Landmark XVI-D Prepared witness response and completion domains Revised proof manuscript — inherited-completion programme unresolved

We establish the prepared response on the counter-propagating pure-trace witness family. The source is smooth and past compact relative to compact observation sets, but is not a compact-interior source. Real polarization and complex bilinear extension produce the response $U_D=-\mathcal{P}{V}_{\lambda }$, where $V_{\lambda }$ is an explicit zero-data time integral. A local energy proof supplies uniqueness among all smooth zero-data solutions on the slab, without a spatial decay assumption.

The spatial traceless response contraction is computed exactly. For the completion, the switched scalar forcing is a compact-interior source, whereas the coordinate forcing is spatially noncompact. An explicit coordinate candidate is nevertheless a unique smooth solution of the required Cauchy problem. We also construct a restricted auxiliary extension of the compact-source wave operator, while keeping its identification with the inherited coordinate prescription conditional.

The single-metric root identity is not directly applied to a polarized source. Its legitimate polarization, and the later root–auxiliary compatibility result of XVI-F-v3, are recorded separately. No compact class kernel, quantum retarded coefficient, or new admission theorem is constructed.

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

Figure fig:scope-XVI-D records that separation. The drawing adds no hypothesis.

and release status.{ The width $\epsilon >0$ is fixed.

Proved here: smooth zero-data uniqueness without spatial decay; support and domain distinctions; real polarization and complexification; the explicit witness response; the spatial traceless response contraction; the compact scalar-leg check; the explicit coordinate candidate; and conditional inherited-completion compatibility.

Not inferred: a failed compact-source hypothesis is not nonexistence of a solution or of an operator extension.

The inherited coordinate prescription must be matched to a specified larger-domain realization before a direct inherited insertion identification is claimed. The class-kernel and quantum-admission programme remains Unresolved. }

The available material comprises the supplied XVI-A, revised XVI-B, original XVI-D, XVI-E, and XVI-F-v3 texts, together with predecessor statements reproduced in those texts and the supplied coverage material.

Complete XI-B, XII-D, and XVI-C sources are not supplied here. References to their internal labels identify reported passages; they do not certify a full-file or full-series audit.

Four kinds of statement are distinguished:

1. Imported formula or definition: retained with its stated convention and domain.

2. Imported theorem: used only within the explicitly recorded quantifier.

3. Proof in this manuscript: a calculation or argument given below.

4. Compatibility obligation: an identification not supplied by the preceding proof.

The main dependency structure is noncircular:

Only the source calculation of XVI-B is used as a premise. Its later-results summary of XVI-D is not used to prove XVI-D.

General wave-equation background is available in BGP; the no-decay uniqueness argument needed here is given explicitly below rather than left as an external premise.

Let

with

Smoothness on the closed slab means smoothness up to its time boundaries; all Cauchy traces used below are therefore defined. No spatial boundedness, decay, or finite total-energy condition is imposed on smooth solutions.

Spacetime indices are raised and lowered with $\eta$. Spatial Euclidean contractions use ${\delta }_{ij}$. The frequency parameter is $\lambda$. Repeated spacetime derivative indices are denoted by $\alpha ,\mu ,\nu$, not by $\lambda$.

Fix a real smooth switch

Thus $\sigma$ is flat at $0$, its positive-order derivatives vanish at $\epsilon$, and $\mathrm{supp}{\sigma }^{\prime }\subset [0,\epsilon ]$.

For a symmetric tensor $r$, define

The analogous raised-index formula is used for contravariant tensors. In four dimensions,

These follow directly from ${\eta }^{\mu \nu }{\eta }_{\mu \nu }=4$. Trace reversal has constant coefficients and commutes with $\square$.

The selected free operator and compact-test inverse are

The coefficient is $-1$, not $-4$ XIB. The placement of $\mathcal{P}$ in this notation does not enlarge the domain of the scalar operator.

