Landmark IX-A An off-shell admission domain and boundary-sensitive topology for sharp Cauchy relational insertions

We construct an off-shell polynomial admission space for the exact sharp zero-Cauchy harmonic insertion $B_f=-\int f({\mathsf{E}}_{0}Ch{)}^{\mu }{\partial }_{\mu }\varphi$. The construction retains compact distributional kernels and uses actual functionals, rather than independent symbols with unspecified relations. A boundary-trace map on linear metric derivatives is well defined modulo microcausal linear kernels. Requiring this map for every directional derivative gives a domain containing the bulk polynomials, the sharp insertions, their linear coordinate descendants, and arbitrary finite pointwise words. The domain is stable under smooth directional differentiation and complex conjugation.

An explicit Hausdorff locally convex topology combines strong compact-distribution convergence with all these boundary traces. The sharp insertion varies continuously with its observation test, whereas its unchanged smooth-kernel approximants do not converge to it. This failure is proved for every nonzero real observation test by a nonvanishing advanced Cauchy trace. Consequently, the domain does not reproduce the normal-topology admission hypothesis of the mixed-Wick obstruction.

Classical pointwise multiplication is separately continuous. No joint continuity or hypocontinuity is inferred, and no renormalized quantum product is constructed. We prove faithful bulk inclusion, exact off-shell classical normalization, and classical presentation independence. We also identify selected already-defined Wick coefficients that belong to the admission space, without claiming quantum closure. The result closes IX-A only: graph extensions, quantum associativity, quantum presentation independence, interactions, and the switched-gauge gate remain separate tasks.

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

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

claim: IX-A only This manuscript constructs a classical admission domain and a boundary-sensitive topology for a subsequent relative quantum deformation problem. It does not construct that deformation. The notation “admission” does not mean that all Wick contractions are defined.

A compact distributional kernel need not belong to a chosen quantum product domain. Admitting the kernel to a larger vector space does not prove that its contractions close there. Finally, choosing a finite part for a numerical coefficient does not prove that the finite parts arise from one associative product.

This manuscript addresses the first of these steps and its topology. It is the IX-A module of a larger coherent-product problem.

The sharp insertion is kept fixed:

No field equation is imposed on $h$ or $\varphi$. The observation test $f$ is smooth, compactly supported, and strictly to the future of the initial surface.

The required change from normal-topology admission is explicit. The new topology records the Cauchy trace responsible for the sharp surface term in metric derivatives. Regular kernels have zero such trace. The sharp insertion need not. Therefore the exact sharp insertion is not declared to be the continuous limit of its unchanged regular approximants.

The following data are retained:

1. the off-shell functional (eq:B);

2. the zero-Cauchy prescription defining $L$;

3. the ordinary bulk polynomial kernels;

4. the fixed bulk Wick product wherever it is already defined;

5. exact equality of classical functionals, including boundary terms arising from integration by parts.

The following assertions are not retained or assumed:

1. normal-topology continuity of sharp admission;

2. convergence of unchanged regular approximants in the new admission topology;

3. a quantum multiplication on the enlarged domain;

4. quantum closure under contractions or retarded products.

The fundamental operations on prescribed-wavefront spaces require their own topological theorems. In particular, continuity and hypocontinuity are distinct claims BDH. We do not replace such a theorem by a choice of notation.

1: algebra-admission gate The present result supplies an off-shell carrier and topology. It does not close the associative quantum-product problem. A later construction must define complete quantum products, prove their identities, and preserve the fixed bulk operations.

2: switched-gauge gate A gauge-compatible interacting realization requires its own intertwining theorem. Classical plateau covariance does not imply a quantum Ward identity, and the existence of a free admission space does not repair a first-order compact-switch defect.

The distinction between a free product, time-ordered products, and the interacting construction is standard in perturbative algebraic quantum field theory FR. No interacting map is used below.

