Em01-E Complete-Contraction Decision Templates and Verification Certificates Fixed-kernel convergence, raw-prescription failure, and authorized characteristic extensions

We formulate and prove coefficient-level decision theorems for a fixed retained contraction whose crossed array is regularized in its actual eight relative variables. A complete finite power–logarithm expansion with a compact-uniform vanishing remainder determines raw convergence, fixed-test raw failure, and the mathematical finite part. The nonsummable coefficients and finite part are unique. The principal theorem does not require a homogeneous translation germ.

Three sufficient verification routes are distinguished: scale-integrability, direct complete-family expansion, and a logarithmically homogeneous translation-origin certificate. Localization and Taylor estimates connect explicit calculations to these routes. Physical identification, witness evaluation, reference agreement, subtraction authorization, normalization, and microlocal output are separate records.

An evaluated complete degenerate specialization of the retained seven-row architecture is calculated on arbitrary compact joint tests. Five rows vanish by its declared observations; both remaining internal rows and all three vertex types are retained. The apparent order ${\delta }^{-2}$ vanishes, a nonzero order ${\delta }^{-1}$ survives, and the subtracted family converges strongly with an order-six $O(\delta )$ remainder. A further normal-derivative cancellation is exposed in the complete inverse-scale coefficient. The singular coefficient and finite part have normal orders at most two and four, respectively, and wavefront sets contained in the conormal bundle of their smooth attachment. The finite part is independent of the specified even factorized profile class.

The specialization is not identified with the physical Einstein–scalar contraction. A physical identification ledger and a six-task verification protocol state what is still required for that application. No physical characteristic decision, authorized physical counterterm, or quantum admission is inferred from the mathematical example.

Figure fig:scope-Em01-E records that separation. Figure fig:split-Em01-E records one view of the same separation. Figure fig:chain-Em01-E records one view of the same separation. Figure fig:pair-Em01-E records one view of the same separation. Figure fig:gate-Em01-E records one view of the same separation. The drawings add no hypothesis.

of the mathematical result{ This paper contains proved decision templates and one evaluated complete mathematical specialization. It does not fill missing physical inputs with convenient substitutes.

A singular finite distribution is not itself an obstruction. Raw-prescription failure means that a nonzero complete coefficient multiplies a scale factor with no finite limit.}

The fixed-kernel problem is

The internal distribution is kept exact. Only the grouped crossed array is smoothed.

The supplied Em01-B and Em01-C excerpts describe positive-scale regularization, moderate growth, characteristic coordinates, and an exact translation representation Em01B,Em01C. The supplied project proposal attributes a homogeneous-origin route to Em01-D-v2. These are statements of project provenance: no unprovided predecessor proof is needed for the results below. All defining formulas and analytic arguments used here are stated in this manuscript.

The scale-integrability and homogeneous-origin methods are infrastructure, not claimed as new analytic discoveries. The contribution is their coefficient-level organization, the operational verification procedures, and the evaluated complete specialization.

The principal deliverables are:

1. the complete-expansion decision theorem (Theorem thm:decision);

2. the exact-family verification protocol and its local-to-global requirements (Sections sec:routes and sec:protocol);

3. the evaluated expansion, fixed detecting test, and conormal coefficient bounds (Section sec:application).

The ordinary multiplication theorem used here is sufficient, not a converse nonexistence theorem BDH (Theorem 6.1). Thus the appropriate conclusion from a verified cancellation configuration is:

The displayed wavefront configuration prevents direct application of the ordinary-product criterion; the complete contraction and any alternative definition require separate analysis.

The algebraic distinction between an open and a closed attachment pattern can be seen from

whereas

Whether the latter nonzero covectors actually belong to the specified kernels' wavefront sets depends on their prescriptions and endpoint orientations. This algebra is not a calculation of the complete interaction coefficient.

The analytic and identification records are parallel:

After these stages, reference agreement, authorized subtraction, normalization, overlaps, residual projection, and the required quantum carrier must still be checked.

Raw convergence, raw failure, and an authorized extension are not a mutually exclusive trichotomy. Raw failure and an independently authorized extension may coexist.

