Landmark VII-I Finite-time harmonic-coordinate insertions: three repair routes, finite-width free products, and sharp-prescription incompatibility

We examine the passage from a finite-time relational scalar to a quantum insertion, compatible products, and an interacting physical observable. The fixed classical prescription uses coordinate-time Cauchy derivatives; replacing them by derivatives along a metric-dependent unit normal would change the observable.

All three proposed repair routes are developed. Field-argument convolution gives a regular free algebra at positive width, but an explicit positive first-order defect excludes covariance under the unchanged plateau gauge action. A smooth divergence-form density preparation gives compact microcausal polynomial insertions through second coordinate order. A tree-composition argument supplies their free products, adjoints, and ordering-stable generated sector. Its sharp limit, however, has a calculated nonzero transmission velocity and is not the original zero-Cauchy prescription.

For the sharp route we construct a minimal boundary-sensitive classical carrier and prove that unchanged regular approximants do not converge in its topology. Separately, the VII-C/VII-D numerical asymptotic gives a quantitative incompatibility between old-product agreement, convergent sharp admission, bounded-set hypocontinuity, and continuous extraction of the triangle coefficient. A finite part cannot satisfy all those requirements.

Auxiliary coordinates are classically equivalent to the prescribed Cauchy construction when both equations and data are imposed. Quantum elimination requires inherited cross-contractions; an independent central-coordinate prescription fails that test. Linear sharp coordinates do have exact smooth representatives on the free shell, yielding a genuine linear free quantum sector.

Presentation descent, associativity, and relative classical gauge closure are proved at their stated levels. The sharp nonlinear interacting quantum observable, its quantum differential, and its complete normalization catalogue remain unresolved. No missing characteristic composition is supplied by a classical carrier, an internal tree kernel, or a numerical subtraction.

and proof boundary{ Constructed: two explicit smoothing comparisons; a finite-width microcausal insertion and free polynomial algebra; a minimal boundary-sensitive classical carrier; classical auxiliary equivalence; and a linear free-shell quantum repair.

Proved incompatible: the unchanged off-shell sharp insertion with the microcausal polynomial domain; the convolution comparison with the unchanged classical gauge action; and the specified continuous sharp Wick-extension requirements.

Unresolved: an associative renormalized sharp nonlinear sector outside those requirements, mixed time-ordered products, an interacting quantum gauge differential, and a physical correlation of the original sharp relational observable.

All positive constructions are labelled by the observable and domain they actually use.}

The desired chain is

These implications require different theorems.

We retain the restricted gauge domain of Cheung's VII-A CheungVIIA. The boundary non-admission in VII-B, the normal-topology obstruction in VII-C, and the numerical finite part in VII-D concern different requirements CheungVIIB,CheungVIIC,CheungVIID. Their coexistence is not a contradiction.

Relational-coordinate calculations and causal propagator constructions provide comparison methods FrobCorrelation,FrobLima. Equicausal functionals provide uniform control for their admitted functional class HRV; they do not admit a kernel already violating the underlying microcausal exclusion. Off-shell deformation quantization, causal perturbation theory, and renormalized BV theory provide the surrounding frameworks Duetsch,BDF,FRBV,BFR,FrobWard, not an automatic construction for the present sharp insertion.

1. An insertion is an admitted algebra element with a specified classical symbol.

2. A free sector has defined products, associativity, adjoint, closure, and stated ordering comparison.

3. An interacting physical sector additionally has the required mixed time-ordered products and a nilpotent quantum gauge differential whose cohomology contains the insertion.

4. A state-level prediction requires a specified state or representation and a controlled evaluation in that sector.

A coefficient distribution is not an insertion, and an insertion is not automatically an interacting cohomology class.

Throughout our constructions,

Put $M_{+}^{\circ }=\{t>0\}$. The retarded and advanced inverses of $\square$ are ${E^{\mathrm{R}}}_0,{E^{\mathrm{A}}}_0$, and $E_0={E^{\mathrm{R}}}_0-{E^{\mathrm{A}}}_0$.

