Landmark XVI-I Reduced insertion graphs, singular loci, and extension freedom Revised proof manuscript Classical admission and graph existence under stated inputs; normalization and interacting observability unresolved

We distinguish the admitted classical prepared insertion from a reduced graph sector involving that insertion and one compactly switched cubic interaction. The supplied XII-D admission excerpt concerns the same polarized kernel represented in revised XVI-H. Under that additional inherited premise, direct microcausal membership and a nonempty free-shell microcausal class follow. This does not repair the different frozen class excluded by XI-G.

For the graph calculation, we choose an ambient massless time-ordered graviton kernel and identify its restriction with the combination $H+\mathrm{i}{G}^{\mathrm{A}}$ in the stated normalization. Its off-diagonal wavefront relation proves that the two differentiated edges can be multiplied. The cubic Hessian and the insertion source jets then define the reduced graph without an unspecified tensor expansion. An explicit spacelike evaluation proves nonvanishing of the open-index vertex block when the interaction switch is nonzero; no nonvanishing of an arbitrary source-contracted or root-attached sum is inferred.

The relative-coordinate building blocks have finite scaling degree. Including both insertion and interaction derivatives gives an upper bound of eight in four relative dimensions. Extensions exist, but their local coefficients are not normalized. Under the explicit root wavefront bound, the compressed root attachment satisfies the distributional product criterion and has proper support. It therefore yields a family of smeared coefficients indexed by the extension choices, not a selected interacting quantum observable. BRST/BV closure and the retarded interacting coefficient remain unestablished.

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

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

and release status.{ The ambient manifold is $M={\mathbb{R}}^4$. The preparation width $0<\epsilon <T$ is fixed. The preparation data $f,g,{\sigma }_{\epsilon }$ are those of revised XVI-H.

The interaction switch ${\chi }_{\mathrm{I}}\in {C_c}^{\infty }(M;\mathbb{R})$ is separate from the preparation data. It is not a past-to-future switch.

Established under explicit inherited inputs: classical microcausal membership of the matching prepared kernel; the punctured two-edge distribution; its finite scaling-degree bounds; extension existence; and a family of compressed root-attached distributions.

Not selected or established: extension normalization, a retarded interacting coefficient, BRST/BV closure, or a physical quantum observable.

No new classical trisolution wavefront calculation is performed. The classical cone consequence is an imported consequence, stated with its hypotheses.

No sharp-width limit, global adiabatic limit, ultraviolet completion, or all-orders finite-parameter claim is made. No ultraviolet regulator-removal procedure is performed. }

This manuscript revises the supplied XVI-I.tex. Its classical references are the supplied revised XVI-H.tex, revised XVI-E.tex, revised XVI-F.tex, and current XVI-G.tex.

An additional supplied excerpt from XII-D, thm:admission and eq:t-bounds, contains

and the associated microcausal admission statement. That microlocal result was not a proved premise of revised XVI-H. It is added here explicitly, rather than attributed retroactively to that manuscript.

Complete XII-D, XI-G, and XI-B sources have not been independently audited. References to their internal labels identify supplied excerpts or formulas reproduced in the supplied manuscripts.

The proof has four separate stages:

1. match the classical kernel and state the narrow admission implication;

2. define and verify the ambient propagator and reduced two-edge distributions;

3. extend the relative-coordinate building blocks, without choosing their normalization;

4. test compressed root attachment by explicit pullback, product, and proper-pushforward conditions.

The last stage advances beyond the original draft's unspecified root-product stop: with the imported $Z_2$ root bound, the compressed product condition can be checked. This establishes attachment existence for an extension family, not a normalized quantum coefficient.

Let

and set

Fix

The fixed preparation ramp is

with $\chi$ real and smooth.

The metric perturbation is a covariant symmetric tensor. Indices are raised only with $\eta$. We use a fixed basis of the ten-dimensional symmetric-tensor fiber and its dual, with component indices $I,J,K,L$. Thus summation over field components counts independent symmetric components once. It does not introduce a second factor for off-diagonal spacetime components.

The fixed Minkowski fiber pairing identifies the tensor bundle and its dual. Trace reversal is

Its two-leg kernel matrix, with the fixed fiber identifications understood, is denoted by ${\mathbb{P}}^{IK}$. All component contractions below use that convention.

The volume density is ${\mathrm{d}}^{4}x$. Distributional pairings are bilinear, without conjugation.

The classical completion is the stated XII-D insertion

with the response normalization $u_{\mathrm{D}}=-\mathcal{P}{E_0}^{\mathrm{R}}{\mathcal{J}}_{\mathrm{D}}$ and preparation source

Its real polarization is the one represented in revised XVI-H by a compact kernel $t_{\mathrm{D}}$, symmetric in its two tensor slots:

In the chosen component frames, its general test-function action is

The coefficients are smooth and may be chosen symmetric:

The compact root kernel $B_A(y,x)$ is the same root kernel used in the XII-D source-root construction.