Use

Let $U⋐M^4$ be open. Let $E_{\mathrm{out}}$ be a fixed finite-dimensional complex vector space and set

All component pairings are complex-bilinear, with a fixed trivialization and the displayed flat coordinate densities.

The test space is

For $\mathrm{supp}\Phi \subset L⋐U$, define

No restriction is made to factorized tests, selected witness families, or normal annuli.

Strong convergence in ${\mathcal{T}}^{\prime }$ means uniform convergence on bounded subsets of $\mathcal{T}$. A bounded subset has common compact support and bounded derivative seminorms of every order.

When wavefront-constrained spaces are used, the normal topology means the strong distributional seminorms together with the localized Fourier seminorms outside the declared cone BDH (Section 3).

For local coordinate Fourier transforms, fix

where $w\cdot \xi$ is the coordinate covector pairing. Minkowski index raising is separately performed with $\eta$. An identification with another convention must record the corresponding sign conversion.

Let $I$ run over a fixed finite internal component set and let

For a smooth symmetric array ${H_I}^{\mu \nu }$ on ${\mathbb{R}}^8$, define

Fix a smooth observation $g$ and mass $m\ge 0$. Define

Its test action is

The three vertex types are kinetic, trace kinetic, and mass. The internal row $K_{9,\mathrm{mass}}$ and the crossed-vertex mass term are different entries.

A package is

with the following fixed data.

1. $K_{\mathrm{ret}}$ is an actual complete internal distribution, with tensor slots and component duality specified.

2. The grouped crossed array is

   $${C_I}^{\mu \nu }={H_{T,\gamma }}^{\mu \nu }{}_I\otimes H_{T,\varphi }-{H_{\gamma }}^{\mu \nu }{}_I\otimes H_{\varphi }.$$

   Its four input distributions, relative arguments, prescriptions, and tensor conventions are specified. No undeclared mixed products are introduced.

3. The vertex is (eq:vertex), including fixed $m,g$.

4. $\rho \in {C_c}^{\infty }({\mathbb{R}}^8)$ is real and has integral one. Nonnegativity is not required. The cutoff $\chi \in {C_c}^{\infty }({\mathbb{R}}^8)$ equals one near $\mathfrak{f}(\overline{U})$.

5. When a reference is available, $V_{\mathrm{ref}}\subset U$ is open and $t^{\circ }\in {\mathcal{D}}^{\prime }(V_{\mathrm{ref}};E_{\mathrm{out}})$ is specified. Any claimed agreement with it requires a proof, not merely its designation as a reference.

6. ${\mathcal{C}}_{\mathrm{allowed}}$ specifies permitted scale-dependent subtractions, finite additions, and normalization rules. These are not inferred from the existence of a finite part.

7. A requested output operation includes its spaces, support conditions, maps, residual bundle, and required closed wavefront cone ${\Gamma }_{\mathrm{out}}$.

Every entry is independent of $\delta$.

Set

This is the fixed-kernel family also denoted ${\mathsf{F}}_{\delta }(K_{\mathrm{ret}})$, with $C,\rho ,\chi$ held fixed.

For every $\delta >0$, $t_{\delta }$ is defined on all of $\mathcal{T}$. It is strongly smooth on compact positive-scale intervals.

It depends on the actual internal distribution, not on a presentation. In particular, reconstruction-null presentations give the zero positive-scale family.

Proof. The convolution of the compact distribution $\chi C$ with a compact smooth profile is smooth. Formula (eq:vertextest) therefore pairs each internal distribution with a smooth compact test.

On a compact positive-scale interval, the profiles and all scale derivatives have common compact support and bounded test seminorms. Their convolutions have locally bounded smooth seminorms. Local finite-order estimates for $K$ justify differentiation, uniformly on bounded joint test sets.

All operations start from the actual $K$. A zero reconstructed distribution has zero pairing with every resulting test.

The regularized mathematical family can exist when its identification with a proposed physical construction remains unproved. This distinction is part of the package, not an analytic defect.

A physical identification record must identify the source, propagators, observations, root operator, density derivative, tensor duality, and correction rules. A required relation such as

cannot be replaced by an independently convenient $F$ while retaining the same physical claim.