The zero-data solution operator is

On future smooth sources it is

For compact observation sets, no spatial decay is required: finite propagation reduces each calculation to a compact causal region BGP.

Let

The coefficient $k$ is metric-jet bookkeeping, not a second independent quantum metric field.

Set

The coordinates solve

Their formal expansion is

The data in (eq:coordinates) are not equivalent to fixing ${n_q}^{\alpha }{\partial }_{\alpha }{{\mathsf{X}}_{\mathsf{q}}}^{\mu }={{\delta }^{\mu }}_0$ on $\Sigma$, where $n_q$ is the future unit normal. For the latter prescription,

Proof. For the surface $t=0$,

Expanding the fixed normal derivative, and using ${\partial }_{\alpha }{x}^{\mu }={{\delta }_{\alpha }}^{\mu }$, gives ${\partial }_t{w}^{\mu }+{n_1}^{\mu }=0$. The inverse-metric expansion gives the displayed $n_1$.

Thus no argument below substitutes normal-derivative data for the coordinate-time prescription inherited from VII-A.

For real $f\in {C_c}^{\infty }(M_{+}^{\circ })$, define formally

The leading map is the identity, so formal inversion is unique. Its coefficients are

All integrals in this paper use ${\mathrm{d}}^{4}x$ unless indicated otherwise.

For $K=\mathrm{supp}f$, the metric dependence lies in

a compact set. Nested retarded operations do not enlarge this overall causal past.

Through order ${\kappa }^2$,

The determinant has leading term $1$; no absolute-value ambiguity arises in the formal orientation-preserving expansion.

Proof. Formal change of variables gives the identity. Explicitly,

Multiply this by the Taylor expansion of $f(x+\kappa w+{\kappa }^{2}v)$. At order one, integration by parts gives $-\int f{w}^{\mu }{\partial }_{\mu }\varphi$. At order two, two integrations by parts give precisely (eq:observable-coefficients).

The observation factor has compact support strictly above $\Sigma$. These integrations occur in the observation variable. They do not delete the initial-surface terms obtained when a coordinate kernel is subsequently transposed in a metric variable.

Fix a scalar-action switch $g$, a nonempty plateau $U$, and

The group is generated by flows of these vector fields. Its second-order formal pullbacks are

They are the identity near $\Sigma$.

For the unswitched coordinate equation,

by naturality and Cauchy uniqueness. Therefore

For the switched scalar solution with fixed free input, the additional plateau intertwiner is the theorem of VII-A CheungVIIA. It is not unrestricted fixed-switch covariance.

At the background coordinate map, preserving fixed values and fixed coordinate-time derivatives requires

at first order. Vanishing only of $\xi {|}_{\Sigma }$ is insufficient. Support in $U$ supplies a stronger sufficient condition.

Let ${\mathcal{P}}_{\mathrm{mc},\mathrm{pol}}$ be the finite polynomial space with compact, species-symmetric kernels $u_n$ satisfying

where each cone contains its zero covector and the total zero covector is omitted.

For a linear coordinate test $a_{\mu }\in {C_c}^{\infty }(M_{+}^{\circ })$, put

The formal adjoint of $C$ is

The compact distribution $q_a=L^{\ast }a$, $L={\mathsf{E}}_{0}C$, is

There is no ${\delta }^{\prime }(t)$ term. The map $\mathsf{B}$ is injective. If $b(0,{\mathbf{x}}_0)\ne 0$, every nonzero normal covector $(0,{\mathbf{x}}_0;\tau \mathrm{d}t)$ belongs to $\mathrm{WF}(q_a)$.

Proof. Transpose ${E^{\mathrm{R}}}_0(\theta Ch)$ against $a$:

The product rule gives (eq:boundary-kernel). Its support is in $J^{-}(\mathrm{supp}a)\cap \{t\ge 0\}$.

The components

prove injectivity.

