Em01-A Conditional Compatibility of Linear Representatives and the Limits of Witness-to-Graph Inference

Two proposed manuscript bridges are examined without identifying their distinct objects or domains. In a fully specified flat linearized-gravity setting, a compact tensor distribution admitting a smooth compact wave-shell representative has a smooth compact conserved representative on a vacuum solution space containing compact pure gauges if and only if the original distribution is conserved. The causal-propagator smoothness required by the conserved-current construction follows from the wave-shell relation once the bundles and operators have actually been identified. An explicit residual-gauge example shows that smooth compact wave-shell representability alone is insufficient.

For the proposed prepared-response-to-joint-graph bridge, we prove precise information-loss, reconstruction-descent, extension, and normal-jet results. Partial witness contraction is noninjective even on smooth compact kernels. Canonical-domain agreement does not establish global descent, and a bound on normal order does not by itself make supported ambiguities finite-dimensional.

These are completed mathematical results about bridge conditions. They neither identify the actual VII-B tensor with a conserved V-current nor construct the unresolved XV joint graph. The paper-specific conclusions are restricted to the supplied passages and registry descriptions.

Figure fig:split-Em01-A records one view of the same separation. Figure fig:chain-Em01-A records one view of the same separation. Figure fig:pair-Em01-A records one view of the same separation. Figure fig:gate-Em01-A records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-Em01-A records that separation. The drawing adds no hypothesis.

The original plan has two different targets:

1. determine when a linear wave-shell representative can be compared with a compact conserved-current representative;

2. determine whether a prepared witness response supplies the missing joint-graph construction.

Neither target permits replacing an unidentified distribution by a similarly named object from another manuscript.

The first target admits a sharp conditional theorem in a common flat setting. The second admits rigorous non-implication and interface theorems, but not a construction of the actual graph from the supplied information.

The principal results are Theorem thm:compatibility, Proposition prop:counterexample, Theorem thm:descent, and Theorem thm:jetextension. Their proofs are given here. They do not depend on treating the registered manuscripts as a single theory.

The supplied passage from V states a smoothing theorem for a compact symmetric conserved tensor distribution, subject to smoothness of its specified causal propagation. Its conclusion is a smooth compact conserved representative on its declared vacuum solution space V (Theorem thm:smoothing, lines 1178–1245).

The supplied VII-B passage states a different result:

and consequently equality of pairings on smooth solutions of $\square h=0$ VIIb (Theorem thm:onshell, lines 1089–1144). The same manuscript excludes the original nonzero-smearing bilinear insertion from its declared off-shell microcausal polynomial domain VIIb (Theorem thm:main, lines 842–903).

For the graph track, XIV-I constructs the named internal amplitude map but explicitly leaves the characteristic composition with the quantum cross-lines unresolved XIVi (Theorems thm:I2, thm:I3). XIV-J constructs a supported-sector reference on its stated off-characteristic domain and leaves the full joint map unresolved XIVj (Theorems thm:reference, thm:J2). The supplied XV-F record and its patches do not pass F0 XVf,Registry.

XVI-D reports a prepared witness response and a conditional coordinate identification. XVI-H constructs a compact three-slot kernel under explicit inherited inputs; its specified partial contraction is not the full kernel or quantum admission XVId,XVIh.

The mathematical proofs below are self-contained relative to the explicitly stated Green-operator package. Comparisons with the registered manuscripts use only the supplied quotations and registry descriptions. No unquoted manuscript theorem, no unregistered manuscript, and no external-model output is used as a premise.

Let

All tensor operators below have constant coefficients. Indices are raised and lowered with $\eta$.

Let $H={\mathrm{Sym}}^2{T}^{\ast }M$. A symmetric contravariant tensor distribution $C$ is identified with its lowered tensor $c$. Its pairing with a smooth covariant tensor $h$ is

In the smooth case this means

Compact support of $c$ makes this pairing meaningful for arbitrary smooth $h$, without a decay assumption on $h$.

Define trace reversal, the gauge operator, and the de Donder operator by

In four dimensions, $T^2=1$. Trace reversal is self-adjoint for the stated pairing.

The linearized Einstein operator used in this paper is

This equation fixes the normalization and sign convention. The formal gauge adjoint is

We call $c$ conserved when $K^{\ast }c=0$.

Define three spaces of smooth tensors:

They are not identified with any manuscript's named solution space without an additional, explicit domain identification.

