Landmark XVI-B Prepared-source diagnostics and the complete-insertion boundary Revised manuscript — programme status: unresolved

We prove the source-level consequences of the supplied density-weighted cubic preparation on the XI-G pure-trace plane-wave pair. The polarized correction is first order in the frequency, its spatial traceless projection vanishes, and the prepared bulk source retains the stated second-order coefficient. We also prove the distinction between comparison with the unramped source and comparison with a sharp-onset source, and derive the weighted-representative residual under compact interior variations.

These results do not identify a source with its retarded response, a response with the completed insertion, or an insertion with a compact free-shell kernel. The XI-G obstruction transfers only after a same-class comparison of actual observable kernels. The supplied later sheets are recorded separately: XVI-D constructs the witness response, XVI-E supplies a polarized root pairing, and XVI-F-v2 supplies an independently defined auxiliary evaluation. Their remaining compatibility checks are not discharged by the source calculation. No new quantum retarded coefficient or admission theorem is claimed.

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

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

and release status.{ The regulator width $\epsilon >0$ is fixed. Elementary source identities and conditional comparison statements are proved below. No new gravitational retarded coefficient is constructed.

X-G and XI-G remain in force on their respective hypotheses. Applicability of XI-G to a prepared observable class is not decided by changing the source. The complete-insertion and class-comparison programme remains Unresolved.

The source, response, root pairing, auxiliary evaluation, and class-kernel pairing are distinct objects. An unverified identification is not an assertion that their values differ. }

This manuscript uses:

• the supplied XVI-B draft and revised XVI-A;

• the supplied XVI-D, XVI-E, XVI-F, and XVI-F-v2 texts;

• the supplied predecessor excerpts and source locators, especially XI-A, XI-B, XI-G, XII-B, XII-C, XII-D, and XII-E;

• the supplied retrieval coverage report.

The coverage report identifies corpus b0c93438, retrieval retrieval-3, and reports retrieved passages across the series through XV-F. That is a report of passage retrieval, not a reproducible, exhaustive search of every complete manuscript. The supplied XVI-C text is absent. Versions of the supplied XVI drafts are not inferred from unprovided metadata.

The distinction between the two supplied XVI-F texts matters. For the support of the integrated kernel and for direct fixed-data asymptotics, this note uses XVI-F-v2.tex, not the earlier assertion that the integrated kernel is automatically compactly supported.

For the documentary question, a replacement would have to be an explicitly identified alternative to:

1. the bilinear source symbol of $J_3$, or

2. the selected inverse $-\mathcal{P}{E_0}^{\mathrm{R}}$,

together with a calculation showing removal of the named XI-G witness mechanism while stating the retained normalization, domains, and observation data.

The preparation $J_D={\sigma }_{\epsilon }{J}_3+{\mathcal{D}}_{\sigma }$ is not described here as a replacement of the frozen bulk symbol or inverse. It is the existing density-prepared source to be analyzed.

The defensible documentary conclusion is:

No verified replacement satisfying this description is identified in the supplied record.

This statement is not a theorem that no such replacement can exist, and it is not a reason to stop analysis of the existing prepared source.

We distinguish:

• Imported input: a formula or theorem reported in supplied predecessor material;

• Proved consequence: a deduction whose proof appears in this manuscript;

• Reported later result: a result stated in a supplied later XVI sheet, not silently imported into an earlier source calculation;

• Open obligation: a missing construction, estimate, or identification.

A citation to a predecessor label does not transfer every theorem of that manuscript.

Work on ambient Minkowski space

and, where a slab is needed, set

Greek indices range over $0,1,2,3$ and spatial indices over $1,2,3$. Spacetime indices are raised and lowered with $\eta$. Spatial Euclidean contractions use ${\delta }_{ij}$.

Let

Write $\sigma ={\sigma }_{\epsilon }$ when no confusion is possible. Smoothness and the constant plateaus imply

No estimate uniform as $\epsilon \downarrow 0$ is claimed.