Write $J_D={\mathcal{J}}_{\mathrm{D}}$ for the designated prepared source. The imported source relation is

It is a real homogeneous quadratic differential polynomial. Its real polarization is

The complex extension is bilinear, not sesquilinear.

Let $|\mathbf{n}|=1$ and, for $\lambda >0$, set

Both $k_a$ are null, so $\square {{\gamma }_a}^{\lambda }=0$.

The source-level result imported from XVI-B is

where

In particular,

Neither $\mathsf{q}$ nor ${\mathfrak{d}}_1$ contains an additional factor of $\lambda$ XVIB.

Equation (eq:Slambda) defines an ambient smooth tensor. It is not merely a slab formula extended discontinuously at an endpoint.

The diagonal response is denoted $u_{\mathrm{D}}[\gamma ]$; the polarized response is $U_D[{\gamma }_1,{\gamma }_2]$. A prospective compact observable kernel is denoted $w_D$. These are not interchangeable.

The sign convention for a diagonal action-equation coefficient is not an unlabelled identification of that coefficient with a response on a different domain. If it is polarized, its polarization is correspondingly the negative of the polarized response.

Define the real compact-interior source class

The complex version consists of real and imaginary parts in ${\mathcal{D}}_{\mathrm{int}}$.

The reported XI-B compact-source theorem states, for $F\in {\mathcal{D}}_{\mathrm{int}}$,

with zero initial data, and the corresponding advanced statement with zero final data. It also gives the causal support bounds

These supports are compact in the closed slab XIB.

The support can meet a time boundary. Compactness in the closed slab is not compact support in the open slab.

The supplied XII-D response statement is:

For every smooth $\gamma$, the response formula defines a unique smooth zero-initial-data solution on the slab.

The reported Section D0 also states:

No spatial decay is required.

Its response formula is

The reported construction is local near compact observation sets and independent of its auxiliary spatial cutoff XIID.

We use its existence assertion on real smooth metric perturbations. We do not infer the scope of uniqueness solely from the phrase concerning spatial decay: the needed uniqueness class is established directly next.

Let $\mu \ge 0$ and let $z$ be a real smooth function on $M_T$. If

then $z=0$.

The same conclusion holds componentwise for complex functions, vectors, and tensors. Consequently, a smooth solution of $LZ=F$ with prescribed initial data is unique whenever it exists. No spatial decay or finite total energy is required.

Proof. Fix $0<t_0<T$, ${\mathbf{x}}_0\in {\mathbb{R}}^3$, and $\rho >0$. For $0\le s\le t_0$, let

and define the local energy

Every integration region lies in a fixed compact set. Thus $\mathcal{E}(s)$ is finite even if the total energy on ${\mathbb{R}}^3$ is infinite.

Writing $e=({z_t}^2+|\nabla z{|}^2+{\mu }^2{z}^2)/2$, the equation gives

Since the boundary of $B_s$ moves inward at speed one, the divergence theorem and differentiation of the moving region give

At each boundary point,

Hence ${\mathcal{E}}^{\prime }(s)\le 0$.

The zero initial data imply $\mathcal{E}(0)=0$: the initial spatial gradient also vanishes because $z(0,\cdot )$ is identically zero. Nonnegativity gives $\mathcal{E}(s)=0$ for every $s\le t_0$. Smoothness then implies $z_t=0$ throughout each $B_s$. In particular $z_t(s,{\mathbf{x}}_0)=0$ for $0\le s\le t_0$, so

The point was arbitrary. Continuity includes $t=T$.

Real and imaginary parts satisfy the same real equation. The argument therefore applies componentwise.

If $LZ=0$, then $\square (\mathcal{P}Z)=0$. The tensor $\mathcal{P}Z$ has zero initial data whenever $Z$ does. The componentwise result gives $\mathcal{P}Z=0$, and ${\mathcal{P}}^2=\mathrm{id}$ gives $Z=0$. Apply this to the difference of two solutions.