The operators above satisfy

Moreover,

Proof. Expanding $DK\xi$, the two gradient-divergence terms cancel, leaving $\square \xi$. Substitution into (eq:einstein-operator), together with commutation of the constant-coefficient operators, gives $\mathcal{L}K\xi =0$.

For any one-form $d$,

Taking the divergence of (eq:einstein-operator) therefore gives

If $Dh=0$, equation (eq:einstein-operator) reduces to $2\mathcal{L}h=-\square Th$. Since $T$ is invertible and commutes with $\square$, the last assertion follows.

The wave equation $\square h=0$ is not the vacuum Einstein equation $\mathcal{L}h=0$ on unrestricted tensor fields. For example, let

Then $\square h=0$, whereas direct substitution gives

Conversely, every compact pure gauge $K\xi$ belongs to ${\mathcal{S}}_{\mathrm{E}}$, but a nonzero compact pure gauge cannot solve the homogeneous wave equation, by uniqueness for zero past data. Thus an on-shell equality on ${\mathcal{S}}_{\square }$ does not automatically give an equality on ${\mathcal{S}}_{\mathrm{E}}$.

The componentwise retarded and advanced Green operators $G_{\mathrm{R}},G_{\mathrm{A}}$ act on compact tensor distributions and satisfy:

1. $$\square G_{\mathrm{R}}a=a=\square G_{\mathrm{A}}a,G_{\mathrm{R}}\square a=a=G_{\mathrm{A}}\square a(a\in {{\mathcal{D}}^{\prime }}_c);$$
2. $$\mathrm{supp}(G_{\mathrm{R}}a)\subseteq J^{+}(\mathrm{supp}a),\mathrm{supp}(G_{\mathrm{A}}a)\subseteq J^{-}(\mathrm{supp}a);$$

3. they commute with constant-coefficient differential operators and with $T$;

4. they take compact smooth sources to smooth solutions;

5. the corresponding retarded, advanced, and smooth Cauchy uniqueness statements hold.

These are the usual componentwise wave Green operators on Minkowski space. We retain them as an explicit analytic input rather than importing an unidentified manuscript propagator.

We use both causal signs:

The wave-shell construction below uses $E_{+}$. The conserved-current formula uses $E_{-}$. This prevents a hidden sign identification between two source conventions.

For $a\in {{\mathcal{D}}^{\prime }}_c(M;H)$,

The same equivalence holds with $E_{-}$.

Proof. If $E_{+}a=0$, then

The support inclusions give

For compact $\mathrm{supp}a$, this intersection is compact in Minkowski space. Hence $r$ is compact and $\square r=a$.

Conversely, if $a=\square r$ with $r$ compact, both Green operators give $r$, so their difference vanishes. The $E_{-}$ statement follows from (eq:causal-signs).

If $c,c_{\mathrm{sm}},r$ are compact tensor distributions and

then

If $c_{\mathrm{sm}}$ is smooth, then

is smooth and spacelike compact.

Proof. Formal self-adjointness and compact support give

Commutation with $T$ and the Green identities give

The last expression is smooth by Assumption ass:green. Its support is contained in the causal hull of a compact set, hence is spacelike compact.

Let $c\in {{\mathcal{D}}^{\prime }}_c(M;H)$, and suppose $u=E_{+}c$ is smooth. Choose a smooth temporal cutoff $\rho$ equal to zero sufficiently far in the past and one sufficiently far in the future. Then

is smooth and compact, and

for some compact tensor distribution $r$.

Proof. The field $u$ solves $\square u=0$ and is spacelike compact. The derivatives of $\rho$ are supported in a bounded time slab, whose intersection with the causal hull of a compact set is compact. Thus (eq:our-wave-representative) is smooth and compact.

Since $c_{\mathrm{sm}}=\square (\rho u)$, uniqueness gives

Therefore $E_{+}{c}_{\mathrm{sm}}=u=E_{+}c$. Lemma lem:compact-primitive supplies the compact $r$.

Let $c\in {{\mathcal{D}}^{\prime }}_c(M;H)$ be symmetric and conserved, and suppose

is smooth. Let $\chi$ be a smooth temporal cutoff, zero sufficiently far in the past and one sufficiently far in the future. Define the covariant representative $j$ and its raised current $J$ by

Then $j$, and hence $J$, is smooth, compact, and conserved. Moreover,

Proof. Since $v=E_{-}Tc$,

Lemma lem:identities gives $\mathcal{L}v=0$. Consequently

The right-hand side is supported in the transition slab of $\chi$. The spacelike compactness of $v$ therefore makes its support compact. Smoothness is immediate, and the Bianchi identity in Lemma lem:identities gives conservation.