For each row below, a completed physical record must give:

1. an exact defining equation or component formula;

2. a manuscript identifier, exact version, and passage;

3. its proof status and source/operator domain;

4. the map identifying it with the mathematical entry.

For the intended physical graph, these rows are not completed by the supplied excerpts alone. For the mathematical application below, they are replaced by explicit mathematical definitions, and that replacement is stated openly.

The preparation materials contain excerpts labelled XI-A, XI-B, XI-C, X-G, and XVI-A through later XVI identifiers, with some claims depending on additional versions or unprovided sources. Here they are documentary context, not independently verified premises.

In particular, this paper does not assume that:

• a gauge-fixed free inverse establishes an invariant sourced Einstein jet;

• a classical tree construction establishes a closed time-ordered contraction;

• admission of a Wick insertion establishes admission of its insertion–vertex product;

• one witness contraction determines the complete kernel;

• a source modification changes its free-shell class;

• a selected two-edge sector is the full retained contraction.

Any future use of a predecessor theorem must first match its actual hypotheses, definitions, version, and operation domain.

Registration and a retrieval restatement are not proof checks. Conversely, the absence of registration is not a mathematical nonexistence theorem.

For one fixed package and profile,

where

The following requirements hold.

1. ${\mathcal{E}}_{-}\subset (-\infty ,0)$ is finite, with distinct real exponents. The exponent and logarithmic-degree inventory is one fixed finite inventory on $U$, independent of the compact test support.

2. Every $J_{\lambda },J_0$ belongs to ${\mathbb{N}}_0$.

3. The coefficients and $T$ are actual distributions in ${\mathcal{T}}^{\prime }$, not unverified formal expressions.

4. For every $L⋐U$, there are $m_L\in {\mathbb{N}}_0$, $C_L>0$, and ${\omega }_L(\delta )\ge 0$, tending to zero, such that

   $$|R_{\delta }(\Phi )|\le C_L{\omega }_L(\delta )\Vert \Phi {\Vert }_{m_L,L},\mathrm{supp}\Phi \subset L,$$

   for all sufficiently small scales, uniformly over these tests.

5. The calculation proves equality for the complete positive-scale family: every retained row, every vertex type, both terms of (eq:crossed), and every required localization sector are included.

Equal scalar scale factors have been collected.

Under Assumption ass:expansion:

1. $t_{\delta }-S_{\delta }\to T$ strongly in ${\mathcal{T}}^{\prime }$, with the compact-local rate (eq:remainder);

2. the raw family converges strongly exactly when all coefficients in (eq:singular) vanish;

3. if a complete nonsummable coefficient survives, there are one fixed compact joint test ${\Phi }_{\ast }$, an index $({\lambda }_{\ast },k_{\ast })$, and $c\ne 0$ such that

   $$t_{\delta }({\Phi }_{\ast })=c{\delta }^{{\lambda }_{\ast }}(\log \delta {)}^{k_{\ast }}(1+o(1)),$$

   where either ${\lambda }_{\ast }<0$, or ${\lambda }_{\ast }=0,k_{\ast }\ge 1$. Hence $|t_{\delta }({\Phi }_{\ast })|\to \infty$;

4. the nonsummable coefficients and $T$ are unique in this finite real power–logarithm class and intrinsic to the actual positive-scale family.

Proof. Every bounded test set has common compact support and bounded $\Vert \cdot {\Vert }_{m_L,L}$. Thus (eq:remainder) proves (i), and proves raw convergence when all nonsummable coefficients vanish.

Otherwise choose the smallest exponent having a nonzero coefficient, treating the degree-zero logarithms as exponent zero. Within that exponent choose the largest logarithmic degree with a nonzero coefficient. Its distribution is detected by one fixed compact test. Higher exponents are negligible relative to its scalar factor, since positive powers dominate every fixed power of a logarithm. Lower logarithmic degrees at the same exponent are also negligible. The finite term and remainder are negligible. This proves (eq:dominant), and excludes raw convergence even in the weak distributional sense.

For uniqueness, subtract two expansions and combine their finite inventories. A nonzero nonsummable difference would have an unbounded fixed pairing, while the difference of the finite terms and remainders has a finite limit. Thus all nonsummable differences vanish. Taking the limit identifies the finite distributions.