This proves the unrestricted smooth uniqueness needed by the polarization and identification arguments. Even if an inherited construction theorem were formulated using a smaller construction class, two smooth solutions of the present Cauchy problem could not differ outside that class.

An ambient field $F$ is called past compact relative to compact observation sets if

is compact for every compact $K\subset M$. This is the meaning of “past compact” used in this manuscript.

This property is distinct from $F\in {\mathcal{D}}_{\mathrm{int}}$. It does not, by itself, prove an operator identity or existence of a retarded solution.

Let $F$ be an ambient field with $\mathrm{supp}F\subset \{t\ge 0\}$. Suppose a compact observation set $K$ is contained in $\{t\le b,|\mathbf{x}|\le R\}$, where $b,R\ge 0$. Then

and the intersection is compact.

Proof. For $y\in \mathrm{supp}F\cap J^{-}(K)$, there is $x\in K$ such that

Thus $|\mathbf{y}|\le R+b$.

The right side of (eq:pastbound) is compact. The support is closed. Also $J^{-}(K)$ is closed: if $y_j\to y$ and $y_j\in J^{-}(x_j)$ with $x_j\in K$, a subsequence of $x_j$ converges to $x\in K$, and the closed causal inequality passes to the limit. The intersection is therefore a closed subset of a compact set.

For $\lambda >0$, $S_{\lambda }$ is smooth, vanishes for $t\le 0$, is flat at $0$, and is past compact in the sense of Definition def:past. Nevertheless,

At $\lambda =0$, the displayed source is zero.

Proof. Smoothness, past vanishing, and flatness follow from (eq:switch), (eq:Slambda), and (eq:d1). Lemma lem:past proves past compactness.

For $\epsilon <t<T$,

This is nonzero at every spatial point. Thus the spatial support is unbounded and the source is not compactly supported in the open slab.

Every term of (eq:Slambda) has a factor $\lambda$ or ${\lambda }^2$, proving the assertion at zero frequency.

The compact-source identity (eq:compactGreen) cannot be applied to $S_{\lambda }$ merely by citing its compact-interior quantifier. This is a limitation of that theorem, not a nonexistence conclusion.

For real smooth symmetric tensor fields $a,c$, define

Then $B$ is symmetric and real bilinear,

Proof. The imported response-existence assertion applies to $a+c,a,c$. Linearity of $L$ and (eq:sourcepolarization) give the displayed equation. The initial data pass through the same linear combination.

The polarization of a homogeneous quadratic differential polynomial is symmetric and bilinear. Therefore each additivity or real-homogeneity defect of $B$ satisfies the homogeneous equation $LZ=0$ with zero initial data. Lemma lem:uniqueness makes every such defect zero. Symmetry also follows directly from (eq:B).

The auxiliary localization in the inherited response construction is not a cutoff imposed on the witness.

For real smooth $a,b,c,d$, define

This is the unique complex bilinear extension of $B$. It satisfies

and preserves zero initial data.

Proof. Complex bilinearity forces the four-term expansion (eq:BC). Conversely, real bilinearity of $B$ verifies additivity and multiplication by $\mathrm{i}$ in each slot, so the formula defines a complex bilinear map.

The operator $L$ has real coefficients. Apply $L$ to the four terms and use Lemma lem:realB. The result is the same four-term extension of the polarized source. Initial data are preserved term by term.

On the witness, write

The real inputs and their sums are smooth. Thus $B_{\mathbb{C}}({{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda })$ is defined using real response evaluations. It does not require invoking a predecessor theorem directly on complex metric perturbations.

Set

Define the ambient particular field

and set

For negative $t$ the integral is zero, because its source is zero there. This is a smooth past-zero extension, not a sharp multiplication of a nonvanishing solution.

The fields $V_{\lambda }$ and $R_{\lambda }$ are smooth and vanish flatly for $t\le 0$. They satisfy

and have zero initial position and velocity.