For the fixed data of Section sec:fixed, there is an explicitly defined space ${\mathcal{P}}_{\Sigma }$ of actual compact-distribution polynomial functionals with a Hausdorff locally convex topology ${\tau }_{\Sigma }$ such that:

1. ${\mathcal{P}}_{\mathrm{mc}}\subset {\mathcal{P}}_{\Sigma }$ injectively, and the inclusion is continuous on every fixed-support, fixed-wavefront bulk stage carrying its normal topology;

2. every $B_f$ and every linear coordinate functional $\mathsf{J}(a)=\int a_{\mu }(Lh{)}^{\mu }$ belongs to ${\mathcal{P}}_{\Sigma }$;

3. ${\mathcal{P}}_{\Sigma }$ is closed under arbitrary finite pointwise products, smooth directional derivatives, and complex conjugation;

4. fixed-direction differentiation and complex conjugation are continuous, and pointwise multiplication is separately continuous;

5. $f\mapsto B_f$ and $a\mapsto \mathsf{J}(a)$ are continuous linear maps from their test-function spaces;

6. if $f\ne 0$ is real and $B_{f,\epsilon }$ is obtained by smoothing both original field arguments with a compact approximate identity, then

   $$B_{f,\epsilon }/⟶B_{f}in{\tau }_{\Sigma };$$

7. the underlying classical functional is independent of its admissible kernel presentation.

No quantum multiplication on ${\mathcal{P}}_{\Sigma }$ is included in the statement.

The proof occupies Sections sec:adjoint–sec:compatibility. Selected old contraction outputs are treated separately in Section sec:partial.

Let

and

All spaces are complexified when needed. Classical real fields are recovered by complex conjugation.

For a smooth symmetric covariant tensor $h$, set

The zero-Cauchy operator is

At zero spatial frequency the sine quotient has its continuous value $t-u$. Thus

For smooth sources without decay, this formula is interpreted locally by finite propagation and compatible compact cutoffs. All observations in this paper concern compact truncated causal regions. Smooth Cauchy theory and finite propagation justify this use BGP.

For $f\in {C_c}^{\infty }(M_{+})$, put

This is compact. The functional $B_f$ depends on $h$ only in ${\mathcal{D}}_f$ and on $\varphi$ only in $\mathrm{supp}f$.

Let

A configuration is denoted by $\Psi =(h,\varphi )$.

Let ${\mathcal{P}}_{\mathrm{dist}}$ be the space of finite polynomial functionals

with symmetric kernels. Here $u_0\in \mathbb{C}$. Tensor indices are paired in the indicated bundles.

The compact distribution $u_n$ acts on smooth functions by inserting a compact cutoff equal to one near its support. This does not depend on the cutoff.

Equip the coefficient carrier with

using the locally convex direct sum and the strong dual topology of $C^{\infty }$ in each degree.

The symmetric kernels in (eq:polynomial) are uniquely determined by the functional on all smooth configurations. Consequently ${\mathcal{P}}_{\mathrm{dist}}$ may be identified with its coefficient subspace in ${\mathcal{T}}_{\mathrm{coeff}}$.

Proof. Scaling a configuration separates homogeneous degrees. Polarization of a homogeneous polynomial recovers its symmetric multilinear form. If the polynomial is zero, that multilinear form vanishes on every tuple of smooth configurations.

In particular it vanishes on compactly supported elementary tensor tests in each bundle component. Finite sums of such tensor tests are dense in the smooth test space on each compact product region. Continuity of the distribution therefore makes the corresponding kernel zero.

Let ${\mathcal{P}}_{\mathrm{mc}}\subset {\mathcal{P}}_{\mathrm{dist}}$ be the finite polynomial domain whose kernels satisfy

where the closed nonspacelike cotangent cones include zero in each slot and the total zero covector is omitted.

Let ${\mathcal{P}}_{\mathrm{reg}}$ be the subspace with smooth compact kernels.

Only finite polynomials are used here. No closure theorem for arbitrary smooth microcausal functionals is assumed. The distinction between fixed-kernel polynomial classes and general functional classes is relevant to the broader pAQFT domain problem HRV.

Every smooth directional derivative of a member of ${\mathcal{P}}_{\mathrm{mc}}$ is again in ${\mathcal{P}}_{\mathrm{mc}}$. At every smooth configuration, its first metric derivative is a compact tensor distribution with no nonzero nonspacelike wavefront covector.

Proof. Directional differentiation contracts kernel slots with smooth directions. Evaluation at a smooth configuration contracts further slots with smooth fields.

The wavefront bound for these contractions allows zero covectors in all contracted slots. A forbidden nonspacelike covector in the remaining first-derivative slot would therefore produce a forbidden tuple in (eq:MC). The same argument applies to every remaining number of slots.

The pointwise product of bulk polynomials remains bulk: its coefficient kernels are symmetrized exterior tensor products. The wavefront condition (eq:MC) is preserved because a forbidden all-future or all-past tuple would already be forbidden in one of the nonzero tensor factors.