First suppose $h\in {\mathcal{S}}_{\mathrm{dD}}$. Put

Both are smooth and compact. Equation (eq:einstein-operator) gives

Integration by parts yields

Hence $⟨j,h⟩=-⟨F,h⟩$.

As $Tv=E_{-}c$, the retarded and advanced identities are

It follows that

By Lemma lem:compact-primitive, $F+c=\square A$ for a compact distribution $A$. Since $\square h=0$,

Thus $⟨j,h⟩=⟨c,h⟩$ on ${\mathcal{S}}_{\mathrm{dD}}$.

Now let $h\in {\mathcal{S}}_{\mathrm{E}}$ be arbitrary. Solve

A global smooth solution exists without spatial decay: componentwise, the zero-data formula is

For negative $t$, the integral has its usual oriented meaning. Each value depends on a finite spatial region, so arbitrary smooth sources cause no convergence issue.

Set $h_0=h-Kw$. The operator identities give

Thus $h_0\in {\mathcal{S}}_{\mathrm{dD}}$. Compact conservation of $c$ and $j$ implies

The equality on $h_0$ therefore extends to $h$, proving (eq:full-vacuum-representation).

If the convention $E_{+}=G_{\mathrm{R}}-G_{\mathrm{A}}$ is used throughout, the same representative is

Changing the causal-propagator sign without changing this prefactor would reverse the represented functional. Equation (eq:conserved-representative) uses the $E_{-}$ convention fixed here.

A vacuum comparison space is a linear subspace $\mathcal{X}\subseteq {\mathcal{S}}_{\mathrm{E}}$ containing every compact pure gauge:

Let $\mathcal{X}$ be a vacuum comparison space. Suppose $c\in {{\mathcal{D}}^{\prime }}_c(M;H)$ is symmetric and has a wave-shell repair

Then the following are equivalent:

1. $K^{\ast }c=0$;

2. $⟨c,K\xi ⟩=0$ for every compact smooth one-form $\xi$;

3. there exists a smooth compact conserved tensor $j$ satisfying

   $$⟨j,h⟩=⟨c,h⟩(h\in \mathcal{X}).$$

When these assertions hold, the representative in (eq:conserved-representative) is available.

For this representative,

In particular, $c_{\mathrm{sm}}$ and $j$ agree on ${\mathcal{S}}_{\mathrm{E}}\cap {\mathcal{S}}_{\square }$, but their equality on all of $\mathcal{X}$ requires the additional condition

Proof. The equivalence of (i) and (ii) is exactly the definition of the distributional formal adjoint.

If (iii) holds, compact conservation of $j$ gives

because $K\xi \in \mathcal{X}$. Hence (iii) implies (ii).

Assume (i). By Lemma lem:wave-equality,

is smooth. Thus the propagator-smoothness hypothesis of Theorem thm:current-smoothing follows from the wave-shell repair rather than being an additional unchecked assumption. That theorem supplies $j$, proving (iii).

Finally, on ${\mathcal{S}}_{\mathrm{E}}$ the representative satisfies $⟨j,h⟩=⟨c,h⟩$. Pairing (eq:bridge-wave-relation) with $h$ gives

which proves the remaining claims.

To apply Theorem thm:compatibility to VII-B and V, one must first prove that the actual $q=L^{\ast }a$, after the stated bundle and duality identification, is the tensor $c$ in (eq:bridge-wave-relation); that the relevant wave operator is this componentwise $\square$; and that V's selected solution space is identified with the comparison space being used.

Only after these identifications does the theorem reduce the remaining full-space representability test to conservation. The supplied passages do not perform those identifications or prove $K^{\ast }q=0$. The theorem therefore supplies a complete conditional bridge, not an unconditional bridge for the registered observable.

If a proposed comparison space omits compact pure gauges, the necessity argument need not apply. In particular, this theorem does not identify a finite-plane representability construction with a full-space admissibility criterion. V distinguishes those constructions V (Object ledger, lines 247–281).

There exists a smooth compact symmetric tensor $c$ such that:

1. $c-c_{\mathrm{sm}}=\square r$ with $c_{\mathrm{sm}}=c$ and $r=0$;

2. $E_{-}Tc$ is smooth;

3. no smooth compact conserved tensor represents $h\mapsto ⟨c,h⟩$ on ${\mathcal{S}}_{\mathrm{E}}$, or even on ${\mathcal{S}}_{\mathrm{dD}}$.