The superscript $A$ labels a source-tensor component, not an additional field species. The coefficients in (eq:t-test) include the complete preparation correction. They are not replaced by coefficients of ${\sigma }_{\epsilon }{J}_3$ alone.

The action on arbitrary smooth tests is meaningful because $B$, and hence $t_{\mathrm{D}}$, is compactly supported. Revised XVI-H supplies finite-order continuity on $C^{\infty }$, not only on compactly supported configurations XVIH.

Define

Then, on the inherited real smooth completion domain,

For real smooth tensor directions $a,c$,

There is no additional factorial in $F_t$.

Proof. Degree-two homogeneity of the diagonal completion gives

The kernel representation proves the first assertion. Differentiating the quadratic polynomial and using tensor-slot symmetry proves (eq:Dh).

In Cartesian cotangent frames, define

This uses the canonical identifications of cotangent fibers in Minkowski space.

The supplied XII-D admission excerpt applies to the kernel in Input input:kernel, with the same insertion, response normalization, preparation data, scalar slot, and source polarization. It gives

The root statement reproduced in the supplied XVI-I gives

The root represents a compact smooth root map:

in the distributional kernel sense.

For the separate trisolution consequence only, the inherited XI-G transport lemma is used in its stated compact-kernel quantifier: it transports component covectors in Minkowski frames, may move their base points, and restricts each surviving entry to a null or zero covector.

These are additional inherited inputs, not new wavefront calculations. The proof below does not derive (eq:tcone) or (eq:Bcone) from compactness.

Under Inputs input:kernel and input:cones, the kernel to which the supplied XII-D admission statement applies is exactly $t_{\mathrm{D}}$ used here.

Proof. The diagonal insertion is the same ${\mathcal{C}}_{3,\mathrm{D}}={\mathcal{A}}_1(u_{\mathrm{D}};\cdot )$. Its polarization has the same factor $1/2$. The source polarization reproduces the same ${\mathcal{J}}_{\mathrm{D}}(a,c)$, including the ramp-derivative terms. The root kernel and derivative-transfer signs are those of (eq:t-test).

Thus both constructions act on general tests by (eq:t-test). They define the same distribution. There is no change of normalization, tensor species, preparation, or real domain in this matching.

Under the additional microlocal input:

1. The species-symmetrized polynomial $F_t$ is an element of the stated classical microcausal polynomial carrier.

2. The free-shell class of $t_{\mathrm{D}}$ has a microcausal representative, namely $t_{\mathrm{D}}$ itself.

3. Under the separately stated transport premise,

   $$\mathrm{WF}({\mathsf{W}}_t)\cap ({\tilde{V}}_{+}^3\cup {\tilde{V}}_{-}^3)=\varnothing ,{\mathsf{W}}_t=({\mathsf{E}}_0\otimes {\mathsf{E}}_0\otimes {\mathsf{E}}_m)t_{\mathrm{D}}.$$

These are classical kernel and free-shell conclusions. They are not time-ordering, graph normalization, or BRST/BV observability statements.

Proof. The closed future and past causal cones are pointed. A nonzero tuple wholly in either cone cannot sum to zero. Equation (eq:tcone) therefore excludes the forbidden cone products.

Embed the typed kernel in the direct-sum field bundle and apply the normalized species symmetrization. Permutations preserve compactness and the sum-zero covector condition. Evaluating the symmetrized kernel on the repeated multiplet argument gives the same polynomial $F_t$; no extra factorial is inserted. This proves item 1.

Define the free-shell null space in the matching compact-kernel space by pairings with all smooth solutions of $\square a=\square c=0$, $P_m\varphi =0$. The class $t_{\mathrm{D}}+{\mathcal{I}}_{\mathrm{sol}}$ contains $t_{\mathrm{D}}$. Its own cone exclusion proves item 2 directly. It is unnecessary to enlarge the frozen-kernel quantifier of XI-G's printed criterion.

For item 3, the transport premise preserves the component covectors in the fixed Minkowski frames. A surviving tuple therefore still sums to zero and is nonzero. Pointedness again excludes both cone products.

Suppose the matching frozen XI-G kernel $u_0$, for data on which its obstruction theorem applies, satisfies

Then, under Theorem thm:admission,

Thus the admitted prepared class is not that excluded frozen class.

Proof. If the difference were in ${\mathcal{I}}_{\mathrm{sol}}$, the affine classes would coincide. The admitted representative $t_{\mathrm{D}}$ would then contradict the frozen obstruction.

The corollary is conditional on both sets of stated hypotheses, including the matching frozen data. It is not a new evaluation of the frozen witness, and does not extend the obstruction to other data.

The former blanket statement “no admission is claimed” is therefore replaced by the narrow classical conclusion above. The new insertion is admitted under the supplied XII-D premise; the different frozen class remains excluded XIID,XIG.