For later comparison only, fix

where $m\ge 0$ and

No positivity of the graviton block is required.

The old bulk Wick product is

It is retained as an operation on its existing bulk domain. It is not used to define a product on all of ${\mathcal{P}}_{\Sigma }$.

For a compact smooth covector test $a$ supported in $M_{+}$, define

Let ${G_{\mathrm{A}}}^0$ be the advanced inverse of ${\square }_{\eta }$, and set

The superscript distinguishes this massless inverse from any massive scalar inverse appearing in a later interacting construction.

The formal adjoint of $C$ is

The functional $\mathsf{J}(a)$ has the compact distributional kernel

where $n_{\mu }={\partial }_{\mu }t$ and

There is no ${\delta }^{\prime }(t)$ term. Moreover,

Proof. On the future half-space,

Transposition of the retarded and advanced inverses gives

All relevant support intersections lie in a compact truncated causal region.

The distributional product rule yields (eq:qa). The operator $C^{\ast }$ has order one and hence differentiates the step function only once. The coefficients $b$ and $C^{\ast }b$ are smooth. The only possible singular covectors are therefore conormal to $\Sigma$.

The map $v\mapsto \mathcal{B}(v)$ is injective. In the present signature,

If $b(0,{\mathbf{x}}_{\ast })\ne 0$, every nonzero normal covector at $(0,{\mathbf{x}}_{\ast })$ belongs to the wavefront set of at least one component of $q_a$.

Proof. The displayed components prove injectivity.

For the wavefront assertion, choose a component whose delta coefficient is nonzero at the point. After localization that component has the form

with $v,r$ smooth and $v$ not identically zero near the point. At fixed tangential Fourier frequency the delta term has a nonzero normal-frequency coefficient of order zero. The step term is of order at most $|\tau {|}^{-1}$. It cannot cancel the delta coefficient in either nonzero normal direction.

Let

and

All these spaces are complex vector spaces.

There is a well-defined linear map

given by

It commutes with complex conjugation and vanishes on ${\mathcal{M}}_1$.

Proof. Suppose $m+q_a=m^{\prime }+q_{a^{\prime }}$. Then

Proposition prop:adjoint shows that the possible wavefront covectors of $q_{a-a^{\prime }}$ are normal to the spacelike surface $\Sigma$, and hence timelike. Membership in ${\mathcal{M}}_1$ excludes them. Lemma lem:boundary therefore gives

Thus (eq:chargedef) does not depend on the decomposition. Compactness of the trace support follows from finite propagation from the compact source. Reality of the operators proves the conjugation statement.

The map $\mathsf{b}$ records the coefficient responsible for the delta contribution in (eq:qa). It is not an independent boundary field. It is computed from the original off-shell functional.

It does not classify every surface singularity. For example, a kernel with vanishing $b{|}_{\Sigma }$ but nonvanishing normal derivative can still have a step singularity. The present topology uses the trace needed for the stated nonconvergence theorem; it is not claimed to be a complete boundary microlocal topology.

For a smooth direction $\eta \in \mathcal{E}$, write

For a finite ordered list $\mathbf{\eta }=({\eta }_1,\dots ,{\eta }_k)$, put

The empty list means the identity.

A first metric derivative is paired against a symmetric covariant metric variation and is therefore a contravariant tensor distribution.

The space ${\mathcal{P}}_{\Sigma }\subset {\mathcal{P}}_{\mathrm{dist}}$ consists of the functionals $F$ such that, for every smooth configuration $\Psi$ and every finite list of smooth directions $\mathbf{\eta }$,

Its boundary-jet coordinate is

This is an explicit condition on actual distributional polynomial functionals. It neither assigns values to failed contractions nor adjoins independent symbols.

Any compact-distribution polynomial depending only on the scalar field belongs to ${\mathcal{P}}_{\Sigma }$, because all its metric derivatives are zero. This observation is an admission statement, not a Wick multiplication theorem for arbitrary scalar distributions.

The space ${\mathcal{P}}_{\Sigma }$ contains:

1. every bulk polynomial $F\in {\mathcal{P}}_{\mathrm{mc}}$;

2. every $\mathsf{J}(a)$ with $a\in {C_c}^{\infty }(M_{+};T^{\ast }M)$;

3. every $B_f$ with $f\in {C_c}^{\infty }(M_{+})$.

For the sharp insertion,

If $\eta =(a,\sigma )$, then

The first term is a regular scalar linear functional. For ${\eta }^{\prime }=(a^{\prime },{\sigma }^{\prime })$,

and derivatives of order three vanish.