Proof. For $t\ge 0$, differentiation gives

Spatial derivatives vanish. The initial traces are zero. Higher time derivatives at $0$ are zero because $S_{\lambda }$ is flat there. This also proves smoothness of the past-zero extension.

Using (eq:P2),

The constant pointwise map $\mathcal{P}$ preserves smoothness, flatness, and the initial data.

On $M_T$,

Proof. Lemma lem:BC and the source input (eq:Slambda) show that the left side solves $L{U}_D=S_{\lambda }$ with zero initial data. Lemma lem:R gives the same equation and data for $R_{\lambda }$. Their difference is a smooth complex homogeneous zero-data solution. Lemma lem:uniqueness makes it zero.

This identifies the particular field with the polarized response. It does not, solely from the compact-interior theorem, identify $V_{\lambda }$ with the named expression ${E_0}^{\mathrm{R}}{S}_{\lambda }$.

Let

and define

The response satisfies

The correction $\lambda {\mathfrak{d}}_1$ contributes zero to this channel, including after trace reversal.

These are response contractions, not completed insertion values.

Proof. The Euclidean identities are

Thus (eq:q) and (eq:d1) give

Integrating proves the first identity.

On raised spatial components,

The added term is annihilated by $h$. Apply (eq:identify) to obtain the second identity. Any correction entering the trace also multiplies ${\delta }^{ij}$ and is annihilated by the same contraction.

No assertion that $F_{\lambda }(t)$ is nonzero at every time or frequency is required.

Fix

and let $\varphi$ be a real smooth scalar. No scalar free equation is required for the existence checks in this section.

The supplied completion coefficients are

The equality with trace reversal is a pointwise calculation.

The inherited legs are written

These are inherited definitions on a domain where their operator applications are justified XIID.

The corresponding functional is

The diagonal completed insertion is

on its inherited domain.

A direct polarized-response reading would be

but only after both legs have been justified on that response. Its witness value is denoted ${\mathcal{A}}_D(\lambda ;f,g,\varphi )$. This is not a definition of a compact class-kernel pairing.

On $R_{\lambda }$,

Moreover,

The forcing is smooth, flatly zero for $t\le 0$, and past compact relative to compact observation sets.

For $\lambda >0$ it is not in ${C_c}^{\infty }({M_T}^{\circ };\mathbb{C})$. Both its real and imaginary time components are nonzero spatially noncompact functions.

Proof. By (eq:coefficients) and (eq:P2),

All coefficients depend only on time, proving (eq:Fco). Equations (eq:q) and (eq:d1) give ${S_{\lambda }}^{0i}=0$, hence ${V_{\lambda }}^{0i}=0$.

Smoothness and flat past vanishing follow from Lemma lem:R. Lemma lem:past proves past compactness.

On the nonempty interval $(\epsilon ,T)$,

The derivative is not identically zero, so ${F_{\mathrm{co}}}^0$ is not identically zero. Likewise its real and imaginary derivatives are $-2{\lambda }^2\cos (2\lambda t)$ and $-2{\lambda }^2\sin (2\lambda t)$. Neither vanishes identically on that interval. Thus both real and imaginary components are nonzero functions independent of $\mathbf{x}$, and neither has compact spatial support.

Consequently the compact-interior theorem (eq:compactGreen) alone does not justify the named coordinate application ${E_0}^{\mathrm{R}}{F}_{\mathrm{co}}$. This conclusion concerns the hypothesis of that theorem. It does not say that the coordinate Cauchy problem has no solution, or that no operator extension exists.

The factor $g$ does not occur in $F_{\mathrm{co}}$. Compact support of the final test $f$ also does not put an unmodified internal forcing into ${\mathcal{D}}_{\mathrm{int}}$.

The forcing

has real and imaginary parts in ${\mathcal{D}}_{\mathrm{int}}$ and support contained in $\mathrm{supp}g$. Hence

is defined by complex linear extension of the compact-source operator. It is smooth, has zero initial data, and satisfies

Proof. Expanding the divergence gives

Every coefficient apart from $g$ and its derivatives is smooth. The support of every derivative of $g$ is contained in $\mathrm{supp}g$, a compact subset of ${M_T}^{\circ }$. Thus every term is smooth and compactly supported there.