Revised XVI-H, using revised XVI-E/F and current XVI-G, proves

at each frozen $\lambda >0$ for every real smooth scalar evaluated there. It also proves the partial kernel contraction

The obsolete claim that this equality is unproved is not retained XVIH,XVIE,XVIF,XVIG.

The smooth density $j_{\lambda }$ is a partial tensor evaluation, not the full kernel and not the two-edge graph defined below.

This section makes an explicit propagator choice. It is not inferred merely from a free-shell equation.

Let $W_0$ be the massless Minkowski positive-frequency distribution

The integral is distributional. Set $h_0=-W_0$, in the reported XI-E convention.

Let ${E_{0,\mathrm{amb}}}^{\mathrm{A}}$ be the ambient advanced fundamental solution of $\square$. Its spatial Fourier kernel is

with the continuous value at $\mathbf{p}=0$. The retarded kernel is $\theta (t)\sin (t|\mathbf{p}|)/|\mathbf{p}|$. These distributions are translation invariant on ambient $M$.

For a source compactly supported in ${M_T}^{\circ }$, its ambient advanced solution restricted to the slab is

The upper limit is $T$ because the zero-extended source vanishes above $T$. This identifies the restriction with the supplied slab sine integral on that source class. It does not declare an arbitrary slab operator translation invariant.

All graph edges below use the ambient kernel. Thus an interaction point outside the slab is not evaluated by an unjustified slab-only prescription.

Define the ambient tensor distributions

and put

The scalar distribution $d$ is

with its globally defined distributional meaning fixed by (eq:DF). Consequently

on ambient compact tensor tests. In components,

Proof. For $t<0$, the spatial Fourier expression is

For $t>0$, the advanced summand vanishes. This proves (eq:timeordered) and symmetry under $r\mapsto -r$.

The temporal derivative of the two Fourier branches has jump $-\mathrm{i}$ at $t=0$. Hence $\square d=-\mathrm{i}\delta$. Using ${\mathcal{P}}^2=\mathrm{id}$ gives

This is an explicit ambient massless prescription, with the factor $\mathcal{P}$ fixed by the inherited inverse and two-point conventions. No X-sheet normalization is imported.

The symbol $i0$ below denotes its distributional boundary condition, not the preparation width and not an ultraviolet regulator-removal scheme.

Use the mathematical Fourier convention $\hat{u}(k)=\int e^{-\mathrm{i}k\cdot r}u(r){\mathrm{d}}^{4}r$, with the coordinate pairing in that formula. Let

On $r\ne 0$,

and

Away from the nonzero light cone, $d$ is smooth. On a spacelike region,

Proof. The angular Fourier integral in (eq:W) gives the massless rational two-point distribution with the positive frequency temporal boundary condition. Selecting the two temporal branches in (eq:timeordered) gives (eq:rational) off the origin.

At a nonzero null point, $\mathrm{d}q\ne 0$, so $q$ is a local coordinate transverse to the cone. For the stated Fourier convention, $(s-\mathrm{i}0{)}^{-1}$ has the single negative-frequency conormal ray. Pulling it back by $q$ gives (eq:WFd). At $q\ne 0$, the reciprocal is smooth.

This is also the massless specialization of the Feynman wavefront description in BDHintro (Proposition 26). That paper uses the opposite Fourier sign in its physical wavefront convention; reversing the covector sign gives (eq:WFd). The one-ray product test is independent of that sign choice.

For arbitrary multiindices $\mu ,\nu$, the product

exists on $M\setminus \{0\}$. Its pullback to

exists under $r=y-w$. Finite tensor contractions and smooth multipliers preserve definedness.

Proof. Differentiation does not enlarge the wavefront set. At each nonzero null point both factors have covectors on the same open half-ray in (eq:WFd). Two such nonzero covectors cannot be negatives of each other. The distributional product criterion therefore applies. The difference map $(y,w)\mapsto y-w$ is a submersion, so its pullback exists.

The tensor matrices are constant, and contraction is a finite component sum. Smooth multiplication is always defined.

The conclusion includes noncoincident null-related points. The light cone remains singular there, but is not deleted from the distributional domain.

Retain the supplied cubic density representative

with the following atoms:

and

Only coefficients appearing in these finite expressions are used. This is the unintegrated representative supplied in the original XVI-I through XII-C, not an Euler-class replacement.

Each monomial has three field-jet factors and total field derivative order two. The density has jet order at most two. The distinction between total derivative order and the maximum jet order of an individual factor will be used below.

Choose

The preparation correction belongs to the insertion kernel. It is not added again as an interaction vertex.

Write $h_{K,a}={\partial }^a{h}_K$, where the multiindices label independent symmetric jet coordinates. Define

Absent jet variables have zero coefficient.

