Landmark XVI-H Compact polarized kernel and witness recovery Revised proof manuscript Kernel representation and witness matching established; free-shell admission and quantum programme unresolved

We represent the real polarization of the stated single-metric prepared completion by a compact distributional kernel with two symmetric-tensor slots and one scalar slot. The construction is relative to explicit inherited source and compact-root inputs. It is implemented by transferring derivatives from diagonal source deltas onto general test functions and then pairing with the root kernel. This gives a distribution without invoking an unverified product of arbitrary distributions.

The compact kernel yields joint continuity on the compact-set smooth topology, with finite but unspecified derivative orders. We identify the polarization conventions with those of revised XVI-E and XVI-F and import the current XVI-G witness theorem. The resulting complexified kernel, partially evaluated on the specified tensor witness pair, is exactly the compact smooth scalar density $j_{\lambda }$. This closes the former witness compatibility gap without identifying the scalar density with the full kernel or determining a free-shell class from one witness family.

We also give a concise conditional audit of the inherited past-to-future switch representative. No wavefront set of the full kernel or its trisolution is calculated. No cubic admission, BRST/BV observability, or quantum-product construction is established. The regulator width is fixed throughout.

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

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

and release status.{ Established under the stated inherited inputs: the real polarized completion, its compact three-slot kernel, finite-order continuity, the witness compatibility chain, and partial contraction of that kernel to $j_{\lambda }$.

Not established: the full-kernel wavefront/admission condition, comparison with the frozen XI-G free-shell class, BRST/BV observability, or a prepared quantum product.

The sharp-width limit, ultraviolet completion, and all-orders finite-parameter renormalizability are outside the scope of this note, not open problems assigned to it. The admission and quantum programme remains Unresolved. }

This manuscript revises the supplied XVI-H-v3.tex. The current witness references are:

• revised XVI-E.tex, “Polarized root representation and witness-level completion compatibility”;

• revised XVI-F.tex, “Auxiliary/root compatibility and direct witness completion”;

• current XVI-G.tex, “Frequency-dependent witness testing and resonant asymptotics.”

The supplied XVI-H-v3 reproduces additional XII-D source-kernel and root-kernel statements, and XI-G switch identities. The complete XII-D and XI-G manuscripts are not independently available here. Accordingly, those statements are formulated below as explicit inherited inputs. Their use is not represented as a full-file verification.

The current XVI-G cites an earlier XVI-H only as a record of subsequent obligations, not as a premise of its witness proofs. No result of the present kernel construction is used to prove the imported witness chain. Thus witness matching below does not depend circularly on this revision.

Let

with

Fix

Let $E={\mathrm{Sym}}^2{T}^{\ast }M$. The tensor arguments are smooth perturbations in $C^{\infty }(M;E)$, not necessarily decaying at spatial infinity. The scalar is real and smooth unless an algebraic complexification of a test pairing is explicitly used.

Choose a fixed real smooth switch

Its coefficients are ambient smooth and flat at the plateau endpoints. No discontinuous extension at $t=0$ is introduced.

Use the fixed volume density ${\mathrm{d}}^{4}x$ and global Minkowski frames to identify tensor-valued distributional densities with their finitely many component distributions. All pairings are bilinear, without complex conjugation.

Trace reversal is

The selected free tensor operator is

The inverse coefficient remains $-1$, not $-4$. The action-equation convention $r_{\mathrm{D}}=-u_{\mathrm{D}}$ does not rename the response or kernel used here XVIA,XVID.

The frequency is $\lambda$. Repeated spacetime indices are denoted by $\alpha ,\mu ,\nu ,\rho ,\ell$ rather than by $\lambda$.

The designated source is

where

The coefficient formulas are those displayed in Section sec:degree.

The bulk term $J_3$ is a finite local homogeneous quadratic differential polynomial in the metric jet, with coefficients independent of the perturbation. The total number of derivatives hitting its field factors is at most two. This is an imported premise, reported in XII-D as the quadratic coefficient

Its individual curvature monomials are not supplied or expanded in this note.