For this manuscript, the prepared-source symbols are written

as local aliases for the designated source and correction in the supplied XVI and XII material. This convention does not identify unrelated objects in unprovided manuscripts.

The symbol ${\mathcal{D}}_{\sigma }$ is not the de Donder operator applied to $\sigma$.

We reserve:

• $u_{\mathrm{D}}[\gamma ]$ for a diagonal response;

• $U_D[{\gamma }_1,{\gamma }_2]$ for a polarized response;

• $w_D$ for a prospective compact observable kernel;

• $A_D(\lambda )$ for a prospective class-kernel pairing.

Thus no lowercase $u_D$ is used simultaneously for a response and a class kernel.

The selected XI inverse is

Here ${E_0}^{\mathrm{R}}$ denotes the selected XI scalar Green symbol, also printed ${{\mathsf{E}}_0}^{\mathrm{R}}$ in some supplied passages. This notation convention does not enlarge its proved domain. The coefficient $-1$ is not replaced by $-4$ XIB.

Where the response is defined, the action-equation coefficient has the inherited sign

The supplied XII-C source theorem gives the Euler source of the selected weighted cubic density as

For a time-only switch, its reported general layer formula is

The coefficient labels $C_2$ and $M_2$ belong to XII-C. Their general component formulas are not reconstructed here XIIC.

For the calculations below, the necessary explicit input is the reported pure-trace restriction:

In that sector, XII-C reports $C_2=0$, so the coefficient of ${\sigma }^{\prime \prime }$ in (eq:time-layer) vanishes.

These are imported formulas, not a new derivation of the full cubic Euler source. For fixed smooth input fields, the source evaluations considered below are smooth tensor fields.

The quadratic polarization convention is

for each homogeneous quadratic map $B$. Complex inputs are evaluated through complex bilinear extension, not through a Hermitian polarization.

Fix $\mathbf{n}\in {\mathbb{R}}^3$ with $|\mathbf{n}|=1$ and $\lambda >0$. Put

where $k\cdot y={\eta }_{\mu \nu }{k}^{\mu }{y}^{\nu }$.

The supplied XI-G symbol calculation is retained as input:

The tensor $\mathsf{q}$ in these displays is the unscaled coefficient. It does not itself contain $\lambda$ XIG.

A source tensor and an observable kernel are different kinds of object. In particular, changing $J_3$ to $J_D$ does not by itself determine a change of free-shell class.

Let $\mathcal{K}$ be the space of compact distributional kernels on $M^3$ with two symmetric-tensor slots and one scalar slot, symmetric under interchange of the tensor slots. Set $P_m=\square +m^2$, with $m\ge 0$, and define

Compact support makes these pairings with smooth solutions meaningful.

For $w\in \mathcal{K}$, write

Let ${\mathcal{K}}_{\mathrm{mc}}\subset \mathcal{K}$ denote the kernels representing the predecessor's microcausal polynomial target. The reported XI-G conclusion, for its frozen kernel $w_0$ and its stated hypotheses, is

This is the kernel formulation of its stated non-admission into ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ XIG.

Suppose an actual prepared observable kernel $w_D\in \mathcal{K}$ has been constructed. If

then

and (eq:old-obstruction) transfers to $w_D$.

Proof. Linearity of the pairings makes ${\mathcal{I}}_{\mathrm{sol}}$ a vector subspace. Consequently

The asserted obstruction follows from (eq:old-obstruction).

Distinct distributions can satisfy (eq:same-class). Conversely, proving $w_D-w_0\notin {\mathcal{I}}_{\mathrm{sol}}$ would remove this particular same-class transfer, but would not prove that the new class meets ${\mathcal{K}}_{\mathrm{mc}}$.

One differing pairing on a triple of free solutions would prove different classes, once both kernels and the compared evaluations exist. Agreement on only one witness family would not prove (eq:same-class), whose quantifier is all free triples.

Neither the source identity (eq:actualsource) nor the calculation of its value on (eq:witness) constructs $w_D$. XI-G applicability is therefore left unresolved, not categorically asserted or categorically excluded.