For smooth tensor directions $A,C$,

The coefficients satisfy

whenever they are nonzero. They are linear differential expressions in the remaining field $h$. For such a term, the derivative order on that remaining factor is at most $2-|a|-|b|$. Moreover,

No integration by parts is made in this definition.

Proof. The chain rule in the independent jet coordinates gives (eq:Hessian). Equality of mixed partial derivatives gives the symmetry.

Differentiating a cubic jet monomial removes two field factors. The remaining coefficient is linear in one field jet. The original total derivative order was two, so the derivatives on the two varied factors have sum at most two, and the remaining factor has the stated residual order.

The interaction switch is fixed under field differentiation. This proves the last formula without derivatives of ${\chi }_{\mathrm{I}}$.

This defines the full tensor Hessian unambiguously. A flattened table is unnecessary, and no unspecified bidifferential operator is substituted for (eq:L).

For two distinct vertices, let the cross-contraction operator use $D_F$, with one derivative at each vertex per edge. The second term of its exponential carries $1/2!$. The insertion derivative is ${D_h}^2{F}_t=2t_{\mathrm{D}}$. The interaction derivative is the ordinary second derivative (eq:Hessian), with no divided-power convention.

Therefore the numerical multiplier of the two-edge contribution is

The sums over the two labelled legs are retained in the Hessian and the two kernel slots. There is no second division for tensor-slot symmetry.

This is the graph-expansion convention of FR (Eq. (82)), adopted here only for this numerical sector. Coupling powers, the vertex multiplier in an $S$-matrix, and ${\hslash }^2$ are not included in the reduced kernel. Thus the object below is not called a complete first quantum coefficient.

For $|\alpha |+|\beta |\le 2$, define on $\Omega$

The source-contracted reduced kernel is

It still has the source/root index $A$. The scalar insertion slot is not contracted.

The derivatives ${{\partial }_y}^{\alpha }$ and ${{\partial }_y}^{\beta }$ arise from the differentiated source deltas in (eq:t-test). Their displayed sign is the distributional derivative-transfer sign.

In (eq:Vblock), derivatives act only on their designated edge. They do not act on $c$, ${\chi }_{\mathrm{I}}$, or $L$. This fixes the recipe before any extension or integration.

For every smooth real $h$, all blocks ${{\mathcal{V}}^{\alpha \beta }}_{IJ}[h]$ and all ${U_A}^{\circ }[h]$ are distributions on $\Omega$. They are linear in $h$. Their coefficients and derivative placements are exactly specified by (eq:L), (eq:Vblock), and (eq:U).

The intended compressed attachment is

on the full variables $(y,w,x)$, followed by integration in $y$. Its definition is tested separately in Section sec:attachment.

Proof. Using $r=y-w$,

Thus each summand is a constant tensor contraction of one of the products in Proposition prop:product, multiplied by smooth coefficients. Every sum is finite. The Hessian coefficients are linear in $h$.

The entire switched kernel is generally not translation invariant. Only its numerical edge building blocks are distributions of $r=y-w$. Restoring the smooth coefficients in $y,w$ must not be mistaken for a new translation-invariance assertion.

Suppose ${\chi }_{\mathrm{I}}\not\equiv 0$. There are a smooth real background $h$, tensor contractions of the open indices of ${{\mathcal{V}}^{00}}_{IJ}[h]$, and a compact test supported in a spacelike subset of $\Omega$, for which the resulting pairing is nonzero.

This does not imply that every source-contracted kernel ${U_A}^{\circ }[h]$, or the root-attached sum, is nonzero.

Proof. Choose $w_0$ with ${\chi }_{\mathrm{I}}(w_0)\ne 0$, and let

At $w_0$, the field and its first derivatives vanish, while

Take constant tensor directions

We calculate the Hessian at this jet rather than infer it from scaling degree. For the metric

put $a=1+\kappa s$, $b=a-\kappa t$. At $w_0$,

The supplied Ricci convention gives

Therefore the coefficient of $\kappa$ in the signed density $-2{\kappa }^{-2}\sqrt{-q_{\kappa }}R$ is

Indeed,

Thus

Every Hessian coefficient with $|a|+|b|>0$ has its remaining field differentiated at most once. Those coefficients vanish at $w_0$, because $h(w_0)=\partial h(w_0)=0$. Consequently all vertex derivative-edge corrections vanish at that point, not merely in a scaling estimate.

Choose open-index contractions whose images under the invertible matrix $\mathbb{P}$ are $A,C$. Choose

At this spacelike pair the selected component of ${\mathcal{V}}^{00}$ is exactly

The graph is a smooth real function on a neighborhood of this spacelike pair. By continuity, a sufficiently small nonnegative test supported there has a nonzero pairing with that component.

If ${\chi }_{\mathrm{I}}=0$, every reduced block is zero. Even for nonzero ${\chi }_{\mathrm{I}}$, the preparation data may make the later contracted sum vanish; for example $f=0$ makes the root vanish. No universal nonvanishing claim is made.

