Em01-D Conditional Fixed-Kernel Characteristic Asymptotics Complete-sum coefficient extraction, controlled remainders, and the distinction between raw limits and authorized extensions

We formulate the characteristic problem for one fixed retained Einstein–scalar kernel, keeping all seven internal bookkeeping rows, all three mixed-vertex terms, and the grouped crossed-line difference. The actual eight-relative-variable smoothing is represented exactly by a compact, test-dependent translation distribution.

The main result is a conditional decision theorem. A specified finite logarithmically homogeneous origin germ of the complete translation distribution, with a uniformly Hölder continuous remainder, determines every nonsummable coefficient and the finite part. It yields quantitative bounds for both the family remainder and its logarithmic scale derivative. The raw family converges exactly when its complete nonsummable coefficients vanish; otherwise one fixed compact joint test detects unboundedness. Coefficient uniqueness makes the conclusion independent of internal presentations. Support, profile independence, admissible subtraction, and normalization are treated separately.

We reproduce the finite source rule and density derivative, derive the shifted characteristic equations with their correct second-slot sign, and retain the full coordinate density. An appendix evaluates an exact nonphysical convolution diagnostic, including all inverse powers and an order-eight remainder.

The supplied materials do not instantiate the designated physical crossed covariances or verify the complete origin-germ hypothesis for them. Thus the conditional theorem is proved here, but no evaluated convergence, failure, or admissible-extension decision for the designated physical graph is asserted.

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

Figure fig:scope-Em01-D records that separation. The drawing adds no hypothesis.

of the revision{ The primary object is the named fixed-kernel family, not a map on every internal presentation. The physical input identification and the complete origin-germ estimate remain explicit hypotheses.

The nonphysical diagnostic is confined to Appendix app:diagnostic; its assumptions and powers are not used as physical inputs or physical asymptotic orders.}

For one fixed input package, the object is

Here $U$ is a relatively compact joint-coordinate patch, $\rho \in {C_c}^{\infty }({\mathbb{R}}^8)$ has integral one, and the internal kernel is kept exact.

The target is a statement on every compact joint test, including tests meeting the aligned-null locus $N$. No restriction to factorized witnesses or normal annuli is made.

The exact translation distribution ${\Omega }_{\Phi }$ is constructed in Section sec:translation. Its complete origin-germ certificate is defined in Section sec:certificate. The certificate is a condition on the actual complete distribution, not on a selected branch or a polynomial-degree presentation.

Fix the data of Sections sec:inputs–sec:contraction and a normalized compact profile $\rho$. Suppose the complete translation distribution has the origin-germ certificate of Assumption ass:germ.

Then the family has the expansion

where

Absent exponents contribute no terms. All displayed coefficients are determined by

If $0$ is absent, ${A_{0,0}}^{\rho }=0$.

For every compact $L⋐U$, there are $m_L,C_L>0$ and $a_L\in (0,1]$ such that, for sufficiently small scales and $\mathrm{supp}\Phi \subset L$,

Consequently:

1. the specified raw family converges strongly if and only if every coefficient in (eq:singularsum) is zero;

2. if one such coefficient is nonzero, one fixed compact joint test detects an unbounded raw pairing;

3. subtraction of (eq:singularsum) always gives the mathematical finite part $T^{\rho }$ under the certificate, but constitutes an admissible extension only under the additional conditions of Section sec:admissibility.

The proof does not require a continuous characteristic map on the whole internal module.

Theorem thm:main is proved in Section sec:proofmain. Its applicability to the designated physical inputs is not assumed.

Raw failure and existence of a separately authorized extension can occur together. They concern different constructions.

A mathematical input package consists of

Every entry is held fixed as $\delta \downarrow 0$.

The four crossed entries are distributions, with the component and relative-argument conventions printed below. The functions $f,g,F$ are fixed smooth functions; for the designated scalar root its identification is

with the designated advanced Klein–Gordon operator. The estimates proved here use the fixed smooth function $F$, not an alternative choice of its propagator.

The source switch $\sigma$ is smooth, with $\sigma =1$ on $x^0>\epsilon$, and $\epsilon >0$ is fixed. For the local analysis, observations and their derivatives are bounded on all compact sets used.

The supplied materials do not provide a complete identification of the first three rows or the correction package. The theorem therefore applies to a fixed mathematical package (eq:package), and applies to the physical graph only after those identifications and its analytic certificate are proved.

Use Minkowski spacetime

All distributions have complex-bilinear pairings and use flat coordinate densities unless a density conversion is printed.

Let $e_I$ be a basis of covariant symmetric tensors, with algebraic dual $e^I$:

The dual basis is not obtained merely by metric index raising.