X-G's source-preserving variational obstruction is a separate statement. Changing the action or preparation is not, without a comparison theorem, an upgrade of the original model XG,XVIA.

The comparisons in this section are made after evaluation on fixed smooth inputs. They are not prescriptions for multiplying an arbitrary distributional kernel by a discontinuous switch.

Let

be the closed temporal layer. It is not spatially compact.

Assume the imported layer property $\mathrm{supp}{\mathcal{D}}_{\sigma }[\gamma ]\subset {\Lambda }_{\epsilon }$, with the correction vanishing at the plateaus as in the supplied XII-C/XII-D preparation. Then, for smooth $\gamma$,

Comparison with the unramped source gives

It need not be supported in ${\Lambda }_{\epsilon }$.

Define the comparison function

Then the locally integrable tensor

has distributional support in ${\Lambda }_{\epsilon }$.

Proof. For $t\le 0$, $\sigma =0$ and the imported correction vanishes. For $t\ge \epsilon$, $\sigma =1$ and the correction vanishes. This proves (eq:pastzero) and (eq:plateau-source), and substitution proves (eq:unramped-comparison).

For the witness pair, the unramped $00$ source equals $-2{\lambda }^2{e}^{2\mathrm{i}\lambda t}$ even at negative times. Thus (eq:unramped-comparison) is genuinely not a layer-supported comparison on this family.

The coefficient $\sigma -H$ vanishes for $t<0$ and $t>\epsilon$. Since the evaluated $J_3$ is smooth, $(\sigma -H)J_3$ is locally integrable and supported in the closed layer. Adding the layer-supported correction proves (eq:sharp-comparison).

The value assigned to $H$ at $0$ does not affect its associated distribution. The product $H{J}_3$ above is justified by the smooth evaluated source. No product of $H$ with an arbitrary source kernel is declared. General distribution products require a justified construction; smooth multiplication and singular multiplication must not be conflated BDH.

The comparison with $H{J}_3$ does not introduce a sharp variational model or assert a sharp-width limit.

There are smooth scalar forcing terms that agree for $t\ge \epsilon$ but whose zero-data responses do not agree there.

Proof. Choose $F\in {C_c}^{\infty }((0,\epsilon );\mathbb{R})$, $F\ge 0$, $F\not\equiv 0$. The sources $F$ and $0$ agree for $t\ge \epsilon$. Their zero-data response difference is

For $t\ge \epsilon$,

In fact $z(t)>0$ for $t\ge \epsilon$. Thus the responses differ after their sources have become equal.

This scalar example proves the invalidity of the general inference. It is not an evaluation of the gravitational response. A retarded support bound places a response in a causal future, not necessarily in the source's temporal layer XIID.

Here $\mathrm{EL}$ denotes the bulk Euler–Lagrange derivative under compact interior field variations. The weight $\sigma$ is fixed and is not varied.

For local densities related by

one has

whereas

Proof. A total divergence has zero bulk Euler derivative under compact interior variations: its variation is again a divergence whose integral has no boundary contribution.

Use the product rule

Apply the linear bulk Euler derivative. The first term on the right is a total divergence and contributes zero. This proves (eq:boundaryrepresentative).

This is the supplied XII-B weighted-representative relation, now derived from the stated variation convention XIIB.

The XI-A statement that the linear Einstein–Hilbert coefficient is a divergence, together with its exclusion of metric boundary variation, does not supply a boundary action in this setup XIA. The narrow citation question is therefore settled:

The cited XI-A passage does not add a boundary action or authorize a metric boundary variation to the prepared cubic model.

This does not settle every possible boundary problem. XII-C fixes its cubic representative and uses compact interior variations XIIC. Fixing that representative excludes an unannounced change of model; it does not prove the weighted residual in (eq:boundaryrepresentative) vanishes for every other representative.

No boundary charge, boundary action, or additional variational principle is constructed here.