Proof. Bulk admission follows from Lemma lem:bulkderivative. The metric derivative of $\mathsf{J}(a)$ is $q_a$, and further directional derivatives are constants.

For $B_f$, differentiation in the metric gives the first formula by the adjoint identity. Integration by parts in the scalar variation gives (eq:Bphi); it is smooth and compact for each smooth metric configuration.

Differentiation in both fields gives (eq:DB) and (eq:DDB). The coefficient ${\partial }_{\mu }(f(La{)}^{\mu })$ of the scalar linear term is smooth and compact. All metric derivatives involved are therefore either zero, bulk linear kernels, or kernels $q_a$ with a smooth compact covector source.

The notation $-L^{\ast }(f\mathrm{d}\varphi )$ in (eq:Bh) is retained as a distributional descendant. Its surface term is not removed by replacing it with an on-shell-equivalent test.

The space ${\mathcal{P}}_{\Sigma }$ is a unital commutative algebra under pointwise multiplication. It is closed under every smooth directional derivative and under complex conjugation.

Let ${\mathcal{P}}_{\mathrm{gen}}$ be the pointwise algebra generated by ${\mathcal{P}}_{\mathrm{mc}}$, all $B_f$, and all $\mathsf{J}(a)$. Then

and ${\mathcal{P}}_{\mathrm{gen}}$ is itself stable under all smooth directional derivatives and complex conjugation. There is no bound on the number of factors in a word.

Proof. For $F,G\in {\mathcal{P}}_{\Sigma }$, apply the iterated directional Leibniz rule, and then take a metric derivative. Every term is a complex scalar value of a directional derivative of one factor multiplied by a first metric derivative of a directional derivative of the other. The latter belongs to ${\mathcal{S}}_1$. Since ${\mathcal{S}}_1$ is a vector space, (eq:admission) holds for $FG$.

Directional stability follows because a further direction simply adds an entry to $\mathbf{\eta }$. Conjugation follows from reality of $L$ and the conjugation invariance of the wavefront condition defining ${\mathcal{M}}_1$.

For ${\mathcal{P}}_{\mathrm{gen}}$, derivatives of bulk generators remain bulk. Proposition prop:generators expresses derivatives of sharp generators in terms of bulk scalar linear functionals, coordinate generators, and constants. The ordinary Leibniz rule then proves closure for every finite word.

The proof uses exterior tensor products of coefficient kernels, not contractions of two singular kernels at identified variables. It establishes pointwise multiplication only.

In particular, neither ${\mathcal{P}}_{\mathrm{gen}}$ nor ${\mathcal{P}}_{\Sigma }$ is asserted to be closed under the fixed Wick product. Arbitrarily long classical words have a place to live; their quantum products remain to be defined.

Give $C^{\infty }(\Sigma ;T^{\ast }M{|}_{\Sigma })$ its compact-open smooth Fréchet topology. We use this target topology even though each individual boundary trace is compactly supported.

The topology ${\tau }_{\Sigma }$ on ${\mathcal{P}}_{\Sigma }$ is the initial locally convex topology for:

1. the coefficient inclusion

   $${\mathcal{P}}_{\Sigma }⟶{\mathcal{T}}_{\mathrm{coeff}};$$

2. every map

   $${\mathsf{b}}_{\Psi ;\mathbf{\eta }}:{\mathcal{P}}_{\Sigma }⟶C^{\infty }(\Sigma ;T^{\ast }M{|}_{\Sigma }).$$

Equivalently, it is the topology induced by the graph embedding into the product of these spaces.

This topology is explicit. Its additional seminorms measure boundary traces of all directional descendants at all fixed smooth configurations.

The definition does not assert completeness. We do not take a completion that could introduce infinite-degree functionals or undefined products.

The topology ${\tau }_{\Sigma }$ is Hausdorff. Complex conjugation and every fixed-direction operator ${\mathrm{D}}_{\eta }:{\mathcal{P}}_{\Sigma }\to {\mathcal{P}}_{\Sigma }$ are continuous. Evaluation of a polynomial or any of its fixed directional derivatives at a fixed smooth configuration is continuous.

Proof. The coefficient inclusion is injective by Lemma lem:kernelunique and takes values in a Hausdorff space. Hence the graph topology is Hausdorff.