Locally, a nonzero component has the form $\delta (t)v(\mathbf{x})+\theta (t)r(t,\mathbf{x})$, with $v({\mathbf{x}}_0)\ne 0$ and smooth coefficients. After localization, the first term has normal Fourier order zero, whereas the second has order at most minus one at fixed tangential frequency. The leading nonzero term cannot cancel. Thus rapid decrease fails in both normal directions. These are actual timelike conormals, not merely an upper bound.

For every nonzero real $f\in {C_c}^{\infty }(M_{+}^{\circ })$,

No element of ${\mathcal{P}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ]]$ has classical symbol equal to ${\mathcal{B}}_f$ on all smooth configurations.

Proof. The metric derivative is

Choose

For $b={E^{\mathrm{A}}}_0(f\mathrm{d}{\varphi }_f)$, integration of $\square b_0=f{\partial }_t{\varphi }_f$ over space gives

The advanced solution has zero final data, which fixes the first equality. The last expression is nonzero because compact support and ${\partial }_t(tf)=0$ would imply $f=0$.

Proposition prop:boundary gives a timelike wavefront component of the metric derivative. Partial contraction of an admitted microcausal polynomial kernel with smooth configurations cannot produce such a component: adjoining zero covectors in the contracted slots would violate the original cone exclusion. This proves non-admission.

An off-shell-equal classical symbol has the same derivative. Positive powers of $\hslash$ cannot alter that contradiction.

This reproduces the relevant VII-B mechanism with its hypotheses visible CheungVIIB. It does not prove universal non-quantizability.

Functional differentiability, distributional existence, contraction existence, and continuity are different entries in this atlas.

Choose a real nonnegative even Euclidean-radial compact mollifier $\rho$, with integral one. Let

At the level of the coordinate construction, define

This is not graph-only smoothing.

Through coordinate order two, every coefficient of ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{conv}}$, for fixed $ϵ>0$, has a smooth compact kernel. Consequently these coefficients belong to ${\mathcal{P}}_{\mathrm{reg},\mathrm{pol}}$ and have all finite free Wick products in the ordinary polynomial algebra.

On fixed smooth configurations the coefficients converge to their sharp classical counterparts as $ϵ\downarrow 0$. This is not convergence of their quantum products.

Proof. Each sharp coefficient is a continuous finite polynomial in smooth fields, with compact multilinear kernel by the kernel theorem and finite propagation. Convolution in every field slot makes that kernel smooth and enlarges support only by a fixed compact $ϵ$-neighbourhood.

On fixed smooth configurations, convolution converges in all smooth seminorms on the relevant causal compact. Continuous dependence of the coefficientwise Cauchy problems then proves the classical convergence.

The finite-width free construction changes the observable. It cannot be assumed covariant under the original field action.

There are admissible plateau parameters and free massless scalar data for which the first-order variation of ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{conv}}$, under the unchanged pullback action on the original fields, is strictly positive.

In a suitable compact dependence neighbourhood, one can choose

with a compact realization of $\xi \in {\mathfrak{g}}_U$, and obtain

Here $f\ge 0$, $f\not\equiv 0$, and the sampled region is above the initial surface.

Proof. Choose the compact parameter support strictly above $\Sigma$, and $ϵ$ small enough that its smearing still vanishes near $\Sigma$. Constant-coefficient differentiation commutes with convolution. Since $CK\xi =\square \xi$,

The first-order scalar pullback is $\delta \varphi =\kappa {\xi }^{\mu }{\partial }_{\mu }\varphi$. The variation therefore is

Take a compact cutoff equal to one on a neighbourhood of the $ϵ$-sampling region of $\mathrm{supp}f$, and set $\xi =t{\partial }_t$ there. The printed scalar is globally smooth and satisfies $\square \varphi =0$. Evenness and normalization give, on that neighbourhood,

Substitution proves (eq:convdefect). A smooth nonnegative radial mollifier of integral one has $m_{00}>0$.

Thus this route constructs positive-width free products but does not retain the unchanged classical gauge covariance. Quantum-order counterterms cannot cancel the displayed classical coefficient. The analogous distinction between convolution and a density ramp is explicit in Cheung's comparison track CheungXAconv.