The full source has total field-derivative order at most two and vanishes for $t\le 0$ as a polynomial in the field jet.

The coefficient description motivates the imported bulk premise. It is not substituted for a curvature expansion proved here. In particular, the degree audit below proves the assertion for the displayed switch correction and uses the bulk assertion as an input XVIB,XVIHvThree,XIID.

The stated diagonal completion is

It is not defined by independently choosing a source pairing.

Its displayed coefficient maps are

and its two retarded legs are

The divergence retains derivatives of $g$, and the mass term is retained. No factor $g$ is inserted into the coordinate forcing.

For every real smooth perturbation $\gamma$, the inherited response $u_{\mathrm{D}}[\gamma ]$ is a smooth zero-initial-data solution of

Smooth zero-data uniqueness holds without a spatial-decay requirement, as proved in revised XVI-D.

The diagonal completion (eq:CD) is defined on these real smooth inputs. The common retarded domain used in this definition is linear for the finite response combinations occurring in polarization. On that domain, the operators in (eq:Aone) are linear, and ${\mathcal{A}}_1$ is linear in $r$ for fixed $\varphi$ and linear in $\varphi$ for fixed $r$.

This input is the direct-completion interface. A compact-source inversion theorem alone would not establish its entire domain. Conversely, the specific witness calculation in XVI-G does not retract this separately stated diagonal interface XVID,XVIErev,XVIFrev.

Choose an admissible fixed collar

Write

and

There is a real linear compact smooth root $a_{f,g}[\varphi ]$, with the inherited collar and normalization, such that

for the real smooth inputs considered here.

In a fixed source-component frame, indexed by $A$, the root has a real distributional kernel $B_A(y,x)$ satisfying

The joint kernel is compactly supported, with

The root-theorem domain includes the real cosine and sine witness inputs and their required sums, with the same $f,g,\sigma ,m,\vartheta$ used in revised XVI-E/F and current XVI-G.

The compactness in (eq:B-support) is a joint root-kernel input. It is not inferred merely from compact support of the root's output for each scalar. The supplied XVI-H-v3 attributes these statements to XII-D, lem:root-kernel, eq:B-kernel, thm:root, and eq:t-bounds XVIHvThree,XIID.

All the kernel theorems below are relative to Inputs input:source–input:root. They do not purport to re-prove those predecessor interfaces.

To avoid confusing expansion coefficients with response fields or completion coefficients, write the quoted source coefficients temporarily as ${\mathsf{u}}_j$ and ${\mathsf{b}}_j$:

The remaining quoted coefficients are

These are the formulas reproduced in XVI-H-v3. Their homogeneity, rather than a new source derivation, is audited here.

Every nonzero displayed monomial of ${\mathcal{D}}_{\sigma }$ has exactly two metric-jet factors. There is no constant or linear term. Consequently, for every real $s$,

Using the separately imported bulk premise gives

The real source polarization

is symmetric and bilinear.

Proof. A derivative of a metric factor still has field degree one. The coefficients ${\mathsf{u}}_j$ and ${\mathsf{b}}_j$ have field degree $j$. Each summand of $C_r$ has degree $r$, and ${\Gamma }_n$ has degree $n$. Thus $C_2$, $C_0{\Gamma }_2$, and $C_1{\Gamma }_1$ all have degree two. Multiplication by fixed derivatives of $\sigma$ does not change field degree. This proves (eq:Dscale).

The first statement in (eq:Jscale) is the imported bulk premise, not a curvature expansion performed here. Adding the two source contributions proves the second statement.

More than homogeneity is used for bilinearity. Each component of the source is a finite sum of products $L(\gamma )M(\gamma )$ of linear differential monomials. For one such product,

This is symmetric and bilinear. Summing proves the assertion. Degree-two homogeneity of an arbitrary nonlinear map would not justify this step.

The field degree and the total field-derivative order are different counts. The latter is at most two by Input input:source; it does not mean that a linear field term is present.

Under Input input:completion,

The response polarization

is symmetric and real bilinear.