Proof. Choose $g\in {C_c}^{\infty }({\mathbb{R}}^3)$ with $g(x_0)>0$. Let $\zeta$ solve

It is smooth and spacelike compact. For example,

By continuity, ${\partial }_t\zeta >0$ on some nonempty spacetime neighborhood, which may be chosen with $t>0$.

Choose a nonzero nonnegative $\psi \in {C_c}^{\infty }(M)$ supported in that neighborhood. Set

and let $c$ be its lowered tensor. Define

Then

Thus $h\in {\mathcal{S}}_{\mathrm{dD}}$. Nevertheless,

Every compact conserved tensor $j$ annihilates this smooth pure gauge:

Consequently no such $j$ represents $c$ on ${\mathcal{S}}_{\mathrm{dD}}$ or ${\mathcal{S}}_{\mathrm{E}}$. Assertions (i) and (ii) hold because $c$ itself is smooth and compact.

This example disproves an implication between the stated properties. It is not claimed to belong to VII-B's actual range $\{L^{\ast }a\}$, and it is not a counterexample to either manuscript's theorem.

Let $\mathcal{P}$ be an off-shell polynomial domain, and let $\mathcal{B}$ be a functional with $\mathcal{B}\notin \mathcal{P}$. Suppose an admissible quantum algebra requires the classical coefficient of every admitted element to belong to $\mathcal{P}$, as an actual functional on the full off-shell configuration space.

Then no admitted element can have classical coefficient equal to $\mathcal{B}$ on that full space. A coefficientwise formal construction requiring $\mathcal{B}$ as an input cannot be repaired merely by proving an on-shell equality for a linear functional.

Proof. Such an admitted element would have a classical coefficient in $\mathcal{P}$ equal to $\mathcal{B}$, contradicting $\mathcal{B}\notin \mathcal{P}$.

An on-shell linear identity supplies neither membership of the bilinear functional in $\mathcal{P}$ nor equality of off-shell classical coefficients. It therefore does not remove the contradiction.

For the hypotheses of VII-B, the supplied exclusion theorem provides precisely the excluded coefficient for nonzero smearing. Its subsequent formal retarded identity is conditional on existence of its terms and stops at the insertion-domain prerequisite VIIb (Theorem thm:main; consequences, lines 1216–1325). Proposition prop:bilinear preserves that scope. It is not a universal obstruction to all larger domains or modified observables.

A witness response is an observation of a distribution. A joint-graph construction is a distribution, or a map producing one, on a specified full test space. These coincide only if the observation is proved to determine the object.

Let $W$ be a smooth manifold and $V$ a finite-rank residual bundle. A distributional section is a continuous functional on the corresponding compactly supported dual tests. For a linear test subspace $\mathcal{H}$, define

Agreement of two distributions on $\mathcal{H}$ determines their equality exactly when ${\mathcal{H}}^{\perp }=\{0\}$. In particular, density of $\mathcal{H}$ in the full test space is sufficient.

Proof. The difference of the distributions belongs to ${\mathcal{H}}^{\perp }$. This proves the first assertion. If $\mathcal{H}$ is dense, continuity makes that difference vanish on every test.

Let $M_i={\mathbb{R}}^{d_i}$, $d_i\ge 1$, and let $w_1,w_2$ be fixed smooth scalar functions on $M_1,M_2$. For smooth compact kernels $T$ on $M_1\times M_2\times M_3$, define the witness response

Then $\mathcal{W}$ is not injective. Its kernel contains a nonzero smooth compact kernel.

Proof. Choose a nonzero $v_1\in {C_c}^{\infty }(M_1)$ with

Such a function exists: the displayed integral is one linear functional on an infinite-dimensional test space. Explicitly, two linearly independent test functions yield a nonzero linear combination annihilating it.

Choose nonzero $v_2\in {C_c}^{\infty }(M_2)$ and $v_3\in {C_c}^{\infty }(M_3)$, and set

Then $B\ne 0$, but

for every $\phi$. All these kernels are smooth and compact.

A prescribed partial contraction does not, by itself, identify the full kernel, even when compact support, finite-order continuity, and absence of wavefront singularities are already available.

Proof. If a kernel has the prescribed contraction, adding the nonzero $B$ from Proposition prop:partial-contraction preserves that contraction and those regularity properties.