Define the constant-coefficient differential expression

For arbitrary smooth scalar functions ${\psi }_1,{\psi }_2$, with $p={\psi }_1{\psi }_2$, the imported correction satisfies

The same identity holds under complex bilinear extension. The coefficient of ${\sigma }^{\prime \prime }$ is zero on this sector, including its polarization.

Proof. Since $\sigma$ depends only on $t$,

Equation (eq:trace-input) becomes

Apply (eq:polarization). The diagonal terms cancel, leaving

Each summand of ${\mathcal{T}}^{\mu \nu }$ is a first derivative with constant coefficient, so the product rule identifies this with the right side of (eq:trace-identity).

The formula contains no conjugation and is bilinear over $\mathbb{C}$ after complexification.

The reported $C_2$ coefficient vanishes for every pure-trace input. The sum of two pure-trace inputs is pure trace. Consequently its polarization also vanishes.

On (eq:witness), define

Then

In components,

The prepared evaluated source is

The coefficient of ${\sigma }^{\prime \prime }$ in this evaluated correction is zero.

Proof. Put ${\psi }_a=e^{\mathrm{i}\lambda k_a\cdot y}$. Then

Substitution in (eq:trace-identity) gives (eq:Dsymbol). The signature gives $B^{00}=3$, $B^{0i}=0$, and $B^{ij}=-{\delta }^{ij}$, proving (eq:d-components).

Add $\sigma$ times (eq:bulk-eval) to (eq:Dsymbol), using (eq:actualsource). This gives (eq:JDeval).

The coefficient $\mathfrak{d}$ includes $\lambda$. The coefficient ${\mathfrak{d}}_1$ does not. Likewise, $\mathsf{q}$ is not multiplied by an additional frequency factor inside its definition.

At fixed $t$, after removing the common exponential, (eq:JDeval) is a polynomial in $\lambda$. Its degree-two coefficient is $\sigma (t)\mathsf{q}$. Where that coefficient is nonzero, a degree-one correction cannot cancel it as a polynomial family. This is a source statement, not a conclusion about a retarded response or complete insertion.

At $\lambda =0$, the displayed source vanishes. For $\lambda >0$ and $t\ge \epsilon$,

Thus the source is not in ${C_c}^{\infty }({M_T}^{\circ })$: it is nonzero at every spatial point at such a slab time.

Let

This is a Euclidean spatial tensor, not Lorentzian lowering of the spatial components of $k_a$.

The spatial traceless projections and contractions satisfy

Consequently

Proof. Equation (eq:qcounter) gives ${\delta }_{ij}{\mathsf{q}}^{ij}=1-9=-8$. Subtracting one third of that trace gives

Equation (eq:d-components) shows that the correction is spatially proportional to ${\delta }^{ij}$, so its traceless projection is zero.

Also $h_{ij}{\delta }^{ij}=0$ and

The contractions and (eq:source-contraction) follow.

This does not prove that the completed insertion detects this channel. The insertion can select tensor contractions and preparation-dependent terms not determined by this source projection alone.

The phase covectors are the lowered vectors

They are co-oriented future null covectors, one in each metric slot.

Their vector sum is

Therefore

The generated phase is noncharacteristic for the massless wave operator. This does not combine the two metric-slot covectors into a replacement for the three-slot XI-G criterion.

The arithmetic multiplier $-1/(4{\lambda }^2)$ sends a coefficient ${\lambda }^{2}a$ to $-a/4$ and a coefficient $\lambda b$ to $-b/(4\lambda )$. This is only frozen-frequency arithmetic. For time-dependent coefficients it omits coefficient derivatives, initial data, and preparation-generated homogeneous terms. It is not an asymptotic theorem for ${E_0}^{\mathrm{R}}$.

For each fixed $\lambda$, $S_{\lambda }$ is smooth. That does not establish smoothness or microcausality of the distributional kernel of the underlying polynomial functional. Wavefront properties of source evaluations and of functional kernels are different questions BDH.