Choose a smooth ramp

It is flat at the endpoints.

Define a coordinate density

The prepared coordinates solve

with the same coordinate-time Cauchy data as (eq:coordinates).

The coefficients are

The resulting functional ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{D}}$ is defined by formal inverse-coordinate composition, not by assigning a finite value to one graph.

Already at first order,

The prescription ${E^{\mathrm{R}}}_0({\sigma }_{\epsilon }Ch)$ omits the second term. It is a different coordinate construction. The general density/source distinction is also the subject of XII-B CheungXIIB.

For covectors compared in the fixed Minkowski trivialization, put

A balanced line is a bidistribution with wavefront contained in $(x,p;y,-p)$. Derivatives of translation-invariant Green kernels and of $\delta (x-y)$ are examples.

Let a finite tree have internal and external vertices, one balanced line on each edge, and smooth vertex weights. Suppose the joint support of the weighted lines is compact. Then their product is defined, its pushforward over internal vertices is defined, and an $n$-slot output kernel has wavefront contained in $Z_n$.

Differentiation and multiplication by fixed smooth coefficients preserve this conclusion.

Proof. Pull each line back to the full vertex configuration space. Its covector is $p$ at one endpoint, $-p$ at the other, and zero elsewhere.

A forbidden product cancellation would assign line covectors whose signed sum vanishes at every vertex. In a forest a leaf forces the covector on its only incident edge to be zero. Removing that edge and repeating forces every line covector to be zero. This contradicts the nonzero covector required for a forbidden cancellation. Adding lines successively therefore defines the product.

The same leaf recursion bounds every line covector by a fixed multiple of the sum of the vertex covector norms. Thus the covector-sum upper bound is closed; no circulation with unbounded internal covectors is hidden in it.

Compact support makes projection over the internal variables proper. A pushforward wavefront point has zero covectors at all internal vertices. Summing the remaining vertex covectors cancels every line pair, so the external sum is zero. The distributional product, pullback, and pushforward criteria are the standard ones Hormander,BDH.

The lemma is a tree statement. It does not define loops or the later grouped characteristic composition.

For fixed $\epsilon >0$:

1. ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{D},(1)}$ has a compact mixed kernel with wavefront in $Z_2$;

2. ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{D},(2)}[h,0,\varphi ]$ has a compact kernel with wavefront in $Z_3$;

3. the $k$-linear contribution is ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{D},(1)}[k,\varphi ]$;

4. these polynomials and all their fixed smooth directional derivatives belong to ${\mathcal{P}}_{\mathrm{mc},\mathrm{pol}}$.

The coefficient maps are continuous on fixed-support test stages with their corresponding normal-topology kernel stages.

Proof. At order one, integration by parts gives a derivative of

with constant fibre tensors. It is a two-slot balanced kernel.

At order two, the inverse-coordinate formula contains:

1. two first-order coordinate lines ending at the observation vertex and a scalar derivative there;

2. a coordinate line from the observation to a local quadratic metric vertex, with two metric slots joined to that vertex by delta lines;

3. a nested coordinate line, a local metric slot, and a further first-order coordinate line.

Every term is a finite sum of differentiated balanced trees. The $k$-term follows by linearity of $a_1$.

The observation vertex lies in $\mathrm{supp}f$. Every retarded ancestor lies in its causal past, and each coordinate-source vertex has a factor ${\sigma }_{\epsilon }$ or a derivative of ${\sigma }_{\epsilon }$; it is therefore at nonnegative time. Delta-attached field slots coincide with such vertices. All relevant supports lie in a fixed compact causal region. Lemma lem:tree applies.

A nonzero tuple wholly in the future or past causal cone cannot have zero sum. Thus $Z_n$ implies the microcausal exclusion. Smearing a slot against a smooth configuration inserts zero covector in that slot, so the remaining tuple still has zero sum.