The proposition is an explicit information-loss model. It does not assert that such an addition respects the actual XV contact conditions, tensor incidences, or output wavefront restrictions. Those additional frozen conditions may remove ambiguities, but that must be proved.

Accordingly, a relation obtained from XVI-D's witness response, or XVI-H's specified partial contraction, cannot be promoted to equality of the actual graph distributions without an identification theorem or a proved determining test class XVId,XVIh.

Let $\mathcal{P}$ be a vector space of internal presentations and

a linear reconstruction map. Define

A presentation-level candidate has the form

There exists a unique linear map

satisfying

if and only if

If ${\mathcal{K}}_{\mathrm{int}}$ carries the quotient topology induced by $\mathcal{P}/\ker \mathsf{Rec}$, a continuous $\tilde{\mathsf{M}}$ satisfying (eq:descent) induces a continuous $\mathsf{M}$.

Proof. If the factorization exists, every null presentation is mapped to zero.

Conversely, if $\mathsf{Rec}A=\mathsf{Rec}{A}^{\prime }$, then $A-A^{\prime }\in \ker \mathsf{Rec}$. Condition (eq:descent) gives $\tilde{\mathsf{M}}(A)=\tilde{\mathsf{M}}(A^{\prime })$. Thus

is well-defined and linear. Surjectivity onto ${\mathcal{K}}_{\mathrm{int}}$ gives uniqueness. Continuity is the defining universal property of the quotient topology.

This is the same algebraic interface made explicit in the supplied XIV-J passage XIVj (Proposition prop:descent, lines 813–846). Here its topological qualification is stated separately: algebraic descent alone does not prove continuity for an independently prescribed topology on the reconstructed space.

Let $U_{\mathrm{can}}\subseteq W$ be open, and put

No identification of $Z$ with a particular attachment set is assumed.

Agreement with canonical contraction on $U_{\mathrm{can}}$ can show

without proving $\tilde{\mathsf{M}}(A)=0$ on $W$.

Proof. An explicit model suffices. Let $W=\mathbb{R}$, $U_{\mathrm{can}}=\mathbb{R}\setminus \{0\}$, and $\mathcal{P}={\mathbb{R}}^2$. Fix a nonzero smooth distribution $k$, and set

Define

Its restriction to $U_{\mathrm{can}}$ is the zero canonical map. However, $(0,1)\in \ker \mathsf{Rec}$ and

Thus global descent fails despite canonical-domain agreement. All maps in this example are continuous on the finite-dimensional presentation space.

For a normal coordinate $z$,

Consequently a presentation splitting that treats these two summands independently need not descend to distributions.

Proof. For a test $\phi$,

This is $-⟨\delta ,\phi ⟩$. Each summand is nonzero, although their sum vanishes.

Tensoring this identity with an existing internal distribution does not change it. The supplied XIV-J adapted-chart example uses exactly this mechanism to show why presentation degree is not automatically an intrinsic joint-symbol filtration XIVj (Lemma lem:null, lines 853–883).

Let $t^{\circ }\in {\mathcal{D}}^{\prime }(U_{\mathrm{can}};V)$. If one extension $T_0\in {\mathcal{D}}^{\prime }(W;V)$ exists, all extensions are exactly

where ${{\mathcal{D}}^{\prime }}_Z$ denotes distributions supported in $Z$. This statement does not assert existence of an extension.

Proof. Two extensions have difference zero on $U_{\mathrm{can}}$, equivalently difference supported in $Z$. Conversely, adding a distribution supported in $Z$ does not change the restriction.

There is a smooth function on $(0,1)$ that has no distributional extension to $(-1,1)$.

Proof. Take $u(x)=e^{1/x^2}$. Suppose a distribution $T$ on $(-1,1)$ restricts to $u$ on $(0,1)$.

On a fixed compact neighborhood of zero, continuity of $T$ gives constants $C$ and $m$ such that

for tests supported in that neighborhood.

Choose a nonnegative nonzero $\phi \in {C_c}^{\infty }((-1/4,1/4))$, and for small $\epsilon >0$ put

Its support lies in $(0,1)$, so

For sufficiently small $\epsilon$, points in this support satisfy $0<x<2\epsilon$. Hence

On the other hand,

The exponential lower bound contradicts the polynomial upper bound as $\epsilon \to 0$.

This example does not describe the actual graph amplitude. It establishes why an off-characteristic reference and a conditional extension interface cannot substitute for the missing extension estimates.