Proof. The two fields $u_{\mathrm{D}}[s\gamma ]$ and $s^2{u}_{\mathrm{D}}[\gamma ]$ satisfy the same linear equation and zero initial data by (eq:Jscale). Smooth zero-data uniqueness proves their equality. Linearity of ${\mathcal{A}}_1$ in the response proves the second identity in (eq:Cscale).

The field $B_{\mathrm{resp}}(a,c)$ solves

with zero initial data. Each bilinearity defect solves the homogeneous zero-data equation, since the right-hand side is bilinear. Uniqueness makes every defect zero.

For real smooth $a,c,\varphi$, define

This is a polarization of three diagonal completion values, not a definition beginning with an arbitrary polarized response slot.

Under Inputs input:source–input:root:

1. ${\mathcal{O}}_{\mathrm{pol}}$ is symmetric in its tensor arguments.

2. It is real linear in the scalar.

3. The source, response, and diagonal completion have the degree-two homogeneity proved in Propositions prop:monomials and prop:Chom.

4. ${\mathcal{O}}_{\mathrm{pol}}$ is real bilinear in its tensor arguments.

5. Its unique complex-bilinear extension in the tensor arguments, with the scalar kept real, is

   $$\begin{array}{rl}{\mathcal{O}}_{\mathbb{C}}(a+\mathrm{i}b,c+\mathrm{i}d;\varphi )=& {\mathcal{O}}_{\mathrm{pol}}(a,c;\varphi )-{\mathcal{O}}_{\mathrm{pol}}(b,d;\varphi )\\ & +\mathrm{i}{\mathcal{O}}_{\mathrm{pol}}(a,d;\varphi )+\mathrm{i}{\mathcal{O}}_{\mathrm{pol}}(b,c;\varphi ).\end{array}$$

6. With the same data and domain conventions as revised XVI-E/F and current XVI-G, the specified witness pair satisfies, for each $\lambda >0$ and real smooth scalar evaluated there,

   $$\begin{array}{rl}{\mathcal{O}}_{\mathbb{C}}({{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda };\varphi )& =T^{\mathbb{C}}({{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda };\varphi )\\ & =Q_{\lambda }[\varphi ]=I_{\lambda }[\varphi ]\\ & ={\mathcal{A}}_1[f,g](R_{\lambda };\varphi ).\end{array}$$

7. On the common real response domain,

   $${\mathcal{O}}_{\mathrm{pol}}(a,c;\varphi )={\mathcal{A}}_1[f,g](B_{\mathrm{resp}}(a,c);\varphi ).$$

   The kernel construction in Section sec:kernel does not require this response-slot formula.

Proof. Symmetry follows from the definition. Scalar linearity follows from the scalar linearity of the three diagonal values. Homogeneity was proved above.

For bilinearity, Proposition prop:Chom gives the bilinear response polarization. Linearity of ${\mathcal{A}}_1$ on its stated common domain allows the finite response combination in (eq:Opol) to be taken inside ${\mathcal{A}}_1$, proving (eq:O-A) and real bilinearity. Alternatively, polarizing the root identity gives

which proves the same bilinearity from the root interface alone.

Real bilinearity verifies (eq:OC); complex bilinearity forces that formula in any extension, proving uniqueness.

For item 6, the definition of ${\mathcal{O}}_{\mathrm{pol}}$ is exactly the definition of $T_D$ in revised XVI-E, eq:TD, and revised XVI-F, eq:TD, on their common real domain. The polarization factor, diagonal insertion, source convention, and observation data are the same. Their four-term complex extension is exactly (eq:OC). Therefore ${\mathcal{O}}_{\mathbb{C}}=T^{\mathbb{C}}$ on that complexified domain.

The remaining equalities are the witness compatibility theorem of revised XVI-F, thm:compatibility, consolidated in revised XVI-E, thm:compatibility, and stated with the pointwise scalar-family quantifier in current XVI-G, prop:import XVIErev,XVIFrev,XVIG. Their operator and root-domain hypotheses are retained in the inputs of this note.