For continuity, there are finitely many smooth multiplications, compatible fixed-kernel products, differentiations, and support-proper projections. Their normal-topology continuity and hypocontinuity properties apply on the just-checked stages BDH. No global completeness or global hypocontinuity claim is required.

Assume

For the same restricted transformations as before,

and

Proof. Write

On the support of the transformation, ${\sigma }_{\epsilon }=1$, and outside that support the transformation and its derivative are the identity. Hence this density transforms naturally under the restricted pullback. The transformation is the identity near the initial surface, so Cauchy uniqueness proves the coordinate identity. Formal inversion and scalar pullback prove the observable identity.

This is the mechanism of Cheung's density-ramp classical construction CheungXA. It proves a classical identity, not that the same gauge action is a derivation of the free Wick product.

Let $h$ be smooth near the closed future half-space. On every compact subset of $t>0$,

where

In particular,

The prepared limit is generally not the sharp prescription $w={\mathsf{E}}_{0}Ch$.

Proof. Equation (eq:layer-source) gives

On bounded causal regions, ${\sigma }_{\epsilon }Ch\to \theta Ch$. The first response converges to ${\mathsf{E}}_{0}Ch$ away from the initial surface.

For the second response use the smooth Cauchy formula

The positive smooth weight ${\sigma }_{\epsilon }^{\prime }$ has integral one. For $t$ bounded away from zero, continuous smooth Cauchy dependence, with spatial localization on the fixed causal compact, permits passage to the limit in every derivative. The limit is the homogeneous solution displayed in (eq:transmission). A Taylor estimate gives an $O(\epsilon )$ rate on each such smooth seminorm, with a constant depending on finitely many derivatives of $h$.

That homogeneous solution has initial displacement zero and initial velocity ${a_1}^{0\mu }[h](0)$.

The sharp-limit comparison is already decided at first order. No assertion about a second-order limit can make the nonzero initial velocity disappear. This is a genuine model change, not a renormalization convention.

Use the even field pair $(h,\varphi )$ and the fixed Hermitian two-point matrix

with

The scalar mass may be any $m\ge 0$. No positivity is assumed for the graviton block.

For finite polynomials define

The contraction series terminates for each polynomial pair.

At fixed preparation width, the polynomials of Theorem thm:rampadmission, the positive-width convolution polynomials, ordinary compact local polynomial fields and vertices, and their fixed smooth directional derivatives generate a unital associative free $\ast$-algebra in ${\mathcal{P}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]]$.

The adjoint is complex conjugation of symbols:

The quotient by $({\kappa }^3)$ is a consistent coordinate-order truncation. No field-degree truncation is imposed.

Proof. The admitted finite kernels obey the microcausal exclusion. For each contraction, a forbidden cancellation against the oriented Hadamard lines would force one input covector tuple wholly into one of the excluded causal cones. Thus the products and proper contractions are defined and their output kernels remain compact and microcausal.

On three factors, the constant even contraction operators ${\Gamma }_{12},{\Gamma }_{13},{\Gamma }_{23}$ commute. Both associations therefore equal

The same microlocal argument justifies the finite multicontractions. This is the polynomial Wick construction BF,Duetsch,HRV.

Hermiticity of $H$ proves the adjoint identity by reversing the contraction slots. Constants are the unit. Products can increase field degree, but each coefficient is still a finite polynomial. The formal parameter ideal $({\kappa }^3)$, unlike an arbitrary field-degree cutoff, is stable under multiplication.

This theorem closes Level II for a prepared free insertion through second coordinate order. It does not close Level III.

Let $H^{\prime }=H+d$, where $d$ is smooth, symmetric, and compatible with the declared reality convention. Define

Then

For the prepared first and second coordinate coefficients,

where $c_d$ is a scalar and $L_d$ is a compact smooth linear field smearing.

Consequently the generated sector, including ordinary local polynomials and smooth directional descendants, is transported onto the corresponding sector generated by the unchanged prepared formulas.

Proof. The self-contraction generator obeys

Exponentiating the commuting operators proves the intertwining identity and gives inverse ${\alpha }_{-d}$.