Let

where $S_0\subseteq {\mathbb{R}}^d$ is open. Write coordinates as $(s,z)$. Let $A\subseteq S_0$ be relatively closed, and set

For a multi-index $\gamma \in {\mathbb{N}}^k$, define

Thus ${\mathsf{E}}_{\gamma }b=b\otimes {{\partial }_z}^{\gamma }{\delta }_0$.

A distribution $T\in {\mathcal{D}}^{\prime }(U)$ has supported normal order at most $m$ along $S$ if

whenever

This definition includes support in $S$.

This support requirement is essential. Merely asking for an extension of an off-set distribution, without restricting its support or normal-jet class, gives a different problem.

Every distribution of supported normal order at most $m$ has a unique representation

Proof. Choose $\rho \in {C_c}^{\infty }({\mathbb{R}}^k)$ equal to one near zero, and define

For $|\alpha |,|\gamma |\le m$,

Consequently

This proves uniqueness and supplies the coefficient extraction formula

These coefficients are distributions because ${\mathsf{J}}_{\gamma }$ is continuous on test spaces.

For a general test $\phi$, subtract its finite normal Taylor polynomial with the cutoff $\rho$:

The resulting test has all normal derivatives through order $m$ equal to zero along $S$. Applying the defining annihilation property of $T$ proves (eq:normal-form).

Suppose $T^{\circ }\in {\mathcal{D}}^{\prime }(U^{\circ })$ is supported in $S\setminus N$, has supported normal order at most $m$, and has unique coefficients

The following are equivalent:

1. there exists a distribution $T\in {\mathcal{D}}^{\prime }(U)$ of supported normal order at most $m$ satisfying $T{|}_{U^{\circ }}=T^{\circ }$;

2. each ${b^{\circ }}_{\gamma }$, $|\gamma |\le m$, has a distributional extension $b_{\gamma }\in {\mathcal{D}}^{\prime }(S_0)$.

Given (ii), the corresponding extension is

If an extension exists, all extensions in this normal-order class differ by

Proof. Assume (i). Lemma lem:normal-coefficients extracts the coefficients $b_{\gamma }$ of $T$. Testing with ${\mathsf{J}}_{\gamma }\psi$, where $\psi$ is supported in $S_0\setminus A$, proves

Thus (ii) holds.

Conversely, assume (ii) and form the displayed finite sum. It has supported normal order at most $m$. On $(S_0\setminus A)\times {\mathbb{R}}^k$, uniqueness of the normal coefficients gives equality with $T^{\circ }$. At points of $U^{\circ }$ with $z\ne 0$, both distributions are locally zero. These local statements cover $U^{\circ }$, proving (i).

The difference of two extensions has coefficients whose restrictions to $S_0\setminus A$ vanish. Those coefficients are precisely the distributions supported in $A$, proving (eq:jet-ambiguity).

Theorem thm:jetextension constructs an extension from already supplied coefficient extensions. It does not prove their existence, a scaling estimate, a contact normalization, or an admissible graph wavefront bound. For $m=3$, it provides the elementary supported-jet interface relevant to the supplied XIV-J comparison XIVj (Theorem thm:conditionalextension, lines 1427–1470).

For $b\in {\mathcal{D}}^{\prime }(S_0)$, define

For the extension in Theorem thm:jetextension,

Proof. Each summand is $b_{\gamma }\otimes {{\partial }_z}^{\gamma }{\delta }_0$. The tensor-product wavefront bound allows tangential covectors in $\mathrm{WF}(b_{\gamma })$, or the zero tangential covector over $\mathrm{supp}{b}_{\gamma }$, together with normal covectors; the total covector must be nonzero. This gives the displayed set.

Equivalently, after product localization, the Fourier transform is the localized Fourier transform of $b_{\gamma }$ multiplied by a polynomial in the normal frequency. Rapid decay in a tangential cone is preserved where the tangential frequency controls the total frequency. The remaining potentially singular directions are the pure normal ones. Taking a finite sum gives the asserted inclusion.

The bound is an upper bound in terms of the chosen coefficient extensions. It is not a proof that those coefficients satisfy the output cone required by a particular quantum carrier.

A fixed normal-order bound does not, by itself, make the supported ambiguity space finite-dimensional, even when the attachment in $S_0$ is a single point.

Proof. Take $S_0=\mathbb{R}$, $A=\{0\}$, and any $k\ge 1$. For $n=0,1,2,\dots$, let

Every $T_n$ has normal order zero in the $z$-directions and is supported in $A\times \{0\}$.