For $u^{\circ }\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4\setminus \{0\})$, use the usual scaling degree at the origin:

with the normalized rescaling of tests.

The diagonal $y=w$ has codimension four. Use coordinates $(r,w)=(y-w,w)$; $w$ is tangential. The smooth interaction and preparation coefficients are not part of the translation-invariant numerical kernel.

The nonzero numerical products ${p_{\mu \nu }}^{\circ }$ of (eq:p) are homogeneous on the punctured domain of degree

and therefore have scaling degree

For zero products the corresponding inequality suffices.

The vertex-only blocks with $\alpha =\beta =0$ have local transverse scaling degree at most six. Including the insertion source derivatives gives

locally, uniformly on compact tangential sets. This is an upper bound before root attachment, not a sharp degree of the total tensor sum.

Proof. The massless $d$ is homogeneous of degree $-2$. Its derivative of order $|\mu |$ has degree $-2-|\mu |$. The punctured product is already defined by Proposition prop:product; rescaling respects that product. Hence its degree is the sum of the two degrees.

For $\alpha =\beta =0$, the Hessian satisfies $|a|+|b|\le 2$, so the numerical edge degree is at most six. With the insertion derivatives included,

This gives the bound eight.

In relative coordinates, all remaining coefficients are smooth functions of $(r,w)$. Their derivatives are bounded on the compact sets used in a local transverse scaling test. Smooth multiplication cannot increase the bound there. Finite summation completes the estimate.

This proof does not deduce a pointwise-product estimate merely from a derivative lemma. It first proves the product, then uses its explicit homogeneity.

For the finitely many numerical products occurring in the graph, extensions

exist without increasing their scaling degree.

For a fixed pair with $n=|\mu |+|\nu |$, two such extensions differ by

In particular, all building blocks in the source-decorated sector have ambiguity order at most four.

Replacing the numerical products in (eq:Vblock)–(eq:U) by these extensions defines distributions ${\bar{\mathcal{V}}}_{IJ}^{\alpha \beta }[h]$ and ${\bar{U}}_A[h]$ on all $M^2$. No coefficients $a_{\mu \nu ;\gamma }$ are chosen.

Proof. The point-extension theorems for finite scaling degree apply in four relative dimensions BF (Theorems 5.2–5.3). The difference of two extensions is supported at $r=0$. A point-supported distribution is a finite sum of derivatives of $\delta$, and

The scaling bound $4+n$ therefore permits only $|\gamma |\le n$.

Pull back each extension under the submersion $(y,w)\mapsto y-w$, multiply by the displayed smooth coefficients, and take the finite component sums. These operations define the claimed global distributions.

The finite collection of point-extension constants is finite dimensional before additional relations. Edge-exchange symmetry can be retained by choosing ${\bar{p}}_{\mu \nu }={\bar{p}}_{\nu \mu }$, for example by averaging. That algebraic restriction does not fix the remaining constants.

A subtraction projection can be specified by functions $w_{\gamma }(r)$ satisfying

and by the values of the extension on these functions. No such values, covariance conditions, Ward conditions, or renormalization scale are selected here.

There are two different ambiguity statements:

• The numerical translation-invariant building blocks have the constant-coefficient point ambiguities (eq:ambiguity).

• An unrestricted extension of the full distribution in $(r,w)$ can have coefficients depending on $w$; transverse order alone does not make that larger freedom finite dimensional.

The present construction uses the first, explicitly specified extension family. It does not classify every possible tangential or fully root-attached ambiguity.

Let

and define the projections

They are submersions. The proposed compressed graph distribution is

For a general root, the product would require

for every component, followed by properness of $\pi$ on the product support.

Under (eq:Bcone), condition (eq:root-condition) holds for every distribution ${\bar{U}}_A[h]$ constructed above, regardless of its detailed wavefront set.

The product $\mathcal{G}[h]$ is compactly supported. Its pushforward $R[h]\in {\mathcal{E}}^{\prime }(M_w\times M_x)$ is defined. The same construction works on the off-diagonal domain before extension.

No normalized extension is thereby selected.

Proof. The lifted root wavefront covectors have the form

by (eq:Bcone). The lifted reduced distribution has covectors

with $(k,\ell )\ne 0$. For these two covectors to cancel, the $x$-component would force $p=0$, contrary to the nonzero root covector. Thus the product criterion is satisfied.

The root is compact in $(y,x)$. Every reduced term contains the interaction switch ${\chi }_{\mathrm{I}}(w)$, including the diagonal extension terms. Hence its $w$-support is contained in $\mathrm{supp}{\chi }_{\mathrm{I}}$. The product support is a closed subset of the compact set

The projection $\pi$ is therefore proper on that support, and the pushforward is defined.

Neither argument requires the root to be smooth or the reduced graph to be smooth at a null-related pair.