Let $a$ be smooth on a neighborhood of $[0,\epsilon ]$, independent of $\lambda$, and let $\lambda >0$. Define the illustrative integral

Then

Proof. Set $b=a{\sigma }^{\prime }$. By (eq:flat), $b(0)=b(\epsilon )=0$. One integration by parts gives

Since $b^{\prime }=a^{\prime }{\sigma }^{\prime }+a{\sigma }^{\prime \prime }$, multiplication by $-\mathrm{i}\lambda B^{\mu \nu }$ proves (eq:ibp).

The absence of ${\sigma }^{\prime \prime }$ in (eq:Dsymbol) is compatible with its appearance in (eq:ibp). The latter derivative arose from transferring a derivative in an integral, not from changing the source symbol.

This calculation assumes a smooth scalar amplitude. It does not prove that an amplitude obtained from an unidentified singular retarded kernel is smooth, and it does not evaluate the completed gravitational insertion.

For the real switch $\sigma$ and $\lambda >0$, set

Then

On an interval, $v_{\lambda }$ solves $v_{\lambda }^{\prime \prime }={\lambda }^2\sigma e^{2\mathrm{i}\lambda t}$ if and only if $\sigma$ is constant there.

Proof. Direct differentiation gives

and

This proves (eq:ode-residual).

If the residual vanishes on an interval, its imaginary part is $-\lambda {\sigma }^{\prime }=0$, because $\sigma$ is real and $\lambda >0$. Thus $\sigma$ is constant there. Conversely, a constant $\sigma$ makes both derivative terms zero.

The restrictions “real switch” and “$\lambda >0$” are essential to the stated converse. They prevent an overbroad reading of the original scalar illustration.

The function

is a distributional solution of

vanishes for $t<0$, and has zero initial position and velocity.

Proof. Let $b(t)=-e^{2\mathrm{i}\lambda t}/4+1/4+\mathrm{i}\lambda t/2$. Then

Distributional differentiation of $Hb$ therefore produces neither a delta term nor a derivative of a delta:

This proves the equation. The traces at $0$ and the negative-time vanishing follow from the same formulas.

For $t>0$, the terms $1/4+\mathrm{i}\lambda t/2$ solve the homogeneous scalar equation. The linear term is $O(\lambda )$ on a fixed bounded time interval.

This $H$ is not ${\sigma }_{\epsilon }$. Neither Lemma lem:ode-residual nor Lemma lem:sharp-ode is a model or evaluation of the gravitational insertion, a value of the selected graviton inverse, or a cancellation of the XI-G witness.

For an admissible response tensor $r$, the supplied completion uses XIID,XVID

Here

and $\varphi$ is real and smooth in the reported later witness calculations.

The completion functional is

The original diagonal insertion is defined by

on its inherited domain.

The divergence in (eq:scalar-forcing) retains derivatives of $g$. The mass term is retained. No factor $g$ is inserted into the coordinate forcing (eq:coordinate-leg).

The supplied XII-D theorem states a continuous linear compact root map

for fixed observation data and collar, and the root identity

on its stated domain XIID.

The reported existence of this scalar identity is not replaced by the source contraction (eq:source-contraction). Conversely, its polarization should not be rejected merely because the theorem is printed using a single diagonal metric.

Suppose (eq:root-identity) holds for every metric in a real vector space of smooth inputs, with fixed $f,g,\varphi$ and root. Suppose $J_D$ is homogeneous quadratic there.

Polarization of the diagonal insertion then equals the root paired with the polarized source. Its complex bilinear extension satisfies, on the witness,

provided the real cosine and sine inputs and their required sums belong to that real domain.

Proof. Apply (eq:root-identity) to ${\gamma }_1+{\gamma }_2$, ${\gamma }_1$, and ${\gamma }_2$. Subtract and divide by two. Linearity of the integral gives the polarized source (eq:polarization).

For real inputs $a,b,c,d$, complex bilinear extension has the unique four-term form

Apply the same expansion to the polarized source and use Proposition prop:pair.

This lemma is a consequence of a valid root identity on the specified real domain. It does not independently verify the coordinate-field construction in (eq:coordinate-leg), nor extend a compact-test Green theorem to a noncompact forcing.