A quadratic self-contraction is a scalar. A cubic has only a linear correction. For a projected linear kernel to have a nonzero wavefront covector $p$, the original $Z_3$-bounded kernel would need a tuple $(0,0,p)$. The zero-sum condition forces $p=0$. Thus the correction is smooth; compactness follows from proper projection.

Smooth directional differentiation commutes with the self-contraction operator. Local polynomial ordering corrections are again local polynomials with smooth coefficients. These and compact linear fields are already among the generators. The inverse transport gives the reverse inclusion.

Ordering transport compares algebra representations. A state must be transported too; keeping a changed symbol and an unchanged numerical state functional is not physical matching. The related distinction is explicit in XII-A and VII-H CheungXIIA,CheungVIIH.

A classical boundary carrier can be constructed without asserting a quantum multiplication.

Let ${\mathcal{P}}_{\Sigma ,\min }$ be the algebra of actual finite polynomial functionals generated, under ordinary pointwise multiplication, complex conjugation, and all fixed smooth directional derivatives, by:

1. ${\mathcal{P}}_{\mathrm{mc},\mathrm{pol}}$;

2. all sharp ${\mathcal{B}}_f$;

3. all linear sharp coordinate functionals ${\ell }_a$.

Identical functionals on smooth configurations are identified.

The pointwise products are multilocal polynomials. They are not declared local merely because their factors were local.

At a fixed configuration $\Phi$, a metric derivative of an element of ${\mathcal{P}}_{\Sigma ,\min }$ has the form

where $u_{\Phi }$ is a compact distribution with no timelike wavefront covectors and $r_{\Phi },v_{\Phi }$ are smooth near $\Sigma$. The boundary coefficient $v_{\Phi }$ is spatially compact.

The coefficient $v_{\Phi }$ in (eq:charge-decomposition) is unique. Write it as ${\mathsf{ch}}_{\Phi }(F)$. It is complex linear and satisfies

Every microcausal polynomial has zero charge.

Proof. The decomposition holds for the generators by Proposition prop:boundary. A smooth metric directional derivative preserves the same form, and a scalar directional derivative changes only the smooth source in that formula. The ordinary Leibniz rule preserves the decomposition under products.

If two decompositions existed, their difference would express $\delta (t)(v-v^{\prime })+\theta (r-r^{\prime })$ as a distribution without timelike wavefront covectors. A nonzero $v-v^{\prime }$ produces the normal Fourier order-zero term of Proposition prop:boundary; neither the step term nor a distribution microlocally smooth in the normal directions can cancel it. Hence $v=v^{\prime }$.

Uniqueness makes the charge intrinsic to the actual functional and proves linearity and (eq:charge-leibniz). For a microcausal polynomial its first derivative has no timelike covectors, so its charge is zero.

On each fixed-support, bounded-polynomial-degree stage, give ${\mathcal{P}}_{\Sigma ,\min }$ the locally convex topology generated by:

1. the strong distribution seminorms of its symmetric coefficient kernels;

2. every scalar pairing

   $$F⟼⟨{\mathsf{ch}}_{\Phi }(D_{{\chi }_1}\cdots D_{{\chi }_r}F),\zeta ⟩,$$

   for fixed smooth directions ${\chi }_j$, fixed smooth configuration $\Phi$, and compact spatial boundary test $\zeta$.

Assemble the stages by the locally convex inductive-limit topology. Denote it by ${\tau }_{\mathrm{bd}}$.

The space $({\mathcal{P}}_{\Sigma ,\min },{\tau }_{\mathrm{bd}})$ is Hausdorff. Fixed smooth directional differentiation and conjugation are continuous. Pointwise multiplication is separately continuous. Bulk inclusion is injective and continuous on fixed-support, fixed-wavefront normal-topology stages.

For every nonzero real $f$, the unchanged regular approximants

do not converge to ${\mathcal{B}}_f$ in ${\tau }_{\mathrm{bd}}$. No quantum product is included in this theorem.