For real $F_1,F_2\in {\mathcal{D}}_{\mathrm{int}}$, define

This is the unique complex linear extension. The real differential operator $\square +m^2$, the initial data, and the support bounds pass to this extension. Equation (eq:phi-equation) includes the minus sign in (eq:phicand).

Derivatives of $g$ and the mass term have not been deleted. The scalar-leg verification does not verify the coordinate leg.

Define

Then $X_{\mathrm{cand}}$ is smooth, flatly zero for $t\le 0$, has ${X_{\mathrm{cand}}}^i=0$, and satisfies

It is the unique smooth solution of this Cauchy problem on $M_T$.

Proof. The integrand is smooth and flatly zero in the past. Thus the integral has the asserted smoothness and past extension. It is spatially independent, and

The initial traces vanish. The spatial components vanish by Proposition lem:coord. Uniqueness follows componentwise from Lemma lem:uniqueness.

This is the candidate formula also present in XVI-E and XVI-F-v3. It is reproved here; no named Green-operator application is used in its definition.

Let ${\mathcal{H}}_T$ be the real vector space of ambient smooth sources $q(t,\mathbf{x})=q(t)$, independent of space and zero for $t\le 0$, restricted to the slab. Regard ${\mathcal{D}}_{\mathrm{int}}$ as an ambient source by its smooth zero extension. Then

Indeed, a nonzero spatially constant function cannot be compactly supported in space.

On ${\mathcal{D}}_{\mathrm{int}}\oplus {\mathcal{H}}_T$, define

This is a well-defined real linear extension of the compact-interior operator. It gives the unique smooth zero-data solution of

It extends uniquely by complex linearity and componentwise.

On the witness,

Proof. The direct-sum decomposition is unique, so the definition is unambiguous. The compact part has the equation and initial data from (eq:compactGreen). The homogeneous part has them by direct differentiation. The sum therefore solves the stated Cauchy problem. Lemma lem:uniqueness proves uniqueness.

Both witness sources belong to the complexification of ${\mathcal{H}}_T$. The first formula in (eq:restricted-witness) is (eq:Vlambda). For the second, put $H^{\mu }=\sigma {V_{\lambda }}^{0\mu }$. Then ${F_{\mathrm{co}}}^{\mu }=(H^{\mu }{)}^{\prime }$ and $H^{\mu }(0)=0$, so

by integration by parts. Taking the negative gives (eq:Xcand).

The new symbol ${{\mathcal{E}}_{0,\mathrm{res}}}^{\mathrm{R}}$ is essential. Its existence is now proved, so existence of a restricted solution-operator extension is not left open. However, the construction is not a declaration that every inherited occurrence of ${E_0}^{\mathrm{R}}$ already denotes this extension.

The restricted domain contains the polarized witness forcing. It need not contain every forcing arising before polarization from real cosine and sine inputs and their sums. That larger family requires its own domain statement if family-level identification is claimed.

The original XVI-D reasoning must not be read as saying that XI-B defines its Green symbols only on compact-interior sources.

The supplied XVI-F-v3 explicitly reports a preceding XI-B sine-integral definition for smooth sources, with nondecaying sources interpreted locally by finite propagation. It uses that wider defining quantifier to identify the advanced operator on its boundary-flat source class. This is separate from the narrower inversion theorem lem:Green XVIFvThree.

Accordingly:

• failure of the compact-interior hypothesis is proved;

• absence of every wider inherited definition is not proved and is not asserted;

• the remaining retarded-coordinate check is whether the inherited completion uses the reported smooth-source prescription on the required family, with the stated smoothness and zero initial data.

Under that verified wider reading, the following comparison settles the field identity. No further uniqueness premise is needed.