The supplied XVI-E makes precisely this scalar root reading on the real cosine and sine family and denotes the resulting witness value by $Q_{\lambda }$ XVIE.

The following result proves a test-pairing statement, not an admission statement.

Let $a_{\mu \nu }\in {C_c}^{\infty }({\mathbb{R}}^4;{\mathrm{Sym}}^2{T}^{\ast }M)$ be independent of $\lambda$. Define

The integral exists, and

at fixed $\epsilon$, $\mathbf{n}$, and $a$.

Proof. The integrand is smooth and compactly supported. Since $S_{\lambda }$ is spatially independent, put

Both belong to ${C_c}^{\infty }(\mathbb{R})$ and are independent of $\lambda$. Equation (eq:JDeval) gives

For each integer $M\ge 0$, integration by parts yields

and the analogous bound for $B$. Given $N\ge 0$, choose $M=N+2$. For $\lambda \ge 1$,

This proves the stated rapid decay.

For the reported compact root $a=a_{f,g}[\varphi ]$, this reproduces the fixed-data estimate of XVI-E. It does not establish:

• the same estimate for a frequency-dependent $\varphi$;

• uniformity as $\epsilon \downarrow 0$;

• pointwise decay of the response;

• a kernel wavefront bound or quantum admission;

• identification with an independently constructed auxiliary completion.

In particular, rapid decay of a smooth compact test pairing can coexist with a nonzero source or response contraction.

The original XVI-B statement that the entire successor chain was unanswered must be localized. The supplied later sheets report progress on named objects. Those results are not reclassified as absent merely because they were not proved in the original XVI-B draft.

XVI-D constructs

and, using the inherited smooth zero-data uniqueness theorem, identifies

It also computes the spatial traceless response contraction XVID.

These are reported later response results. They do not identify $V_{\lambda }$ with a Green symbol applied outside its proved compact-interior domain.

The scalar completion forcing is compact because every term contains $g$ or a derivative of $g$. The coordinate forcing

has no such factor $g$ and is spatially noncompact. Thus the compact-interior Green lemma alone does not verify its use in the named coordinate leg.

XVI-E constructs the root pairing $Q_{\lambda }$, retains the polarized scalar root identity $T^{\mathbb{C}}=Q_{\lambda }$, and constructs the explicit field

It proves the corresponding wave equation and zero-data uniqueness comparison, but does not identify that candidate with an independently verified inherited coordinate field on the witness family XVIE.

Uniqueness of solutions to a Cauchy problem is not, on its own, a definition of the named Green operator on a larger source space.

XVI-F-v2 independently defines

It proves existence and a source representation involving an integrated kernel. The earlier claim that this integrated kernel is automatically compact was corrected in that version XVIFvTwo.

The relevant elementary support fact is as follows.

Let $\hat{\tau }\in {C_c}^{\infty }(\mathbb{R})$ be supported in $[0,t_{\ast }]$ and define

Then $K_I$ is smooth, $K_I^{\prime \prime }=\hat{\tau }$, and $K_I=K_I^{\prime }=0$ for $u>t_{\ast }$. For $u<0$,

Thus compact support of $\hat{\tau }$ does not imply compact support of $K_I$.

Nevertheless $\sigma K_I$ and ${\sigma }^{\prime }{K}_I$ belong to ${C_c}^{\infty }(\mathbb{R})$. The same statements hold componentwise for tensor coefficients.

Proof. Differentiate under the integral:

The future vanishing follows from the source support. For $u<0$, expand $(t-u)$ over the fixed source support to obtain (eq:affine-tail).

The product $\sigma K_I$ is smooth, vanishes for $u\le 0$, and vanishes for $u>t_{\ast }$. The product ${\sigma }^{\prime }{K}_I$ is smooth and supported in the fixed ramp.