The decisive property is that the root has no nonzero wavefront covector with zero scalar-slot covector:

The stronger $Z_2$ input guarantees it. If that input were withheld, the generic test (eq:root-condition) would remain an unresolved attachment obligation. Compactness alone would not replace it.

This compressed argument does not separately multiply every factor of an unfolded formula

That internal composition is used only through the already constructed root $B$. No new claim about all unfolded partial-diagonal products is made.

For each extension tuple $\{{\bar{p}}_{\mu \nu }\}$, each smooth $h$, and each real smooth scalar $\varphi$, the number

is well defined by compact support. It is linear in $h$ and in $\varphi$. Equivalently,

where compact cutoffs on the fixed effective support are understood.

Proof. Theorem thm:attach gives a compact output distribution, which acts on arbitrary smooth functions. The Hessian coefficients are linear in $h$, and the root is linear in $\varphi$.

To obtain (eq:smearfirst), multiply the root kernel by the smooth scalar, push forward its $x$-slot, and use the inherited root-kernel identity. The resulting $a_{f,g}[\varphi ](y)$ is smooth. The wavefront test just proved, together with proper support, permits the corresponding distributional composition and interchange of these pushforwards. Pairing against the reduced distribution gives (eq:smearfirst).

Thus root attachment is no longer left at an unspecified product obstruction under the stated $Z_2$ premise. What remains unresolved is the selection of the extension tuple and the normalization of the resulting coefficient.

The family $C_{\bar{p}}$ is not asserted to be independent of its extension constants. Its differences are diagonal contact contributions propagated through the root. Locality of a subtraction in $r$ does not make $a_{f,g}[\varphi ]$, or the final coefficient as a functional of the original fields, local.

No identification with ordinary bulk gravitational counterterms is proved.

Every graph formula uses the particular off-shell $t_{\mathrm{D}}$ of (eq:t-test). Free-shell equivalence is not used to change it.

An ideal term such as ${\square }_{1}a$ pairs trivially with a free tensor solution. It need not pair trivially with the chosen propagator, since

Such changes can create contact terms in a graph calculation. It is incorrect to conclude that every nonzero ideal term produces a nonzero graph; the assertion is only that free-shell invisibility does not guarantee time-ordered invisibility.

Declare the preparation data and $t_{\mathrm{D}}$ fixed under the free variation. Adopt the inherited free field formulas

with

Ghost directions may be treated as formal Grassmann-valued smooth directions in the following polynomial identity.

With that convention,

No vanishing of the right-hand side is asserted.

Proof. Only the two tensor arguments vary. The scalar variation is zero, and tensor-slot symmetry makes the two tensor contributions equal. Compact support makes the pairing with smooth directions well defined.

If preparation data are assigned transformations, their variation contributes an additional term. It is not computed here.

The chain-rule identity is not a Ward identity or a proof of BRST/BV closure. No interacting differential, BV antibracket, master action, or compensating graph family is constructed.

For a specified scalar two-point distribution ${\mathsf{H}}_2$,

is definable because $j_{\lambda }\otimes j_{\mu }$ is a compact smooth test density. That is a scalar pairing, not the tensor two-edge product, and it does not determine the interaction Hessian, normalization, or graph extension.

Here the tensor edge prescription has been explicitly chosen. It does not provide a positive state for the full gauge multiplet, ghost and scalar time-ordered blocks, or a complete family of time-ordered products.

No Bogoliubov or retarded interacting coefficient is assembled from the family (eq:coefficient). No extension is selected by a covariance, field-equation, unitarity, Ward, or renormalization condition. No scale derivative is assigned to an unchosen normalized extension.

The boundary-value notation defining $d$ is not a temporary ultraviolet regularization of the graph. The extension problem is treated directly in distribution space.

No global adiabatic limit, sharp preparation-width limit, ultraviolet completion, or all-orders finite-parameter renormalizability theorem is part of this manuscript.

Retain the exact classical matching inputs, the supplied classical and root cone bounds, the ambient propagator prescription, and the declared cubic density representative.

Then:

1. The classical prepared polynomial has direct microcausal membership, and its free-shell class contains its own microcausal kernel.

2. The oriented off-diagonal wavefront relation of the selected propagator defines every differentiated two-edge product occurring in the explicit reduced sector.

3. For nonzero ${\chi }_{\mathrm{I}}$, the open-index vertex block has a nonzero spacelike component for a specified smooth background. No general nonvanishing of the full contracted sum follows.

4. The source-decorated reduced sector has local transverse scaling degree at most eight in four relative dimensions.

5. Its numerical building blocks admit point extensions; their ambiguity orders are at most four. No normalization constants are chosen.

6. Under the root $Z_2$ premise, compressed root attachment and proper pushforward exist for each extension choice. The resulting smeared coefficients form a family.