Suppose the inherited application ${E_0}^{\mathrm{R}}{F}_{\mathrm{co}}$ is defined and is a smooth solution of

with zero initial data. Then

Consequently the direct inherited completion on $R_{\lambda }$ exists and equals

Proof. The first two expressions in (eq:coordinate-compatible) refer to the inherited definition (eq:completion-legs). By the assumed operator property they solve (eq:boxX). Proposition lem:Xcand and Lemma lem:uniqueness give their equality.

The scalar field is already verified by Proposition lem:scalar. Thus both fields in the integrand of (eq:direct-conditional) are smooth. Multiplication by $f$ gives compact support and a finite integral. Substitution in (eq:Aone) proves the formula.

Agreement of two extensions on ${\mathcal{D}}_{\mathrm{int}}$ alone would not prove agreement at $F_{\mathrm{co}}$. Agreement as smooth zero-data Cauchy solutions does prove it.

For comparison, the integral on the right of (eq:direct-conditional) exists independently of the hypothesis of Proposition prop:coordinate-compatibility. When independently defined it is denoted $I_{\mathrm{cand}}$. Its existence does not silently identify it with the inherited completion.

The reported XII-D compact-root theorem supplies, for fixed $f,g$ and collar $\vartheta$, a compact smooth root $a_{f,g}[\varphi ]$ and the diagonal identity

on its stated domain. The root has support in a fixed compact subset of $J^{-}(\mathrm{supp}f)\cap \{t\ge -\delta \}$ XIID.

A polarized source $S_{\lambda }$ is not itself a single metric argument $\gamma$ of this theorem. Thus substituting $S_{\lambda }$ for $\gamma$ is not a direct application of (eq:root-identity). This does not prohibit polarization of valid diagonal values.

Suppose (eq:root-identity) holds on a real vector space containing the cosine and sine witness inputs and their required sums. With $f,g,\varphi$ and the root fixed, the real polarization

satisfies

Its complex bilinear extension has witness value

The integral defining $Q_{\lambda }$ exists whenever the stated compact smooth root is available.

Proof. Apply the diagonal root identity to $a+c,a,c$ and polarize. The common root is independent of the metric input, so linearity of the integral gives the polarized source.

Complex extension uses the same four-term expansion as (eq:BC). The complexified source evaluates to $S_{\lambda }$ by (eq:Slambda), yielding (eq:Q). Finally, the product of the compact smooth root with the ambient smooth source is compactly supported and integrable.

This scalar argument does not independently exhibit an inherited coordinate field. To pass from polarized diagonal values to the direct response reading ${\mathcal{A}}_1[f,g](R_{\lambda };\varphi )$, the domain and linearity of the inherited completion must also hold on the required response family.

The supplied XVI-E retains $T^{\mathbb{C}}=Q_{\lambda }$ through the real root-theorem values and proves fixed-data rapid decay of that scalar root pairing XVIE.

The supplied XVI-F-v3.tex goes further than the earlier XVI-F versions:

• it constructs $I_{\mathrm{cand}}$ independently;

• it proves $I_{\mathrm{cand}}=Q_{\lambda }$ in prop:compat;

• it identifies the advanced root on the slab using the reported smooth-source sine-integral definition;

• it proves direct fixed-data rapid decay of $I_{\mathrm{cand}}$;

• it does not identify the auxiliary coordinate field with the inherited retarded coordinate prescription.

The earlier unresolved residual is therefore not repeated as the current result of XVI-F-v3. That version proves vanishing of the residual pairing; it does not require the discrepancy tensor to vanish at every negative time XVIFvThree.

These are later results on named scalar evaluations. They are not premises of the response proof in Section sec:response. They do not by themselves prove the retarded operator identity in Proposition prop:coordinate-compatibility.

Their rapid-decay assertions hold at fixed regulator and fixed real smooth $\varphi$ independent of $\lambda$. They are not pointwise response decay, a frequency-dependent scalar estimate, or a class-admission theorem.