Fix $|\mathbf{n}|=1$ and use

The source and response are those of revised XVI-B and XVI-D:

Here

and

For reference, define the independently checked witness slot

Equation (eq:witness-chain) identifies this scalar functional with the witness restriction of ${\mathcal{O}}_{\mathbb{C}}$.

The equality holds for arbitrary assignments $\lambda \mapsto {\varphi }_{\lambda }$ of real smooth scalars, at each frozen frequency. No regularity in $\lambda$ is needed, and no decay is implied by the equality alone.

There is no need for another comparison of coordinate prescriptions to prove this witness equality: the definitions match, and the supplied compatibility theorem supplies the operator comparison on the witness slot. This closes the former open item in XVI-H-v3.

The result does not:

• insert a pair as one metric argument of ${\mathcal{C}}_{3,\mathrm{D}}$;

• construct a metric $\gamma$ with $u_{\mathrm{D}}[\gamma ]=R_{\lambda }$;

• establish a general extension of a retarded operator to arbitrary distributions.

The diagonal insertion and its polarization remain differently defined objects with different input structures.

Choose a global component frame for the input tensor bundle, with indices $I,J$. The local polynomial structure of the source yields a finite normal form for its polarized kernel:

The coefficient tensors are smooth and ambient. The source identity is

This is the source-kernel structure attributed to XII-D in the supplied XVI-H-v3. It also follows abstractly from the finite local quadratic-polynomial premise: a monomial $q(y)({\partial }^{\alpha }{a}_I)({\partial }^{\beta }{c}_J)$ is represented by the corresponding differentiated deltas with coefficient $(-1{)}^{|\alpha |+|\beta |}q(y)$.

The normal form may be symmetrized without changing (eq:j-id), so we choose

Its coefficient functions can be chosen to vanish for $y^0\le 0$, since every source contribution contains $\sigma$ or a positive-order derivative of $\sigma$. No coefficient values of the unexpanded $J_3$ are fabricated.

Let $F_{IJ}(y_1,y_2,x)$ be an arbitrary compactly supported smooth test field of the appropriate component type on $M^3$. Define

The right-hand side pairs a distribution with a smooth function. It is well defined because $B$ is compactly supported. Any auxiliary cutoff equal to one near $\mathrm{supp}B$ gives the same value.

For clarity, the corresponding distribution before integrating out $y$ can be defined on a general test $G(y,y_1,y_2,x)$ on $M^4$:

Thus derivatives on the source deltas are transferred to the test before evaluation on the diagonal. They are not silently applied to a singular root kernel.

With

the distribution $\mathcal{K}$ is a finite sum of derivatives in $y_1,y_2$ of the pushforwards ${\iota }_{\ast }(cB)$. Its support is compact, so ${\pi }_y$ is proper on that support. The notation

is understood through (eq:lifted-test). It is this particular diagonal construction, not a claim that arbitrary pairs of distributions may be multiplied.

Under Inputs input:source–input:root, (eq:t-test) defines a real compactly supported distributional kernel on ambient $M^3$, symmetric under interchange of its tensor slots. For real smooth $a,c,\varphi$,

The pairing is meaningful without compact support of the configuration fields.

The representation uses the root identity and does not require the response-slot formula (eq:O-A).

Proof. A derivative of a smooth test followed by evaluation at $(y_1,y_2)=(y,y)$ is smooth. For tests supported in a fixed compact set, each derivative seminorm of this evaluated function is bounded by a finite derivative seminorm of the original test, by the chain rule. Smooth multiplication by the fixed coefficients preserves these estimates. Continuity of $B$ therefore proves that (eq:t-test) is a distribution.

The definition can only detect test germs on the image of $\mathrm{supp}(cB)$ under $(y,x)\mapsto (y,y,x)$. That image is compact by (eq:B-support). Thus $t_{\mathrm{D}}$ is compactly supported. Formula (eq:c-sym) and exchange of $\alpha ,\beta$ in the finite sum prove tensor-slot symmetry.