The distributions $a_n$ are linearly independent: a finite linear combination can be tested against polynomial jets times a cutoff equal to one near zero, isolating the highest derivative and then each lower derivative. Coefficient extraction makes the $T_n$ linearly independent as well.

Additional bounds on tangential order, scaling degree, support geometry, or wavefront sets can reduce this space. None follows from a normal-order bound alone. This result does not transfer an ambiguity classification to the actual XV graph without its frozen source package.

Before a prepared response can be used in the joint-graph problem, the bridge must specify at least:

1. the input and output manifolds and residual bundles;

2. the actual prepared source distribution and support class;

3. the internal presentation space and reconstruction map;

4. the complete retained branch sum, tensor coefficients, vertex operators, and grouped quantum cross-lines;

5. the proved canonical domain, without enlarging it by an unproved complement identity;

6. the characteristic test space, correction parameters, contact conditions, and required output wavefront cone.

A statement of equality must then compare distributions on the full specified test space. Equality after a witness contraction is only an observation of that statement.

The supplied XIV-I passage explicitly distinguishes its reconstructed internal sum from the missing operation of composition with the grouped cross-lines XIVi (Theorem thm:I3, lines 1459–1548). The supplied XIV-J reference is for the supported sector away from its stated characteristic set, not the full six-branch target XIVj (Theorem thm:reference, lines 1215–1299).

A graph bridge certificate consists of:

1. an explicit identification of the prepared-source object with the frozen graph input package;

2. a presentation-level construction on characteristic tests, together with global reconstruction descent;

3. a resulting full joint map agreeing with canonical composition on its proved domain and retaining the complete frozen contraction;

4. the required section and controlled remainder;

5. uniform estimates in the transverse variables required by the graph interface;

6. contact compatibility and the complete output wavefront bound;

7. the independently required proof of membership and normalization in the stated quantum output space.

This is a list of obligations, not a construction. Descent belongs to defining a map on actual reconstructed kernels: it is not an optional conclusion to be postponed after presentation-level formulas have been treated as intrinsic.

A response identity obtained by a fixed partial witness contraction does not, as a matter of mathematical implication, supply a graph bridge certificate.

Proof. Proposition prop:partial-contraction gives distinct smooth compact kernels with identical partial witness data. Hence that identity alone does not determine the required full distribution.

Even canonical-domain agreement does not remove the descent issue, by Proposition prop:restriction-descent. An extension still requires existence and normalization; Proposition prop:affine-extension only classifies its differences if it exists, and Theorem thm:jetextension requires actual coefficient extensions. Finally, Proposition prop:jet-wavefront makes the output bound depend on those coefficients, not on the witness identity.

Thus none of the missing distributional identifications, descent conditions, or output obligations is a formal consequence of a partial witness equality.

Proposition prop:no-witness-certificate does not prove that the actual prepared source cannot participate in a future graph construction. It proves that a stated witness identity is not itself that construction. A sufficiently strong additional identification and composition theorem could change the status.

The supplied record must not be summarized by saying that every manuscript XV-A through XV-F has the identical F0 ending. The individual descriptions have different stops: XV-A retains J2; XV-B records B1; XV-C, XV-D, and XV-E record C0, D0, and E0; XV-F and its patches retain F0 XVrecord,XVf,Registry.

Likewise, the phrase “XV-B remains closed” in the supplied XIV-I and XIV-J releases concerns the downstream admission exit. It must not be read as denying the existence or technical content of the separately registered XV-B manuscript XIVi (Theorem thm:release) XIVj (Corollary cor:noadmission) XVrecord.

A constructed internal kernel is therefore not automatically a product with the quantum cross-lines. An off-characteristic reference is not automatically a characteristic extension. A classical response is not automatically a quantum coefficient.

Under the explicit common-setting assumptions of this paper:

1. a compact symmetric tensor distribution with a smooth compact wave-shell repair has a smooth compact conserved representative on a vacuum comparison space if and only if it is conserved;

2. the possible difference between that conserved representative and the wave-shell representative on the full vacuum space is exactly $-⟨r,\square h⟩$;

3. smooth compact wave-shell representability alone does not imply conserved-current representability, even on the de Donder wave shell;

4. partial witness contraction is noninjective on smooth compact kernels;

5. a presentation-level linear candidate defines a map on reconstructed kernels exactly when it annihilates every null presentation;

6. extension within a fixed supported normal-order class is equivalent to extension of its unique coefficient distributions, and normal order alone does not make the supported ambiguity finite-dimensional.