Assume the explicit source input (eq:Slambda)–(eq:d1), the reported existence of the real diagonal responses (eq:response), and the compact-interior scalar Green theorem (eq:compactGreen).

For every $\lambda >0$ and $0<\epsilon <T$:

1. $S_{\lambda }$ is past compact relative to compact observation sets but is not a compact-interior source.

2. Real polarization has a unique complex bilinear extension, and its witness value is $U_D[{{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda }]=-\mathcal{P}{V}_{\lambda }$.

3. The exact spatial traceless response contraction is (eq:contraction).

4. The scalar completion field is verified on the witness.

5. The coordinate Cauchy problem has the unique smooth solution (eq:Xcand), and the restricted auxiliary extension (eq:restricted-extension) exists.

6. If the inherited retarded coordinate prescription satisfies the hypothesis of Proposition prop:coordinate-compatibility, its direct witness completion equals $I_{\mathrm{cand}}$.

The conclusions require no spatial decay of the witness or of the compared smooth solutions.

Proof. Assertion (1) is Proposition prop:source-domain. Assertion (2) follows from Lemmas lem:realB, lem:BC, and Proposition lem:identify, using the independent uniqueness proof Lemma lem:uniqueness. Assertion (3) is Proposition lem:contraction. Assertions (4) and (5) are Propositions lem:scalar, lem:Xcand, and prop:restricted-extension. Assertion (6) is Proposition prop:coordinate-compatibility.

The theorem is about smooth responses and explicitly specified completion routes. It is not a construction of a compact observable kernel.

In particular, a response tensor $U_D$ is not a kernel $w_D$ on two metric slots and a scalar slot. If such a compact kernel is later constructed, comparison with the frozen XI-G kernel $w_0$ must use the relevant free-shell null space

within the specified compact kernel space.

The condition $w_D-w_0\in {\mathcal{I}}_{\mathrm{sol}}$ transfers the frozen obstruction. Changing a source does not establish or exclude that condition. A different-class proof would not itself prove admission XVIA,XVIB,XIG.

The separate characteristic quantum-product programme described in the supplied coverage material is not discharged by these smooth witness-response calculations.

{}1.1

Verify, for the inherited retarded prescription used in (eq:completion-legs), one of the following:

1. an applicable smooth-source definition or extension that gives a smooth zero-data solution at $F_{\mathrm{co}}$;

2. agreement there with the explicitly constructed restricted extension; or

3. a larger-domain theorem covering the real response family needed before polarization.

The new restricted extension already proves that such an extension is mathematically possible on the witness source space. The remaining question is its match to the inherited prescription and, when required, the prepolarization family.

The hypothesis of Proposition prop:coordinate-compatibility is sufficient for the direct witness reading. Family-level polarization additionally requires the relevant diagonal coordinate fields and linearity on their domain.

Status of that inherited identification in this manuscript: Unresolved.

The minimal order of further work is:

1. verify the inherited retarded-coordinate realization on the required source domain;

2. compare it with $X_{\mathrm{cand}}$ using the already proved uniqueness theorem;

3. identify the direct inherited completion and its relation to the reported root and auxiliary scalar values;

4. formulate any further high-frequency, compact-kernel, and free-shell comparison tasks for that actual identified observable.

There is no need to re-prove uniqueness or to leave $I_{\mathrm{cand}}=Q_{\lambda }$ open after importing XVI-F-v3. Conversely, neither result replaces the inherited retarded domain check.

The revision corrects the following points:

• Smooth uniqueness is proved in the relevant unrestricted class, rather than inferred from an ambiguous reading of a localization statement.

• Past compactness is defined explicitly and is not confused with compact support in the open slab.

• The repeated derivative index is $\alpha$, avoiding confusion with the frequency $\lambda$.

• Coordinate forcing is proved spatially noncompact on the nonempty interval $(\epsilon ,T)$; both real and imaginary components are checked.