For a decomposable test,

Consequently (eq:j-id) and (eq:B-root) give

The last expression is (eq:O-root).

Compactness supplies a cutoff-independent pairing with the smooth, possibly noncompact fields. This justifies the same calculation for all inputs in the stated real smooth domain.

Let

The coefficient support and (eq:B-support) imply

Both sets on the right are compact in ambient $M$. Indeed, if $K_f$ lies in $\{t\le b,|\mathbf{x}|\le R\}$, then

The causal past of a compact set is closed in Minkowski space, so $C_{\delta }$, and hence $C_0$, is compact. The set $O_{f,g}$ is compact by its definition. The right side of (eq:t-support) is the image of $C_0\times O_{f,g}$ under $(y,x)\mapsto (y,y,x)$.

The support may reach $t=0$. Ambient compactness does not require support bounded away from that surface. The following statements remain separate:

• $t_{\mathrm{D}}$ is defined on ambient $M^3$;

• its support is compact there;

• its regularity or singularity at $t=0$ has not been determined.

Flatness of the switch guarantees smooth coefficient functions; it is not a theorem removing boundary singularities of the resulting kernel.

The real smooth tensor and scalar spaces carry their compact-set derivative Fréchet topologies. For a compact set $K\subset M$, define componentwise

The test space ${C_c}^{\infty }$ carries its usual LF topology, not the subspace topology inherited from $C^{\infty }$.

There are compact sets $K_1,K_2,K_3$, finite orders $r_1,r_2,r_3$, and a constant $C$ such that

Thus ${\mathcal{O}}_{\mathrm{pol}}$ is jointly continuous on the chosen smooth configuration spaces.

The orders are finite but unspecified. One may take $r_1=r_2=r_3=r$ for a finite distributional order bound $r$ of $t_{\mathrm{D}}$. If $N$ is a finite order bound for $B$ on a suitable compact neighborhood, the construction provides an order bound at most $N+2$ for $t_{\mathrm{D}}$. No numerical or optimal value of $N$ or $r$ is asserted.

The constant may depend on the fixed observation data, switch, mass, collar, coefficient functions, and root kernel. No estimate uniform as $\epsilon \downarrow 0$ is claimed.

Proof. A compactly supported distribution has finite order. Choose compact neighborhoods $K_1,K_2$ of $C_0$ and $K_3$ of $O_{f,g}$. There are $r,C$ such that

for every smooth $F$, using the cutoff-independent extension of the pairing.

For $F=a\otimes c\otimes \varphi$, differentiation in the three independent variables gives

Together with (eq:O-t), this proves (eq:target).

Alternatively, an order-$N$ bound for $B$ applied to (eq:t-test) differentiates the evaluated test at most $N$ additional times. The source deltas have at most two derivatives. The chain and product rules therefore require at most $N+2$ derivatives of $F$; derivatives of the fixed coefficients are absorbed into the constant.

The product seminorm estimate proves joint continuity of the trilinear map.

The source derivative order two is not the order of the entire kernel. The root distribution contributes as well. This continuity result is also distinct from the sharper fixed-witness scalar estimate of XVI-G. Neither estimate alone proves a microcausal cone exclusion.

Let ${t_{\mathrm{D}}}^{\mathbb{C}}$ denote the complex-linear extension of the real distribution $t_{\mathrm{D}}$ to complex test fields. Its evaluation on complex tensor arguments and a real scalar implements the four-term extension (eq:OC).

Current XVI-G, prop:kernel-slot, supplies

where

Here

and

Its extension to ambient space is smooth because the support is compact-interior.

For each $\lambda >0$, partial evaluation of the compact kernel on the two witness tensors defines a compact scalar distribution $q_{\lambda }$ by

It satisfies

with the fixed volume convention. In particular,

Proof. The witness tensors are smooth, although not compactly supported. Compactness of $t_{\mathrm{D}}$ makes (eq:q-partial) meaningful.