No member of that family is selected as a normalized retarded interacting coefficient or a BRST/BV observable.

Proof. Theorem thm:admission proves item 1. Proposition prop:product and Theorem thm:reduced prove item 2. Proposition prop:nonzero proves item 3. Proposition prop:sd proves item 4. Theorem thm:extension proves item 5. Theorem thm:attach and Corollary cor:coefficient prove item 6. The unchosen normalizations and unconstructed interacting identities are recorded in Section sec:limits.

{}1.12 {@{}p{0.30}p{0.65}@{}} Object or result & Current status

Object or result & Current status, continued

Classical polarized kernel & Established under the revised XVI-H root/source inputs. Its general test action is (eq:t-test).

Matching XII-D admission kernel & Exact matching under the explicitly supplied domain and normalization premises: Proposition prop:match.

Direct classical admission & Established as an imported-premise consequence: Theorem thm:admission. The old blanket non-admission disclaimer is removed.

Prepared free-shell class & Contains the admitted representative $t_{\mathrm{D}}$. On data with the frozen obstruction, it cannot be the excluded frozen class: Corollary cor:oldclass.

Trisolution cone exclusion & Consequence of the separately stated transport premise and imported $Z_3$ bound. No new classical wavefront calculation is performed.

Witness matching to $j_{\lambda }$ & Established in revised XVI-H using current XVI-E/F/G. It is partial evaluation, not the graph kernel.

Ambient tensor propagator & Explicitly chosen and identified: Proposition prop:DF. $D_F=\mathcal{P}d$, with $(-\square \mathcal{P})D_F=\mathrm{i}\mathrm{id}$. The slab restriction is justified on interior sources.

Off-diagonal edge product & Proved for the chosen propagator, including nonzero null-related pairs: Proposition prop:product.

Cubic Hessian and derivative recipe & Defined by exact jet differentiation: (eq:L)–(eq:Hessian). Total vertex derivative order is at most two.

Reduced graph sector & Defined and proved distributional on $y\ne w$: (eq:Vblock)–(eq:U), Theorem thm:reduced.

Nonvanishing & Proved for an open-index vertex block when the switch is nonzero: Proposition prop:nonzero. Not asserted for every contracted or attached sum.

Scaling degree & Vertex-only bound six; source-decorated bound eight. Local transverse upper bounds, not a sharp total degree: Proposition prop:sd.

Diagonal extension & Existence and point-building-block ambiguity established: Theorem thm:extension. Normalization constants unchosen.

Compressed root attachment & Product and proper-pushforward tests proved under $\mathrm{WF}(B)\subset Z_2$: Theorem thm:attach. No root singular support is deleted.

Smeared root-attached coefficient & A family exists for extension choices: Corollary cor:coefficient. No normalized member is selected.

Unfolded root graph & No separate product theorem for every internal $z$-factor is claimed. The compressed inherited root is the route used.

Free variation & Chain-rule formula proved with fixed preparation: Proposition prop:s0. Closure is not asserted.

Retarded interacting coefficient & Not assembled. Time-ordered normalization and the interacting identity remain unresolved.

BRST/BV observability & Not established. No interacting differential, Ward identity, or master construction is supplied.

Sharp-width and ultraviolet claims & Outside scope. No regulator-removal or all-orders theorem is performed.

The principal corrections to the original XVI-I are:

1. Admission implications are stated honestly. The additional $Z_3$ premise concerns the matching prepared kernel and implies narrow classical admission. It is not simultaneously imported and denied.

2. The ambient propagator is actually specified. Its translation-invariant definition is separated from a slab inverse. Algebraic inversion alone is not used as a wavefront theorem.

3. The two-edge recipe includes insertion derivatives. The source-delta derivatives and their signs are included in (eq:Vblock), not omitted from the power count.

4. Total derivative order is audited. The cubic Hessian places at most two derivatives in total on its two varied legs, not independently four. The insertion can add another two, yielding the conservative bound eight.

5. Nonvanishing has an off-diagonal test. The switch must be nonzero, and the derivative corrections vanish at the chosen jet. Scaling degree alone is not a nonvanishing proof.

6. Point and tangential ambiguities are separated. A finite point-extension family is not a classification of arbitrary tangential counterterms.

7. Root attachment is tested rather than merely withheld. The compressed product condition follows from the supplied root cone bound. The normalized coefficient nevertheless remains unselected.

8. Version references are synchronized. The current witness chain is not described using superseded XVI-E/F/G stops.

Choose the extension constants and impose the desired covariance, derivative identities, field equations, exchange symmetry, and any Ward conditions. Prove compatibility of those choices before calling $C_{\bar{p}}$ a normalized insertion coefficient.

Extension existence does not choose these conditions or their solutions. Status: Unresolved.

Specify the complete perturbative normalization, coupling and $\hslash$ orders, relevant time-ordered and retarded products, and companion sectors needed for an interacting identity. Then formulate and prove the appropriate gauge or BRST/BV statement.