Evaluation and directional contraction with fixed smooth fields are continuous on each strong compact-distribution coefficient space. To see this directly, a bounded set of smooth target tests is mapped to a bounded set of smooth source tests by multiplication with the fixed field or by insertion in a kernel slot. Strong-dual seminorms therefore give the required bounds. The degree shifts define continuous maps on the locally convex direct sum.

For boundary coordinates,

Thus differentiation shifts the already included coordinates. Conjugation sends the coordinate at $(\Psi ,\mathbf{\eta })$ to the conjugate coordinate at $(\overline{\Psi },\overline{\mathbf{\eta }})$. These operations are continuous in the smooth target spaces.

For each fixed $G\in {\mathcal{P}}_{\Sigma }$, the maps

are continuous on $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. No joint continuity or hypocontinuity is asserted.

Proof. At coefficient level, multiplication by the fixed polynomial $G$ is a finite sum of symmetrized exterior tensor products with its fixed compact distributions.

Exterior tensor product with a fixed compact distribution is continuous in the strong topology. Indeed, partial application of that fixed distribution maps bounded sets of smooth joint tests to bounded sets of smooth tests in the remaining variables. This gives the strong-dual estimate. Permutation and finite symmetrization are continuous as well.

For a boundary coordinate, apply the directional Leibniz rule. After the metric derivative, the boundary trace of each term has the form

Here $I,J$ range over the finitely many partitions of the direction list. The values of derivatives of $G$ and their boundary traces are fixed. Evaluations of $F$ are continuous scalar maps, and its boundary traces are coordinates of ${\tau }_{\Sigma }$. Multiplication of a fixed smooth function by a scalar is continuous. Every coordinate of multiplication by $G$ is therefore continuous.

The theorem is a statement about multiplication by a fixed factor. Its proof does not give uniform control over arbitrary bounded sets of varying factors. Consequently it is not a hypocontinuity theorem.

This distinction is essential: the normal-topology mixed-Wick obstruction uses a stronger multiplication hypothesis. Nothing here asserts that the classical topology already supports such a quantum multiplication.

The maps

and

are continuous linear maps.

For fixed compact supports of finitely many observation and coordinate tests, every finite pointwise word in these generators depends jointly continuously on those tests in their fixed-support smooth topologies. No joint-continuity statement for unrestricted support variation is included.

Proof. For coordinate functionals, the source-to-advanced-solution map and its trace at $t=0$ are continuous on each fixed compact source stage. This follows from smooth finite-time Cauchy estimates and finite propagation. Formula (eq:qa) then gives continuous dependence of the compact distribution kernel in its strong topology.

For the sharp mixed kernel, use

on a compact truncated causal region. The product with the step function exists: its wavefront covectors have zero $x$-component, whereas nonzero wavefront pairs of the wave kernel have nonzero $x$-component. Move $C$ to the kernel and the scalar derivative to the $x$-kernel. Multiplication by $f(x)$ and finite differentiation give the kernel of $B_f$.

On a fixed source support, these operations against bounded smooth joint tests are bounded by finitely many smooth seminorms of $f$. Thus $f\mapsto B_f$ is continuous into the strong coefficient space.

The nonzero boundary coordinate with no directions is

With one direction, only the scalar direction contributes to the first metric derivative, and the same formula holds with that direction replacing $\varphi$. Higher such coordinates vanish. Multiplication by a fixed smooth scalar and advanced trace are continuous on each compact test stage. This proves continuity of all boundary coordinates. The test-function LF topology is final with respect to its fixed compact support stages, so the linear maps are continuous on the full test spaces.

For a finite word with fixed compact test supports, coefficient kernels are exterior tensor products of the fixed distributional kernels multiplied by smooth test coefficients. Finite-order distribution estimates on the common compact region give joint continuity. Boundary coordinates follow from the finite Leibniz formula and continuous evaluations and traces.

The inclusion

is injective. On each fixed-degree, fixed-support, fixed-wavefront bulk stage with its normal topology, it is continuous.

All boundary-jet coordinates of a bulk polynomial vanish. The topology induced on the bulk is its strong coefficient topology; it is not asserted to coincide with the finer normal topology.

Proof. Injectivity follows because both spaces consist of the same actual functionals with unique kernels.

Lemma lem:bulkderivative puts every relevant first metric derivative in ${\mathcal{M}}_1$. The map $\mathsf{b}$ vanishes there. Thus every boundary-jet coordinate of a bulk polynomial is zero.

The remaining topology on the bulk is the strong coefficient topology. The normal topology contains the strong distribution seminorms, so its inclusion into this topology is continuous. No converse continuity is needed or claimed.