More explicitly, choose smooth compact cutoffs ${\rho }_1,{\rho }_2$ equal to one near the first and second support projections of $t_{\mathrm{D}}$. Then

This is independent of those cutoffs and continuous in $\varphi$. Equivalently, multiply ${t_{\mathrm{D}}}^{\mathbb{C}}$ by the smooth witness coefficients, contract its tensor indices, and push forward to the scalar slot. The pushforward is proper because the kernel is compact.

By Theorem thm:trep and algebraic complexification, the left side of (eq:t-to-j) equals ${\mathcal{O}}_{\mathbb{C}}({{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda };\varphi )$. Theorem thm:Opol(6) identifies this with $I_{\lambda }[\varphi ]$. Equation (eq:j-pair) proves equality for every real smooth scalar.

For a complex compact test, apply that equality separately to its real and imaginary parts. Complex linearity proves (eq:q-equals-j) as a distributional equality.

The cutoffs in this proof only implement pairing with a compact distribution. They do not replace the witness by a different metric configuration.

limitation of the matching. Equation (eq:q-equals-j) identifies a partial evaluation of $t_{\mathrm{D}}$, not the full kernel with $j_{\lambda }$. It is not a reconstruction or uniqueness theorem for a three-slot kernel from one witness family. It is not an all-free-triples comparison.

The full kernel representation is established by Theorem thm:trep; the witness identities are used only to identify its specified partial contraction.

For a fixed compact set $L\subset {M_T}^{\circ }$ containing $\mathrm{supp}f\cup \mathrm{supp}g$, current XVI-G proves

The constant is independent of $\lambda ,\varphi$ at fixed observation and background data.

By (eq:t-to-j), this estimate applies to the same witness contraction of ${t_{\mathrm{D}}}^{\mathbb{C}}$. It is imported from XVI-G, not derived from the unspecified finite-order estimate (eq:target).

The established rapid-decay cases are:

1. one real smooth scalar independent of $\lambda$;

2. sets bounded in every $p_{r,L}$, with decay uniform on the stated set;

3. families satisfying $p_{r,L}({\varphi }_{\lambda })\le C_r{\lambda }^{\rho r+a}$ with $0\le \rho <1$.

They are current XVI-G, prop:fixeddecay, prop:bounded, and prop:slow XVIG.

For real fixed $b\in {C_c}^{\infty }({M_T}^{\circ })$, set

The resonant theorem of XVI-G gives

The order-$\lambda$ contribution is scalar-leg only; the coordinate contribution is bounded. No integrals or remainder estimates are recomputed here.

When ${\rm Y}[b]\ne 0$, these are nonzero leading linear terms. When it vanishes, only $O(1)$ is concluded. The number ${\rm Y}[b]$ is not a wavefront covector and does not decide admission.

The compact kernel permits a free-shell class to be defined. This does not determine its admission or its relation to the frozen XI-G class.

Let $\mathcal{K}$ be the matching space of compact three-slot kernels with the stated tensor symmetry. Define

Since $\mathcal{P}$ is invertible and constant, $-\square \mathcal{P}a=0$ is equivalent to $\square a=0$. Compactness makes these pairings with smooth solutions well defined.

The class of the constructed kernel is

Comparison with a matching frozen kernel $u_0$ requires determining whether

If that condition holds, equality of the affine classes follows by linearity; the applicable frozen obstruction then transfers. A different-class proof would remove only that transfer, not establish admission XVIA,XVIB.

The inherited target ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ uses compact graded-symmetric polynomial kernels satisfying

Where the XI-G separation theorem applies with its complete hypotheses, it tests the trisolution

through

The propagators here are ambient causal propagators, not the slab retarded inverses by themselves XIG.

Under the inherited compact-distribution propagator interface, ${\mathsf{W}}_{t_{\mathrm{D}}}$ is an abstractly defined distributional trisolution. No explicit expression, wavefront estimate, or exclusion test for it is supplied in this note. Neither $\mathrm{WF}(t_{\mathrm{D}})$ nor $\mathrm{WF}({\mathsf{W}}_{t_{\mathrm{D}}})$ is calculated.

boundary.{ The compact kernel and its witness matching are established under the stated root/kernel assumptions. The existence gate is no longer left open for this polarized completion.