• “The compact-source theorem does not apply” replaces any wording suggesting that every inherited construction fails.

• The existence of a coordinate solution and of a restricted extension is affirmative, not left as a supposed nonexistence issue.

• The broader smooth-source definition reported by XVI-F-v3 is acknowledged; its retarded use must be matched to the inherited completion.

• The root limitation is a limitation on direct substitution, not a prohibition on valid polarization.

• The current XVI-F-v3 compatibility result replaces the earlier unresolved-residual status.

No claim is made of:

• repair of X-G or XI-G;

• new admission into a microcausal polynomial class;

• construction of a compact class kernel or a quantum retarded coefficient;

• a general distributional operator-domain extension;

• a cutoff-modified witness;

• a sharp-width limit or uniform estimates as $\epsilon \downarrow 0$;

• an interacting BRST/BV construction, loop, state, or ultraviolet subtraction.

The auxiliary extension is a specified smooth-source construction, not a distributional extension theorem. The inverse coefficient $-1$ is not $-4$. A response contraction is not an insertion, and a smooth response is not a class kernel. No successful compiler execution or full-corpus verification is represented.

release statement.{ The prepared witness-response theorem is proved, including no-decay uniqueness, polarization, explicit identification, the traceless contraction, and the scalar completion check.

The coordinate Cauchy problem has a unique explicit solution, and a restricted auxiliary operator extension exists. Their identification with the inherited coordinate prescription is stated precisely and remains conditional.

The supplied XVI-F-v3 equality $I_{\mathrm{cand}}=Q_{\lambda }$ is recorded as proved there. It is not misreported as an unresolved residual, nor used to silently rename the coordinate field.

The inherited-completion and class-admission programme remains Unresolved. This manuscript is complete as a response proof and a domain-sensitive compatibility interface, not as a proof of the unresolved quantum coefficient. }

XVIA

Landmark XVI-A. Supplied revised XVI-A.tex. Used for the separate retarded-product obligations and the conditional free-shell class-transfer statement.

XVIB

Landmark XVI-B. Supplied revised XVI-B.tex. Source-level locators: prop:trace, prop:pair, eq:JDeval, and cor:tf. Only the source-level results are premises of the response proof; its summary of later XVI-D results is not used circularly.

XVIE

Landmark XVI-E. Supplied XVI-E.tex. Locators: lem:Q, lem:TD, lem:TC, lem:Xcand, lem:compat, and cor:decay. Used for the later root reading and its stated compatibility boundary, not as a premise of witness-response identification.

XVIFvThree

Landmark XVI-F. Supplied XVI-F-v3.tex. Locators: prop:Icand, prop:kernel, prop:formula, prop:compat, and prop:decay. This version proves witness root compatibility and reports the wider smooth-source sine-integral definition. Its result supersedes the earlier unresolved root–auxiliary residual.

XIB

Manuscript XI-B, version 1. Supplied identifier 862627cd. Reported locators: lem:Green, thm:main(ii), thm:graviton, and eq:Ggr. The compact-interior Green statements are reproduced in the supplied XVI texts. The wider smooth-source defining paragraph is reported in XVI-F-v3; the complete XI-B source is not independently certified.

XIID

Manuscript XII-D, version 1. Supplied identifier e3e3f67d. Reported locators: Section D0, eq:response, thm:response, rem:forcing-sign, eq:completion-legs, eq:Aone, def:completed, eq:CD, thm:root, and eq:root-identity. Used only through the supplied statements and formulas.

XIG

Manuscript XI-G, version 1. Supplied identifier 3cae2be9. Reported locators: lem:qvalues, eq:qcounter, def:class, and thm:obstruction. The frozen obstruction is not transferred to a source tensor.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, arXiv:0806.1036 [math.DG] (2008). General background on wave equations, Green operators, and Cauchy problems. The local-energy uniqueness argument used in this manuscript is proved explicitly in Lemma lem:uniqueness.