precomposition proposition is reproduced partially.

Ijoint

Manuscript XIV-I, version 1. Joint-map interface, Proposition prop:canonical, and Theorem thm:I3, lines 1459–1548. Direct excerpt supplied.

Jdescent

Manuscript XIV-J, version 1. XIV-J.tex, identifier 3c48e041. Proposition prop:descent, “Descent criterion for a coefficient-level construction,” lines 813–846. Direct excerpt supplied.

Jnull

Manuscript XIV-J, version 1. Lemma lem:null, “A normal identity invisible to polynomial-degree bookkeeping,” lines 853–883. Direct excerpt supplied.

Jreference

Manuscript XIV-J, version 1. Theorem thm:reference, “Supported-sector reference distribution,” lines 1215–1299. Direct excerpt supplied. Used only on its stated off-characteristic domain.

Jconditional

Manuscript XIV-J, version 1. Theorem thm:conditionalextension, “Conditional supported realization,” lines 1427–1470. Direct excerpt supplied. The local formulation in this manuscript retains an explicit response-support condition.

Jverdict

Manuscript XIV-J, version 1. Theorem thm:J2, “Exact J2 verdict,” lines 1608–1649. Direct excerpt supplied.

Jrelease

Manuscript XIV-J, version 1. Theorem thm:release, “XIV-J mathematical release,” lines 1894–1939. Direct excerpt supplied. The predecessor exit language uses the earlier downstream numbering convention.

Jadmission

Manuscript XIV-J, version 1. Corollary cor:noadmission, “No quantum admission in this release,” lines 1808–1822. Direct excerpt supplied.