If all complete nonsummable coefficients vanish, then

This alone gives neither profile independence, global overlap compatibility, a required wavefront cone, nor quantum admission.

A surviving complete nonsummable coefficient proves failure of the specified unmodified family on one fixed test. It does not prove failure of every extension or every other construction.

Presentations of the same actual internal distribution have the same extracted coefficients and finite part whenever Assumption ass:expansion holds. A reconstruction-null presentation has all these coefficients equal to zero.

Proof. Their positive-scale families agree by Proposition prop:positive. Apply uniqueness.

A finite limit $a(\tau ){\delta }_N$, or a finite transverse derivative of such a distribution, is compatible with raw distributional convergence. A singular coefficient is an obstruction here only when it is nonzero and multiplied by one of the nonsummable scale factors in $S_{\delta }$. The finite part itself may be singular.

The routes below are alternatives. They do not classify every possible scale behavior. Failure to verify one route does not prove failure of the contraction.

Suppose for every $L⋐U$,

with $b_L\ge 0$. Then $t_{\delta }\to t_0$ strongly for some distribution $t_0$, and

Proof. Positive-scale smoothness gives

The estimate makes the pairings Cauchy uniformly on bounded test sets. A fixed positive-scale member plus the integrable increments supplies a finite-order local bound for the limit. It is therefore a distribution. Letting ${\delta }_1\downarrow 0$ gives the tail estimate.

This is the scale criterion described in Em01-C Em01C. It can be preferable to coefficient extraction when complete cancellation is estimated directly.

Put ${\delta }_n=2^{-n}{\delta }_0$. Suppose for each compact $L$ there are $e_n,h_n\ge 0$ such that

Then the full family converges strongly. On that scale interval, its error is bounded by

Proof. Sum the dyadic increments. Their finite-order bounds give a distributional limit and uniform convergence on bounded test sets along the dyadic sequence. The interval estimate connects every intervening scale to that sequence.

Dyadic convergence without interval control is not convergence of the full family.

Route B verifies Assumption ass:expansion directly from an exact integral or defined distributional representation. Changes of variables, derivative transposition, and tail control must concern the actual smoothing.

Fix $L⋐U$. Suppose the exact localized complete family has a finite representation

where ${\beta }_{\ell }\in \mathbb{R}$ and $k_{\ell }\in {\mathbb{N}}_0$. Assume:

1. each $D_{\ell }$ is a fixed compact integration domain, $a_{\ell }$ is integrable, and the $x_{\ell ,i}$ are bounded;

2. every $w_{\ell ,i}>0$;

3. $F_{\ell }[\Phi ]$ is linear in $\Phi$, smooth in its last variables on neighborhoods containing the Taylor segments, and its derivatives through order $N_{\ell }+1$ are uniformly bounded by $C_L\Vert \Phi {\Vert }_{m_L,L}$;

4. $${\beta }_{\ell }+w_{\ell ,\min }(N_{\ell }+1)>0,w_{\ell ,\min }=\underset{i}{\min }{w}_{\ell ,i}.$$

Taylor expansion gives, on this fixed support, a finite power–logarithm expansion with continuous coefficient functionals and a remainder $C_L{\delta }^{a_L}\Vert \Phi {\Vert }_{m_L,L}$, for some $a_L>0$.

A global expansion additionally requires compatible local coefficient distributions and one common finite inventory.

Proof. The Taylor polynomial is

Its uniform remainder is bounded by

Integration against $a_{\ell }$ preserves the bound. The prefactor leaves a positive remainder power, with at most a fixed logarithm.

Collect negative exponents and degree-zero logarithms into $S_{\delta }$, degree-zero constants into $T$, and positive powers into the remainder. A smaller positive power absorbs every finite logarithmic factor. The test-jet bounds give coefficient continuity on the fixed-support test space.

The lemma does not justify (eq:weightedrep); obtaining that exact representation is a separate task. It also does not control omitted sectors.

For a compact joint test, define compact relative distributions

Their pushforwards are defined because the distributions being pushed forward are compactly supported. Set

For $\mathfrak{d}(x,v)=x-v$, put

The compact distribution ${\Omega }_{\Phi }$ exists, and

For

one has

These are complete-family identities and respect reconstruction.