Proof. Assertions (i) and (ii) are Theorem thm:compatibility. Assertion (iii) is Proposition prop:counterexample. Assertion (iv) is Proposition prop:partial-contraction. Assertion (v) is Theorem thm:descent. Assertion (vi) follows from Theorem thm:jetextension and Proposition prop:infinite-ambiguity.

For V to VII-B, the completed deliverable is a sharp conditional compatibility theorem. In a genuinely common flat setting, the compact wave-shell relation supplies the propagator-smoothness input, while conservation is the decisive full-space condition. The actual conservation and bundle/domain identification for VII-B's $L^{\ast }a$ remain unproved by the supplied record. No repair of the original off-shell bilinear insertion is asserted.

For XV-F to XVI-D, the completed deliverable is a precise non-discharge result: a smooth witness response does not supply a full distributional identification, reconstruction descent, characteristic composition, or output admission. XVI-H's compact three-slot construction does not change that inference without the missing bridge certificate.

Consequently this paper releases neither the actual XV joint graph nor an admitted quantum retarded coefficient. It claims no interacting Ward identity, sharp-width limit, or ultraviolet completion.

V

Cheung, N. V, version 1, V.tex. Enter Calculus Companion. Supplied passages: “Two questions and five distinct objects,” lines 219–281; scope, lines 362–367; Theorem thm:smoothing, lines 1178–1245; Theorems thm:rawclassification and thm:harmonic, lines 1254–1336. Registry: https://enter.grok.me/.

VIIb

Cheung, N. VII-B, version 1, VII-B.tex. Enter Calculus Companion. Supplied passages: Theorem thm:main, lines 842–903; Theorem thm:onshell, lines 1089–1144; “Consequences for second-order retarded products,” lines 1216–1325. Registry: https://enter.grok.me/.

XIVh

Cheung, N. XIV-H, version 1, XIV-H.tex. Enter Calculus Companion. Supplied passages: Proposition prop:K5exact, lines 1135–1202; Theorem thm:H2, lines 1539–1582. Registry: https://enter.grok.me/.

XIVi

Cheung, N. XIV-I, version 1, XIV-I.tex. Enter Calculus Companion. Supplied passages: Proposition prop:pole, lines 726–826; Theorem thm:I2 and the precomposition formulas, lines 1288–1378; Theorem thm:I3, lines 1459–1548; Theorem thm:release, lines 1729–1814. Registry: https://enter.grok.me/.

XIVj

Cheung, N. XIV-J, version 1, XIV-J.tex. Enter Calculus Companion. Supplied passages: Proposition prop:descent, lines 813–846; Lemma lem:null, lines 853–883; Theorem thm:reference, lines 1215–1299; Theorem thm:conditionalextension, lines 1427–1470; Theorem thm:J2, lines 1608–1649; Corollary cor:noadmission, lines 1808–1822; Theorem thm:release, lines 1894–1939. Registry: https://enter.grok.me/.

XVrecord

Cheung, N. XV-A, XV-B, XV-C, XV-D, and XV-E, version 1. Enter Calculus Companion. The supplied registry descriptions are used only for the respective audit and stopping-status statements, not as proofs of unquoted analytic claims. Registry: https://enter.grok.me/.

XVf

Cheung, N. XV-F, version 3, XV-F-v3.tex; XV-F-patch, version 1; XV-F-patch-ii, version 1. Enter Calculus Companion. The supplied registry and retrieval summaries record the incomplete package, the F0 ending, and the patch-ii Q2 qualification. They are not treated as constructions of the full joint graph. Registry: https://enter.grok.me/.

XVId

Cheung, N. XVI-D, version 1, XVI-D.tex. Enter Calculus Companion. The supplied registry description and response-package summary are used for the prepared-response scope and conditional identification statement. No unregistered continuation is imported. Registry: https://enter.grok.me/.

XVIh

Cheung, N. XVI-H, version 1, XVI-H.tex. Enter Calculus Companion. The supplied registry description and thm:main summary distinguish the compact three-slot kernel, its specified partial contraction, and the unresolved wavefront, class-comparison, observability, and quantum-product questions. Registry: https://enter.grok.me/.

Registry

Enter Calculus Companion. Registered-manuscript descriptions and scope notices, as supplied with the question. These descriptions distinguish the manuscript tracks, record that the XV-F patches do not pass F0, and state that the registered XVI notes do not construct the retarded coefficient. https://enter.grok.me/.