The full-kernel wavefront test and the comparison with the frozen free-shell class remain Unresolved. The XI-G admission criterion is not applied. No cubic admission or obstruction is asserted. }

The supplied XII-E handoff reports kernel and admission results for its stated insertion and admissibility sheet. That report is not rescinded. Its complete theorem and hypotheses are not reproduced here as an admission premise. The XII-D kernel representation used in this note is not, by itself, a verification of every hypothesis on the XII-E admissibility sheet XIIE.

The scalar partial evaluation is different: $j_{\lambda }$ is compact smooth, so $\mathrm{WF}(j_{\lambda })=\varnothing$. It satisfies the degree-one scalar kernel exclusion. That fact does not imply the cubic exclusion (eq:mc) for $t_{\mathrm{D}}$.

This section concerns the representative formula reproduced from XI-G in XVI-H-v3. It is not needed for the kernel construction, continuity, or witness matching.

The following are inherited operator inputs:

1. The ambient causal propagator is

   $${\mathsf{E}}_{\mu }={{\mathsf{E}}_{\mu }}^{\mathrm{R}}-{{\mathsf{E}}_{\mu }}^{\mathrm{A}}.$$

   Both Green inverses invert $P_{\mu }=\square +{\mu }^2$ on the appropriate compact distributional inputs. Hence ${\mathsf{E}}_{\mu }{P}_{\mu }r=0$ for $r\in {\mathcal{E}}^{\prime }(M)$.

2. The distributional slot convention is

   $$⟨{\mathsf{E}}_{\mu ,i}w,\phi ⟩=-⟨w,{\mathsf{E}}_{\mu ,i}\phi ⟩.$$

   This is the transpose convention for the causal propagator, not an additional sign in ${\mathsf{E}}_{\mu }{P}_{\mu }r=0$.

3. The temporal functions ${\zeta }_i$ are past-to-future switches:

   $${\zeta }_i(t)=0(t\le -T_0),{\zeta }_i(t)=1(t\ge T_0).$$

   They are not compactly supported.

4. The inherited slot calculus supplies the Green identities, causal support, and commuting distinct-slot operations used below for compact kernels.

These are the reproduced interfaces attributed to XI-G, lem:spcompact and lem:kernel. The complete XI-G source is not independently checked here XVIHvThree,XIG.

Let $u$ be a compact kernel in that inherited slot calculus, and put

Choose the switches with transition times covering the relevant compact support projections. Then

is compactly supported and satisfies

The intermediate switched trisolution need not be compact. These conclusions alone are not an admission result.

Proof. For a compact input $v$ in one slot, the inherited Green identities give

The causal support bounds and the opposite temporal truncations make $r_i$ compact: the retarded contribution is truncated above, and the advanced contribution below, while the source is compact.

Therefore

is compact, and ${\mathsf{E}}_i{T}_{i}v={\mathsf{E}}_{i}v$ because ${\mathsf{E}}_i{P}_i{r}_i=0$. There is no additional minus sign in this equality.

Apply the three operations successively to compact inputs. The inherited distinct-slot commutations identify their composition with (eq:tilde). Equivalently, telescoping the three differences gives

for compact corrections at the successive stages. Applying all three causal propagators annihilates each term, proving (eq:W-preserved).

Compactness follows already from the successive compact representatives. It can also be read from

using the trisolution equations. Each commutator is supported in its slot's transition region, and the inherited causal support controls the spatial projections there.

The switch convention is essential. If the entire intermediate product were compact and the formula were read merely as $P_1{P}_2{P}_3$ applied to a compact distribution, its trisolution would vanish by the annihilation identity. Past-to-future switches avoid that mistaken compact-intermediate reading.

The representative can participate in an admission argument only after the needed wavefront hypotheses are separately established. Preservation of the trisolution does not prove those hypotheses. No such argument is performed for $t_{\mathrm{D}}$ here.

With $\omega =2\lambda$,