The admitted classical insertion and one reduced graph do not supply that construction. Status: not established.

No selected counterterm table, scale derivative, retarded interacting coefficient, or physical observable is asserted. No repair of the frozen XI-G obstruction is claimed.

The manifold, preparation width, graph prescription, component conventions, and extension domains have been fixed explicitly. The manuscript is a proof of the stated reduced construction and its extension family, not a proof of the unresolved interacting programme.

No successful compiler execution, full-series search, or machine verification is represented.

release statement.{ The additional supplied XII-D premise admits the matching classical prepared kernel; the old frozen class remains excluded on its own hypotheses.

The chosen ambient propagator defines the reduced two-edge sector off coincidence. Its source-decorated relative scaling degree is bounded by eight. Point extensions exist with unfixed local coefficients.

Under the root $Z_2$ bound, compressed attachment and smearing are defined for each extension choice. The result is a family, not a selected normalized quantum coefficient.

Normalization, the retarded interacting construction, and BRST/BV observability remain Unresolved. }

XVIH

Landmark XVI-H. Supplied revised XVI-H.tex, “Compact polarized kernel and witness recovery.” Locators: thm:Opol, thm:trep, prop:estimate, prop:tmatch, and thm:main. Used for the compact classical representation, continuity, and current witness matching. Its unproved full-kernel wavefront gate is not retroactively reported as proved there.

XVIE

Landmark XVI-E. Supplied revised XVI-E.tex, “Polarized root representation and witness-level completion compatibility.” Locators: lem:TD, lem:TC, and thm:compatibility. Used for matching the polarization convention and current witness compatibility.

XVIF

Landmark XVI-F. Supplied revised XVI-F.tex, “Auxiliary/root compatibility and direct witness completion.” Locators: prop:compat, thm:compatibility, and cor:family-pointwise. Used for the current auxiliary/root and direct-completion record, not a superseded residual stop.

XVIG

Landmark XVI-G. Supplied current XVI-G.tex, “Frequency-dependent witness testing and resonant asymptotics.” Locators: prop:import, prop:kernel-slot, and prop:resonant. Its scalar density is not substituted for the graph kernel.

XIID

Manuscript XII-D, version 1. Supplied identifier e3e3f67d. Additional supplied excerpts: def:completed, lines 556–569; thm:root, lines 628–674; thm:admission and eq:t-bounds, lines 805–841. The root $Z_2$ statement is reproduced in the supplied original XVI-I as lem:root-kernel. These interfaces are used as explicit inherited premises; the complete source is not independently audited.

XIIC

Manuscript XII-C, version 1. Supplied identifier e0611fd7. Density formulas reproduced in the supplied original XVI-I: eq:ell3, eq:finite-curvature, and eq:J3. Additional source excerpt: thm:actualsource, lines 636–678. The declared unintegrated density representative is retained.

XIG

Manuscript XI-G, version 1. Supplied identifier 3cae2be9. Supplied excerpts: def:class, lines 494–507; thm:criterion, lines 954–987; thm:obstruction, lines 1107–1134. The compact-kernel transport premise is reproduced in the supplied original XVI-I. The printed frozen-kernel criterion is not silently promoted to a universal theorem.

XIB

Manuscript XI-B, version 1. Supplied identifier 862627cd. Interfaces reproduced in the supplied XVI documents: the quadratic metric operator, lem:Green, thm:graviton, and the free differential eq:s. Used for the normalization and free-variation convention.

XIE

Manuscript XI-E. Two-point formulas reproduced in the supplied original XVI-I: eq:W, eq:h, eq:Heven, and eq:antisym. The massless $W_0$ and $h_0=-W_0$ conventions are retained. No complete XI-E source verification is asserted.

BF

R. Brunetti and K. Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Communications in Mathematical Physics 208 (2000), 623–661; arXiv:math-ph/9903028. Section 5.1 and Theorems 5.2–5.3 are used for point extensions of finite scaling degree. The full interacting construction is not imported.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428v2. Equation (82) fixes the adopted identical-edge factorial convention. Sections 6–7 provide graph and extension background; no complete time-ordered-product family is transferred to the present nonlocal insertion.

BDHintro

C. Brouder, N. V. Dang, and F. Hélein, A smooth introduction to the wavefront set, Journal of Physics A: Mathematical and Theoretical 47 (2014), 443001; arXiv:1404.1778. Theorem 13 and Proposition 26 supply the product criterion and the Feynman wavefront description. The Fourier-sign conversion is stated explicitly in Lemma lem:WF.

BDHops

C. Brouder, N. V. Dang, and F. Hélein, Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker), arXiv:1409.7662v2. Sections 5–6 give the pullback, product, and proper-pushforward interfaces used in the compressed root-attachment test. No stronger unrestricted continuity theorem is asserted.