Consequently, a representation

with fixed smooth coefficients of this type has rapid decay by the same integration-by-parts proof as Proposition prop:root-decay. This is the direct fixed-data mechanism used for $I_{\mathrm{cand}}$ in XVI-F-v2. It does not require transferring decay from $Q_{\lambda }$.

XVI-F-v2 leaves the compatibility residual

unresolved, with

Its explicit advanced-field construction settles a spatial-integral identity for that constructed field. The remaining named check is its identification with the inherited advanced symbol on the boundary-flat source class.

The pairings are not proved equal and are not asserted to differ. Two rapidly decreasing scalar families need not be equal.

The supplied XII-E handoff reports compact microcausal membership of the polarized kernel of the completed XII-D insertion on its stated sheet XIIE. This note neither re-proves nor rescinds that reported theorem.

It is not silently identified with XVI-C's class-kernel pairing, an independently defined auxiliary evaluation, or a quantum retarded coefficient. Any proposed use of it must state the matching object, domain, and normalization.

The original reference to “the complete pairing of XVI-C” is a programme pointer. Since XVI-C was not supplied, this manuscript does not invent its section labels or definitions.

The following chain is a local specification of what must be checked:

A smooth root pairing or auxiliary evaluation can be a separate verification route, but an identification with this chain must be proved.

{}1.12

The complete-pairing test must retain separately:

1. the bulk source and the full correction ${\mathcal{D}}_{\sigma }$;

2. forced response terms and preparation-generated homogeneous terms;

3. both completion legs, including derivatives of $g$ and the mass term;

4. the tensor contractions actually selected by the insertion;

5. any gauge companion and Einstein-equation term produced by the relevant complete derivation;

6. support, operator-domain, polarization, and complexification hypotheses at each stage;

7. the final comparison with the frozen observable kernel.

The gauge-fixed free-shell condition is not substituted for an Einstein-shell hypothesis. A cancellation is used only after proof. The decisive question is which preparation-dependent terms the complete observable detects, not whether the correction vanishes as an uncontracted source.

Verify the inherited coordinate-field construction on the real cosine and sine response family and its required sums. Prove that its polarization and complexification give the same field used by the independently constructed candidate, if that identification is claimed.

State the exact source space, zero-data prescription, and cutoff-independence or operator-domain theorem used. A compact final observation test does not by itself make the coordinate forcing a compact-interior source.

Status: Unresolved.

Resolve the residual comparison reported in XVI-F-v2. Specify whether the inherited advanced symbol on the boundary-flat class denotes the independently constructed smooth zero-final-data field.

Prove equality of the required pairings if it follows. Do not infer equality from their separate rapid decay, and do not infer inequality from their different definitions.

Status: Unresolved.

Specify the actual observable to be compared with XI-G, its compact kernel $w_D$, its slot structure and symmetry, and the free-shell pairing.

Evaluate whether $w_D-w_0\in {\mathcal{I}}_{\mathrm{sol}}$. Apply Proposition prop:transfer only after that comparison. A different-class result would not, by itself, discharge admission.

Status: Unresolved.

After the actual evaluation is defined and identified, establish its required high-frequency estimate with the complete tensor and branch summation retained.

State whether the scalar test is fixed or frequency dependent, which constants are uniform, and which kernel wavefront or admission conclusion is sought. The fixed-data estimates for $Q_{\lambda }$ and $I_{\mathrm{cand}}$ are not substituted for that conclusion.

Status: Unresolved.

These are compatibility and construction obligations, not proved obstructions. They do not discharge XVI-A's retarded-product obligations 4–6.

The revision makes the following corrections explicit:

1. Class transfer is conditional. A changed source is not proof of a changed free-shell class. The exact same-class transfer is Proposition prop:transfer.

2. The documentary finding is bounded. The supplied retrieval report is not an exhaustive nonexistence theorem.

3. The sharp comparison has a domain. $H{J}_3$ is used only for smooth evaluated sources. No discontinuous multiplier is applied automatically to an unidentified distributional kernel.