Proof. Transposing the derivatives in (eq:vertextest) gives

The kinetic sign becomes $+i$, the trace-kinetic sign $-i/2$, and the mass sign remains $-i{m}^2/2$. Symmetry accounts for the first-line symmetrization.

Substitute $H={\rho }_{\delta }\ast (\chi C)$. The tensor pairing is against ${\rho }_{\delta }(x-v)$. Both tensor factors are compactly supported, so the pushforward in (eq:Omega) is defined and yields (eq:translation).

Differentiation gives $\delta {\partial }_{\delta }{\rho }_{\delta }={\theta }_{\delta }$. Integration of $-\mathrm{div}(h\rho )$ gives $\int \theta =0$. Proposition prop:positive justifies the scale identity. Every construction is linear in the actual internal distribution.

For a distribution $H$ on ${\mathbb{R}}^8$, set

A chain $H_0,\dots ,H_J$ of real degree $\lambda$ satisfies

The chain consists of distributions defined at the origin as well as away from it.

Suppose, on a common neighborhood of zero for tests supported in each fixed $L$,

Use one common finite inventory on $U$. Assume:

1. the $H_{\lambda ,j}[\Phi ]$ are linear, with local finite-order bounds in the joint-test and translation-test arguments, and form the displayed homogeneous chains;

2. $W_{\Phi }$ is a continuous function near zero, linear in $\Phi$, and for some $a_L>0$,

   $$|W_{\Phi }(0)|+\underset{0<|h|<r_L}{\sup }\frac{|W_{\Phi }(h)-W_{\Phi }(0)|}{|h{|}^{a_L}}\le C_L\Vert \Phi {\Vert }_{m_L,L}.$$

Then Assumption ass:expansion follows with

and

where the degree-zero term is omitted when absent. The family remainder is $O_L({\delta }^{a_L})$.

A separately calculated logarithmic scale derivative of the subtracted family has the same power bound.

Proof. For small scales the profile samples only the specified germ. Each chain contributes its exact finite power–logarithm polynomial. The continuous remainder contributes

The Hölder estimate bounds the latter integral.

For the derivative statement use (eq:scaleprofile), not differentiation of a remainder inequality. The chain terms differentiate their exact polynomials, while the remaining term is

It has the same bound since $\int \theta =0$.

This route is stronger than the principal family-level hypothesis. The application below verifies Route B directly; no homogeneous-origin certificate is claimed for it.

For

and an orthonormal spatial transverse frame $E_A(\mathbf{n})$, use

Work on a sphere subchart with positive lower bounds on $\alpha ,\beta -\alpha ,\beta ,\beta -u$. The aligned-null locus is

The exact cone equations and density are

For $s=b-a$, the inverse has

For $d=y-a$,

These formulas prove the local coordinate property. The null-frame identities prove (eq:cones). The two endpoint Jacobians are $2(\beta -u{)}^2$ and $2$; the full map is block triangular. This proves (eq:geometrydensity) and compact-local derivative bounds. The convention agrees with the supplied Em01-C coordinate excerpt Em01C.

Strict separation of $a,y,b$ does not remove $z=a$. The supported incidence must remain in the complete calculation.

Write

For the true displacement $\delta (H_1,H_2)$,

In particular,

The sign must be converted when the second cone is written in terms of $b-y$.

The suggested core

is not a complete localization proof. It supplies no common scaling for $r$, no tail estimate, and no physical asymptotic coefficient by itself.

Nor is this weighted scaling automatically Dang's Euler scaling. An Euler field acts as the identity on the normal ideal modulo its square Dang (Section 1). On the $(u,v,\xi )$ slice the weighted generator

sends ${\xi }^A$ to ${\xi }^A/2$, not to ${\xi }^A$ modulo quadratic terms. Applying an ordinary Euler-scaling extension theorem therefore requires an additional transfer argument.

1: keep the full inventory. Retain

with both crossed terms. The row definitions are printed in Appendix app:kernels.

2: cover all sectors meeting the patch. Besides the rescaled core, account for the supported response incidence, the $b-z$ cone in $K_{47}$, the $a-z$ and $z-a$ cone tips in the two $K_5$ placements, and the complementary sectors on each compact joint support. A core or annular calculation is not a complete estimate.