This statement preserves bulk functionals and bulk operations. It does not silently assert preservation of every old microlocal topological feature.

Let $f\in {C_c}^{\infty }(M_{+};\mathbb{R})$ be nonzero. Set

Then

In particular, $b{|}_{\Sigma }\ne 0$.

Proof. Let

On bounded time intervals the supports are uniformly spatially bounded. Since ${\square }_{\eta }{b}_0=a_0$, spatial integration gives

The advanced solution has zero data at sufficiently late times. Therefore

Put $u={\partial }_t(tf)$, so ${\varphi }_f=u$. Compact integration by parts gives

If $u=0$, then $tf$ is independent of $t$. Compact time support forces $tf=0$, hence $f=0$, a contradiction.

The sign in (eq:witness-positive) uses ${\square }_{\eta }=-{{\partial }_t}^2+\Delta$. Copying the witness sign from the opposite signature without recomputing the advanced moment would be incorrect.

For every nonzero real $f$,

The map $f\mapsto B_f$ is injective.

Proof. At $(0,{\varphi }_f)$, the metric derivative is $-q_a$. Lemmas lem:witness and lem:boundary give a nonzero timelike wavefront component, forbidden for a bulk first derivative. It is also a nonzero derivative, so $B_f\ne 0$.

For complex $f$, reality of $L$ and separation into real and imaginary parts show that $B_f=0$ forces both parts to vanish.

Choose a real even compact smooth approximate identity

and define

If $b_f$ is the mixed kernel of $B_f$, the approximating mixed kernel is

It is smooth and compact, so $B_{f,\epsilon }\in {\mathcal{P}}_{\mathrm{reg}}$.

For every nonzero real $f$,

More precisely, at the fixed configuration $(0,{\varphi }_f)$,

whereas

There is a continuous scalar seminorm coordinate that separates the proposed limit from every approximant.

Proof. Every approximant is regular and hence bulk. Its boundary coordinates vanish by Proposition prop:bulk. The charge of $B_f$ is (eq:Bcharge), and Lemma lem:witness makes it nonzero.

Choose a compact spatial test $\chi$ equal to one on the compact support of $b_0{|}_{\Sigma }$. The continuous functional

is zero on every approximant and nonzero on $B_f$. Convergence is therefore impossible.

For completeness, define the translation-invariant angular cone in the mixed variables $(x,z)$ by

The case $\xi =0$ is included in the second set.

The mixed kernel $b_f$ satisfies $\mathrm{WF}(b_f)\subset {\Gamma }_B$. On a common compact support stage containing a neighborhood of $\mathrm{supp}{b}_f$,

Proof. The retarded wave kernel has diagonal conormals and direct null propagation covectors $(\xi ,-\xi )$. Multiplication by $\theta (z^0)$ adds boundary conormals in the $z$-slot, giving

At the boundary, the observation variable has strictly positive time, so the bulk diagonal does not meet that boundary branch. Differentiation and multiplication by the smooth observation test do not enlarge the wavefront bound.

Strong distribution convergence of compact convolution approximants is immediate from convergence of the corresponding smooth test functions on bounded sets.

For the Fourier seminorms, compactness and the translation-invariant angular bound give arbitrary rapid decrease in every closed cone separated from the angular cone in (eq:Gamma). Convolution multiplies the Fourier transform by bounded smooth multipliers converging to one on fixed frequency balls. Split into a fixed ball and its complement; uniform convergence handles the ball and arbitrary rapid decrease handles the tail.

Localized seminorms are treated by convolution with the rapidly decreasing Fourier transform of the localization. Angular separation controls the far-frequency part, while a slightly larger separated cone controls the near-frequency part. This is the usual compact approximate-identity argument for the normal seminorms.

The identity sending a mixed kernel to its actual polynomial functional is not sequentially continuous from the normal stage of Lemma lem:normal into $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$ at $b_f$.

Thus the present admission maps do not satisfy the combination:

Proof. Lemma lem:normal gives the convergence in the source topology. Theorem thm:IX1 excludes the convergence of its images.

This is not a subtraction of the old regular products. Those products remain fixed. Nor is it a proof that any new quantum multiplication exists. It is the precise compatibility boundary required before attempting that construction.

The conclusion completes the topological part of Theorem thm:main.

IX-A must distinguish three claims:

Only selected instances of the first two are proved here.