4. Temporal support is not compact support. A closed ramp layer is spatially unbounded.

5. Late source agreement is not response agreement. Lemma lem:memory gives an explicit counterexample to that inference.

6. The boundary conclusion is narrow. No boundary action follows from the cited XI-A passage. Weighted representative changes still have (eq:boundaryrepresentative).

7. Frequency factors are separated. $\mathsf{q}$ and ${\mathfrak{d}}_1$ are independent of $\lambda$; $\mathfrak{d}=\lambda {\mathfrak{d}}_1$.

8. The ODE converse has its hypotheses. The switch is real and $\lambda >0$.

9. Polarization is not rejected on typographical grounds. A valid diagonal root identity on a real vector space polarizes by Lemma lem:root-polarization. That scalar operation does not independently verify an unconstructed coordinate-field representation.

10. Later progress is not erased. XVI-D's response and XVI-E's root pairing are recorded. XVI-F-v2 supplies direct auxiliary decay, not a transfer from the root.

11. The integrated support error is corrected. A compact source can generate an affine past tail. Effective switched amplitudes can be compact even when the integrated kernel is not.

12. No stale hypothesis is cited. No null-survival assumption is imposed. The supplied revised XVI-A contains no adopted hyp:null-survival gate.

13. No compilation result is fabricated. This is a single-file pdfLaTeX source. Compiler execution and full-corpus verification are not claimed.

This manuscript does not claim:

• a repair of X-G or XI-G;

• a new admission into ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ or ${\mathcal{F}}_{\mathrm{pol}}$;

• a new quantum retarded coefficient or a completed second-order relational construction;

• an interacting BRST differential or an unrestricted quantum master equation;

• a general extension of the named Green operators from compact-interior tests to every forcing;

• equality or inequality of $I_{\mathrm{cand}}$ and $Q_{\lambda }$;

• a complete inherited insertion evaluation from the source-level contraction alone;

• a sharp-width limit or estimates uniform in $\epsilon$;

• a loop, state, positivity theorem, or ultraviolet subtraction.

The selected inverse retains coefficient $-1$, not $-4$. Full Einstein–Hilbert density weighting is not substituted for the selected cubic preparation. The source $\sigma J_3$ alone is not the prepared model XIIC,XVIA.

release statement.{ The source-level identities, support comparisons, weighted-representative residual, and elementary diagnostics are proved from the stated inputs.

The supplied later response, root, and auxiliary results are recorded on their named objects and domains. Their unresolved identifications are not replaced by a source symbol argument.

X-G and XI-G retain their original quantifiers. Applicability of XI-G to an actual matching prepared class requires a kernel comparison.

The complete-insertion, class-comparison, and admission programme remains Unresolved. This is a completed source-diagnostic proof manuscript, not a completed proof of the unresolved retarded coefficient. }

XVIA

Landmark XVI-A. Supplied revised XVI-A.tex. Used for the declared normalization, forcing sign, conditional class-transfer correction, and the unresolved retarded-product obligations. No external publication status is asserted.

XVID

Landmark XVI-D. Supplied XVI-D.tex. Internal locators: eq:Slambda, eq:Vlambda, lem:identify, lem:contraction, lem:coord, and lem:scalar. Used as a supplied later response and completion-domain record.

XVIE

Landmark XVI-E. Supplied XVI-E.tex. Internal locators: lem:Q, lem:TD, lem:TC, lem:Xcand, lem:compat, and cor:decay. Used for the root pairing, conditional polarized root reading, and delimited coordinate compatibility.

XVIFvTwo

Landmark XVI-F, revised supplied text. Filename XVI-F-v2.tex. Internal locators: prop:Icand, prop:kernel, eq:Kaffine, prop:residual, prop:sEA, prop:sint, prob:missing, and prop:decay. The earlier supplied XVI-F.tex is not used for its superseded compact-support assertion.

XIA

Manuscript XI-A, version 1. Supplied identifier 0662c6c3. Supplied locator: “Quadratic gravitational coefficient,” prop:FP, lines 404–445. The boundary-variation and divergence statements are used only within the compact-interior setup reported in the supplied XVI-B material.