3: standardize conventions. Use one tensor pairing, derivative placement, density, relative-slot sign, and coefficient-distribution representation. For example,

does not mean differentiation of the whole incidence product.

4: calculate and collect before deciding. Produce exact local representations, extract candidate orders, transpose to one common test-jet convention, and sum all row–vertex contributions at each order. A nonzero branch coefficient is not a complete obstruction.

5: certify the complete remainder. State one finite derivative order on each fixed compact support. Include the tails and any partition derivatives. A leading algebraic zero without a vanishing complete remainder does not prove convergence.

Let $L⋐U$, and let a fixed finite smooth partition ${{\sum }_{\ell =1}}^N{\zeta }_{\ell }=1$ hold near $L$. Suppose each exact localized complete family $t_{\delta }({\zeta }_{\ell }\Phi )$ has an expansion with a compact-uniform vanishing finite-order remainder.

Their sum, with equal factors collected, gives such an expansion on tests supported in $L$. Its inventory is finite on this support.

Proof. Sum the exact localized identities. Multiplication by the fixed cutoffs is bounded in every required test seminorm. Take the maximum derivative order and sum the finitely many remainder envelopes. The resulting envelope tends to zero.

Let an open cover of $U$ carry verified expansions of the restricted actual family. Suppose all local expansions use one common finite power–logarithm inventory, allowing zero coefficients, and have compact-local bounds of the form (eq:remainder).

Then the local coefficients and finite parts agree on overlaps, define distributions on $U$, and give Assumption ass:expansion globally.

Proof. On an overlap, uniqueness in Theorem thm:decision, applied to compact tests there, identifies the coefficient restrictions. The distributions therefore glue.

For any compact $L$, select a finite subordinate localization near $L$. Lemma lem:assembly gives the local remainder bound for the glued expansion. The common inventory is already finite and independent of $L$.

Overlap uniqueness does not manufacture a common global finite inventory from inventories that are only locally finite. A scale-dependent partition also requires separate uniform estimates.

A verification record must distinguish:

1. Fixed-test certificate: one ${\Phi }_{\ast }\in \mathcal{T}$, independent of scale, with a proved dominant asymptotic such as (eq:dominant).

2. Bounded-set certificate: a specified bounded set $B\subset \mathcal{T}$ and uniform control of ${\sup }_{\Phi \in B}|(t_{\delta }-T)(\Phi )|$, or a proved failure of that control.

3. Moving-witness calculation: a family ${\Phi }_{\lambda }$, with its support and derivative growth recorded, and the behavior of its pairings.

For example, if

then its derivatives have polynomial frequency growth. For nonzero $b$, differentiation in a direction pairing nontrivially with $\xi$ shows that this is not a bounded test family.

The distinction is not cosmetic. On $\mathbb{R}$, take $A={\partial }_x{\delta }_{\mathrm{D}}$, and $\psi =1$ near zero. The constant family $t_{\delta }=A$ converges strongly, but

Thus growth on an unbounded moving test family can coexist with strong convergence of the distributional family.

Witness identities can check signs and selected coefficients. A specified partial contraction is not automatically injective on kernels: already a single nonzero test functional on ${C_c}^{\infty }$ has a nontrivial kernel. Adding a tensor product with a function in that kernel preserves the partial evaluation while changing the full kernel.

These tasks do not add assumptions to the decision theorem. They specify how to supply its existing hypotheses.

1. Logical certificate audit.

   Deliverable: the table of Section sec:introduction, with each claimed conclusion assigned to its actual certificate.

   Limit: a failed sufficient product test is not a nonexistence theorem.

2. Physical identification freeze.

   Deliverable: the completed ledger of Section sec:identification, including exact formulas, versions, domains, and proof status.

   Acceptance condition: no required physical entry is replaced by a convenient independent mathematical choice.

3. Narrow but complete localization.

   Deliverable: an exact localized representation on arbitrary joint tests in one strictly separated characteristic patch.

   Acceptance condition: every sector meeting the patch is included, including the incidence $z=a$.