Proof. Symmetric coefficient kernels determine an actual polynomial uniquely by polarization. Their continuous map into the Hausdorff distribution spaces therefore separates points.

Differentiation contracts coefficient kernels with fixed smooth directions, and sends each charge seminorm to another seminorm in the defining list. Conjugation has the same property.

On a fixed stage, multiplication by a fixed polynomial uses finite tensor products with fixed compact distributions and finite symmetrization. These are continuous on the relevant strong fixed-support spaces. Charges of every directional derivative of a product are finite Leibniz sums of evaluations and charges, which are continuous there. The inductive-limit universal property gives global separate continuity.

The normal topology includes the strong distribution seminorms. Bulk charges vanish, so fixed bulk-stage inclusions are continuous. They are injective because actual functionals are not quotiented further.

Use the witness configuration of Theorem thm:sharp. The charge of ${\mathcal{B}}_f$ is a nonzero smooth boundary tensor. Choose a compact boundary test detecting it. Every ${\mathcal{B}}_{f,ϵ}$ has a smooth kernel, hence zero charge at every configuration. The continuous charge pairing separates the proposed limit from all approximants.

The construction is a minimal comparison carrier, not an identification with every stage of IX-A. Its mechanism agrees with the boundary-sensitive nonconvergence explicitly recorded there CheungIXA. It changes the convergence hypothesis used in VII-C; it does not construct the missing sharp multiplication on the IX carrier.

This section uses one explicitly stated numerical input from the supplied VII-C/D sources. Its proof is not replaced by a citation to a general extension theorem.

VII-C/D use ${\eta }_{\mathrm{C}}=-\eta$ and ${\square }_{\mathrm{C}}=-\square$. The real field identification

gives

Their sharp insertion therefore pulls back to our ${\mathcal{B}}_f$. Their middle observable ${h_{00,\mathrm{C}}}^2$ pulls back to ${h_{00}}^2$. The graviton two-point tensor has two metric factors, so its displayed numerical tensor is unchanged. Both insertion signs in the triple occur twice.

Let

be understood first on product tests and then as the supplied joint distribution on $X=(M_{+}^{\circ }{)}^3$. The field mollifier is used on both slots of both insertions; the middle observable is unsmeared.

There are a nonzero positive residue $r$ and a common distribution $U_m$ such that

On each fixed compact, the error is bounded by the stated factor times a test seminorm of order $64$. Specifically,

There are nonnegative pairwise-disjoint $f,g,f^{\prime }$ for which this pairing is positive.

For a subtraction scale $\mu >0$,

For any two allowed fixed radial profiles,

These are numerical distributional statements, including strong convergence and the full reflected-diagonal intersection CheungVIIC,CheungVIID.

Only this declared input is needed for the following theorem.

There is no Hausdorff locally convex algebra over $\mathbb{C}[\hslash ]/({\hslash }^4)$ satisfying all of:

1. the regular Wick products of the displayed approximants and $W_g$ agree with the old products through ${\hslash }^3$;

2. each unchanged ${\mathcal{B}}_{f,ϵ}$ converges to an admitted sharp ${\mathcal{B}}_f$;

3. multiplication is hypocontinuous with respect to bounded sets;

4. extraction of the field-independent ${\hslash }^3$ coefficient is continuous and has finite complex values.

For a separated positive-residue test with coefficient $C>0$, the violating sequence has the quantitative lower bound

for all sufficiently small $ϵ$.

Proof. Choose the separated tests from Input theorem input:triangle. The approximant families converge by (Q2), hence are bounded. Hypocontinuity implies separate continuity and continuity along pairs of convergent sequences: split the difference of two products and use uniform continuity over the bounded family in the other slot.

Thus

converges. By (Q4), its displayed coefficient converges to a finite number.

By (Q1) it is the old numerical coefficient. Equation (eq:triangle-input) instead gives the stated logarithmic lower bound and divergence. This is a contradiction.

No adjoint or Ward identity can restore all four requirements: the contradiction occurs before either additional condition is used. The boundary topology of Theorem thm:bdcarrier avoids (Q2), exactly as declared.