For the covariance (eq:Hhh),

The induced $ww$ covariance is zero. The mixed component selected by $h_{00}$ is

where $r=|p|$.

Proof. The trace of $P$ in four dimensions is $-{\eta }_{\rho \sigma }$. Substitution into $C$ cancels the trace term and gives the first identity. Applying $C_y$ cancels the mixed second derivatives and gives the second.

For the $ww$ statement, the contracted massless momentum numerator vanishes before either zero-Cauchy operator is applied. The smeared covariance is therefore zero.

For $K$, solve the zero-data equation on a mode:

Multiply by the derivative $-ir$ and the spectral density $1/(2r)$, and separate the two sine frequencies. This gives (eq:K). The infrared factors $r^{-1}$ and $r^{-2}$ are locally integrable in three dimensions.

These tensor identities are not an algebra-closure theorem. In particular, the reflected term in (eq:K) is retained.

The following already-defined coefficient functionals belong to ${\mathcal{P}}_{\Sigma }$:

1. the coefficients of the products of $B_f$ with a smooth bulk linear field, in either order;

2. the pointwise term $B_f{B}_{f^{\prime }}$ and the scalar-contracted self-product coefficient

   $$T_{f,f^{\prime }}[h]=\int f(x)f^{\prime }(x^{\prime })w^{\mu }[h](x)w^{\nu }[h](x^{\prime }){{\partial }_{\mu }}^x{{\partial }_{\nu }}^{x^{\prime }}{\Delta }_m^{+}(x-x^{\prime }){\mathrm{d}}^{4}x{\mathrm{d}}^4{x}^{\prime };$$

3. the coefficients of $B_f$ with $W_g=\int g{h_{00}}^2$, in either order, when only this two-factor product is taken.

The assertion does not include the subsequent multiplication by another sharp insertion.

Proof. For a smooth linear metric test $a$, $H_{hh}a$ is smooth. The single contraction with $B_f$ is a regular scalar linear functional with coefficient obtained from $L(H_{hh}a)$. For a scalar linear test $u$, the contracted scalar factor is smooth and the output is a coordinate functional $\mathsf{J}(a_u)$ with smooth compact source. Reverse ordering has the same admission properties.

For (eq:selfT), the differentiated scalar covariance is paired against a smooth compact function for each smooth $h$. Polarization and the kernel theorem give a compact quadratic metric kernel.

Its first metric derivative is obtained by varying one of the two $w$ factors. In each term, the other $w[h]$, multiplied by its compact test, is smoothly smeared against the differentiated scalar covariance. That convolution is smooth. The varied factor therefore has the form $\mathsf{J}(a[h])$ with a smooth compact source. At every fixed configuration and after every directional derivative, the first metric derivative is a finite sum of $q_a$'s. Thus $T_{f,f^{\prime }}\in {\mathcal{P}}_{\Sigma }$. The possible double contraction vanishes because $H_{ww}=0$.

For $W_g$, the single graviton contraction leaves a functional of the form

up to its fixed combinatorial coefficient and sign. For fixed $\varphi$, smearing the $x$-slot gives a smooth function of $y$. One justification is to write it as a bulk covariance smeared with a kernel $q_a$: the latter has only timelike wavefront covectors, whereas the nonzero covariance covectors are null. The resulting composition is smooth. Equivalently, the Fourier expression (eq:K), smeared in $x$, has rapid high-frequency decay, including its reflected branch. After multiplication by $g$, the first metric derivative is a smooth compact kernel. Further directional derivatives have the same property. The output therefore belongs to ${\mathcal{P}}_{\Sigma }$.

Let

The finite contraction formulas for $B_f$ and $Q_q$, in either order, define compact polynomial coefficients belonging to ${\mathcal{P}}_{\Sigma }$.

This does not assert closure for arbitrary local quadratic polynomials or arbitrary later multiplications.

Proof. A graviton-only contraction leaves a scalar quadratic functional with a compact distribution kernel. Such a functional belongs to ${\mathcal{P}}_{\Sigma }$ because it has no metric dependence.

A scalar-only contraction leaves a quadratic metric functional of the form

with tensor indices understood and $D$ a differentiated scalar Hadamard distribution. Varying $w[h]$ gives $q_a$ with a smooth compact source: the $y$-factor is a smooth compact smearing at each fixed configuration. Varying $h(y)$ gives a smooth compact metric kernel obtained by smearing $D$ in $x$. Further directional derivatives have the same two forms. The output is therefore admitted.