4. Complete analytic calculation.

   Deliverable: a complete expansion with a compact-uniform vanishing remainder, or a complete scale-integrability estimate.

   Limit: Route C is not required if Route A or B suffices; a branch calculation or moderate-growth bound is not a decision.

5. Correct test certificate.

   Deliverable: if divergence survives, one fixed compact joint test with a justified dominant asymptotic. Record any bounded-set or moving-witness claims separately.

   Limit: selected witness agreement is not full-kernel reconstruction.

6. Extension and admission audit.

   Deliverable: reference agreement, subtraction support, wavefront bounds, authorized corrections, normalization, overlaps, and residual projection.

   Limit: a mathematical finite part is not automatically an admitted quantum coefficient.

The next physical milestone is deliberately narrow:

For one physically frozen package on one strictly separated characteristic patch, derive an exact complete representation and prove raw convergence or fixed-test raw divergence with a compact-uniform remainder.

The supplied excerpts do not complete that milestone. The application below demonstrates the procedure for its own explicit mathematical inputs.

This section decides one complete family. It does not change its density map to isolate a difference. Its degeneracy is explicit: five internal rows and the second crossed tensor product are identically zero.

Choose a symmetric-tensor basis with $e_0=\eta$, completed by the six off-diagonal symmetric tensors and the three trace-free diagonal tensors

Use its algebraic dual. The residual fiber is ${\mathbb{C}}^{10}$.

Fix

Then

Let $U$ be a sufficiently small relatively compact neighborhood of

Require disjoint closures of the $b$- and $y$-projections, the time inequalities $b^0>y^0>a^0$, $b^0>z^0$, and, in the fixed-frame coordinates below,

Thus $A<B$.

Choose fixed compact smooth ${\zeta }_b,{\zeta }_y$, equal to one near the respective endpoint projections, with disjoint supports. Set

Hence $F(b)=b^0$, $g(b)=1$, and $g(y)=y^0$ on the relevant neighborhoods.

Define the quadratic source

Its derivative is

The source array used here encodes this derivative, not its half-polarization. Its sole nonzero coefficient is

with both derivative multiindices zero.

Choose the even normalized bump

and set

For $h=A\ell +uk+X$, $X\in {\mathbb{R}}^2$, define

Since ${\mathrm{d}}^{4}h=2\mathrm{d}A\mathrm{d}u{\mathrm{d}}^{2}X$, these profiles have integral one.

The crossed inputs are

Fix any compact $\chi =1$ near $\mathfrak{f}(\overline{U})$.

Define the smooth attachment and reference domain by

The reference is zero, as proved below. Set ${\mathcal{C}}_{\mathrm{allowed}}=\{0\}$: no nonzero scale-dependent subtraction or finite addition is authorized for this package. No restrictive quantum output condition is imposed.

This is not the physical Einstein–scalar source or crossed prescription. In particular, its independently specified $F(b)=b^0$ is not ${E_m}^{\mathrm{A}}f$, since $f=0$. The specialization is complete for its declared data, but does not exercise a nonzero seven-row physical graph.

The tensor maps of Appendix app:kernels give

Put

The source response is $T^{0,0,0}=-D$. Therefore

and all other components vanish. The first five internal rows vanish exactly because $f=0$. Thus

The actual convolution has the exact local value

Indeed the $\kappa$-marginal of ${\rho }_4$ is $p$. The product profile gives the two four-variable convolutions. For sufficiently small scales, the cutoff is invisible near $\mathfrak{f}(\overline{U})$, including for derivatives.

Set

The kinetic crossed term is $+{\ell }^{\nu }{A}_{\delta }$, the trace-kinetic term is $-2{\ell }^{\nu }{A}_{\delta }$, and the trace is $4E_{\delta }$.

The $\ell$-derivative annihilates both profile arguments. Here $K$ is independent of $y$, and ${\ell }^{\nu }{\partial }_{\nu }g(y)=1$. Consequently

All six nonzero row–vertex cells are:

The other fifteen cells vanish by the fixed data $f=0$. The second crossed term vanishes by (eq:applines). Neither vanishing is an asymptotic omission.

Expanding the $b^0$-derivatives gives

Use the fixed-frame coordinates