The VII-D finite part cannot be the continuous value of the old triple under (Q1)–(Q4). A counterterm supported only on the ordinary observation diagonals cannot cancel its logarithmic residue on all tests.

Proof. Assigning the finite part only at the sharp endpoint leaves the divergent regular sequence unchanged. Subtracting it along that sequence changes the already prescribed products in (Q1).

For the second assertion, every ordinary-diagonal-supported distribution pairs to zero with the pairwise-disjoint test bank of (eq:triangle-residue), while $r$ has positive pairing there.

The operator monomial at this graph order is the identity: the three quadratic insertions have six legs, and the reflected triangle contracts all six. This identifies the numerical coefficient's place in the old word. It does not construct the other output coefficients or the next binary multiplication.

IX-B explicitly preserves this non-factorization boundary CheungIXB.

Suppose a proposed sharp product has defined bilinear maps

Then associativity is equivalent to

where

If two products agree below order $p$, a comparison $Z=\mathrm{id}+{\hslash }^{p}D(\mathrm{mod}{\hslash }^{p+1})$ must satisfy

Proof. Expand both associations coefficientwise and collect all terms with one occurrence of ${\mu }_n$; the remaining terms are the right-hand side of (eq:associativity-equations). Expanding $Z(F{\star }_{a}G)=ZF{\star }_{b}ZG$ at order $p$ gives (eq:comparison-equation).

These equations apply only when their maps and compositions are defined. The scalar identity ${u_{\mu }}^a-{u_{\mu }}^b=c_{ab}r$ neither defines $D$ nor verifies (eq:comparison-equation) on the generated sector.

A derivative-order bound also need not make the normalization space finite-dimensional: arbitrary smooth tangential coefficients multiplying a normal delta already give an infinite-dimensional space. This distinction is central in the supplied XV-F ambiguity discussion CheungXVF.

Introduce auxiliary smooth variables $w^{\mu },v^{\mu }$, and impose

For every smooth metric jet $(h,k)$, the constraints (eq:auxiliary-constraints) have exactly one solution. Projection from their solution graph to $(h,k)$ is bijective, and substituting that solution into

gives the original sharp ${\mathcal{O}}_f$.

The same statement holds for the prepared density equations, with their corresponding coefficients and zero data.

Proof. The first equation is a linear Cauchy problem with prescribed zero data. Its unique solution is $w={\mathsf{E}}_{0}Ch$. With that solution inserted, the second is another linear inhomogeneous Cauchy problem, whose unique solution is (eq:coordinate-coefficients). The observable equality is substitution into its formal inverse-map expansion. The prepared equations are handled in the same order.

Without the data in (eq:auxiliary-constraints), arbitrary homogeneous coordinate fields remain. Quantizing those fields would not be a quantization of the fixed Cauchy prescription.

At first order, write a linear auxiliary constraint as

The second term is the actual sharp distributional smearing. Its use below is a constraint test, not an assertion that it already lies in the old microcausal algebra.

Suppose an auxiliary Wick construction is defined on the displayed linear constraints and preserves their ordinary graph ideal as a quantum ideal, with no nonzero scalar identified with zero. Then its contractions must satisfy

as identities of their admitted smeared pairings.

For the sharp zero-Cauchy prescription and the fixed de Donder matrix,

Consequently an independent central $w$-sector cannot implement exact elimination while retaining the old $hh$ contractions.

Proof. For a linear field $h(b)$,

The first term belongs to the ordinary graph ideal. If that ideal is also a quantum ideal, the scalar contraction must vanish. This gives the first identity in (eq:forced-covariance). Reverse products give the transposed identity. Applying the same test to an auxiliary linear field gives the last identity.

The tensor contraction $C_x{H}_{hh}{C_y}^{\mathsf{T}}$ is a constant multiple of ${\eta }^{\mu \nu }\square {\Delta }_0^{+}$, hence zero before the two Cauchy solution maps are applied. This yields $H_{ww}=0$.

The mixed component is