For the double contraction, the numerical product is

In both direct and reflected branches of the first factor, the nonzero $y$-slot covector has the same temporal orientation as the $y$-slot covector of the scalar factor. It is nonzero and null. The two factors consequently cannot have opposite total wavefront covectors. The distributional multiplication criterion applies. Compact smearing gives a constant, which belongs to ${\mathcal{P}}_{\Sigma }$. Reverse ordering reverses both orientations and gives the same conclusion.

The coefficient maps in the preceding propositions are the old contraction formulas wherever those formulas are defined. No finite-part choice has been introduced.

A future quantum multiplication is required to reproduce these coefficients on this common domain. The present propositions do not define how to multiply their outputs by every other admitted element.

In particular, they do not assign the value of

after either association. That belongs to the subsequent graph and coherence problem.

The notation $\mathsf{B}(f)$ may be used for the admitted element $B_f$, but it is not a new independent field. Its classical functional is exactly (eq:B).

Likewise $\mathsf{J}(a)$ is the actual coordinate functional (eq:Jcoord). It can have nontrivial relations among different source tests. Those relations are determined by equality of the compact kernels $q_a$, not by an arbitrarily chosen quotient.

The construction has the following properties:

1. no two distinct bulk polynomial functionals are identified;

2. if two admissible presentations define the same polynomial functional on every smooth off-shell configuration, they define the same element of ${\mathcal{P}}_{\Sigma }$, with the same topology coordinates;

3. kernel permutations, tensor symmetries, compact integration by parts with all boundary terms retained, and bulk/surface decompositions that preserve the actual functional do not change the admitted element.

Equation-of-motion equivalence is not included.

Proof. The first assertion is the injectivity in Proposition prop:bulk.

For the second, Lemma lem:kernelunique identifies the symmetric coefficient kernels. Equality of the functionals also identifies all their directional derivatives and metric derivatives. Lemma lem:charge then identifies every boundary coordinate.

The listed rewritings preserve the distributional pairing when all terms are retained. For example, the decomposition

is equality of compact distributions. Dropping its delta term is not one of the admitted rewritings.

The theorem establishes presentation independence of the carrier. It does not establish presentation independence of future renormalized contractions.

Once quantum maps are introduced, they must be shown to factor through these actual-functional relations. A graph presentation cannot be granted a different quantum value merely because its classical formula was obtained by another valid integration by parts.

Define the locally convex module

with the finite product topology. Define

The map $\mathrm{cl}$ and field-independent coefficient evaluations are continuous.

The old bulk quantum algebra

embeds linearly and injectively in ${{\mathcal{V}}_{\Sigma }}^{[3]}$. Its old product remains prescribed on that subspace.

boundary{ We write ${{\mathcal{V}}_{\Sigma }}^{[3]}$, not ${{\mathfrak{A}}_{\Sigma }}^{[3]}$. The former is a coefficient module. It becomes a quantum algebra only after a multiplication with the required closure and identities has been constructed.}

For any proposed product

the three preservation requirements are:

Only the carrier needed to state these requirements is constructed here. Equations (eq:bulkpres)–(eq:clpres) are obligations for a future quantum product, not results about an unconstructed map.

Every compact-distribution polynomial output has a canonical place in ${\mathcal{P}}_{\mathrm{dist}}$. To place it in ${\mathcal{P}}_{\Sigma }$, one must verify (eq:admission), including its directional descendants.

Thus the carrier construction gives a concrete output test rather than a promise that every future graph passes it. If a later graph has a first metric derivative outside ${\mathcal{S}}_1$, then that graph requires a further domain enlargement or supplies an admission obstruction for the current domain.

No unconditional ${\hslash }^3$ output-closure theorem follows from the module (eq:module).

Let a proposed relational functional have bare coefficients

If the relevant retarded products exist, its first interacting coefficient is

and its second is

This is coefficient bookkeeping, not a construction of the maps.

{}1.2

The order ledger is

A truncation in $\hslash$ does not bound the number of pointwise factors or the field degree.

The membership of $B_f$ gives no theorem admitting $D_f$. Likewise, the selected old Wick coefficients do not imply that a retarded product with $B_f$ exists.

No gauge transformation is needed to construct ${\mathcal{P}}_{\Sigma }$ or ${\tau }_{\Sigma }$. Nevertheless the distinction from an interacting relational realization can be stated directly.

For a compact gauge vector $\xi$ supported strictly after $\Sigma$,

Hence the sharp bare dressing changes by