One concrete basis uses $I=(i,j)$, $0\le i\le j\le 3$:

Thus off-diagonal component factors are fixed by the duality, not inserted independently.

Define

The notation ${\mathsf{A}}^{\mu \nu }{}_L$ means evaluation on $e_L$; similarly ${\mathsf{b}}_L={\mathsf{b}}^L$. These role labels do not instruct another fiber-metric multiplication.

For tests supported in $L⋐U$, put

Strong distributional convergence means uniform convergence on bounded test sets, which have common compact support and uniform bounds on all derivatives.

For a formal metric $q_s=\eta +s\gamma$, fix

and $R(q)=q^{\mu \nu }{R}_{\mu \nu }(q)$. Declare

No integration-by-parts replacement of the density is made before this variation.

Let $X_{I,\gamma }$, $|\gamma |\le 2$, denote ordinary, not factorial-normalized, derivative jets:

Mixed derivatives use their common multiindex.

The coordinate Euler components are

Let

The propagated source components are

Thus the coordinate Euler gradient is not substituted for the metric-valued propagated source.

The coefficients in

are given by the finite formula

They satisfy

They vanish for $|\alpha |+|\beta |>2$, and transform covariantly under a simultaneous change of basis and dual basis.

Proof. The coefficient ${\ell }_3$ is cubic in field jets, and each monomial has total derivative weight two. Euler differentiation removes one field factor. The derivatives transferred by the Euler operator restore exactly the derivative weight removed from that factor; they may also differentiate $\sigma$. The resulting source is quadratic and has total derivative weight two when switch derivatives are included. In particular the field derivative weight is at most two.

For any quadratic polynomial $\mathcal{J}$,

Expanding this identity in independent ordinary jet coordinates gives (eq:sourcehessian). Commutativity of polynomial differentiation gives slot symmetry and all repeated-variable factors. The underlying bilinear tensor map is unchanged by a consistent component transformation.

On a neighborhood where $\sigma =1$, the full source and its coefficient jets coincide with the unweighted source. This is a local equality, not a gain in smoothing order.

Write $t_r={\mathrm{tr}}_{\eta }r$ and $G:h=G_{\mu \nu }{h}^{\mu \nu }$. For the declared density

the quadratic coefficient is

Its directional derivative is

If the designated ${\dot{a}}_2[G;h]$ is identified with this derivative, then

For every symmetric $h$ and scalar $\psi$,

Proof. The inverse metric has coefficients ${\eta }^{\mu \nu },-r^{\mu \nu },r^{\mu }{}_{\lambda }{r}^{\lambda \nu }$. The determinant square root has coefficients

Their product gives (eq:a2), and ordinary differentiation gives (eq:Da2).

For $G=\psi \eta$, the first two terms in (eq:Da2) give $2\psi h^{\mu \nu }$; the next terms give $-2\psi h^{\mu \nu }-\psi t_h{\eta }^{\mu \nu }/2$, and the last term gives $\psi t_h{\eta }^{\mu \nu }/2$. The sum is zero. The quadratic identity follows directly.

The symmetric polarization

satisfies $D{a}_2[G][h]=2a_2[G,h]$. This factor must be settled by the physical identification. The density derivative is not assigned to the physical graph merely because it is a familiar formula.

Work on

with separated endpoint regions and

The source switch is one throughout the retained variables.

Define

Put

The fixed retained kernel is

where

These formulas adopt the retained architecture supplied in Em01-B and Em01-C Em01B,Em01C. The seven rows are bookkeeping rows, not seven new interaction branches.

All displayed internal distributions exist. For $D=R,B$,

Every coefficient derivative generated by this formula is retained. For $L⋐W$,

Proof. Move the derivatives on the two incidence deltas to their external variables. The undifferentiated integration is

which gives (eq:response).

The nonsupported seeds are tensor-product pullbacks under the invertible linear maps

The source incidence is the differentiated pullback under the submersion $(v,a,z)\mapsto (v-z,v-a)$. The response integrations are locally proper because the incidence forces $v=a=z$ on compact external supports.

Both $R$ and $B$ are locally finite measures. Each row is a finite sum of at most four derivatives of such tensor-product seeds, multiplied by smooth coefficients. Transferring derivatives to coefficient–test products proves the estimate.

Declare

Both tensor products exist. Their difference is kept grouped. No undeclared mixed products are introduced.

The relative map is

For smooth symmetric $H$, define

The complete vertex is

Its arbitrary-test action is

Taking the product of these three types with the seven rows in (eq:Ksum) is the complete branch–vertex inventory. The internal $K_{9,\mathrm{mass}}$ and crossed-vertex mass term are distinct and both remain present.

Fix $U⋐W$ and $\chi \in {C_c}^{\infty }({\mathbb{R}}^8)$ equal to one near $\mathfrak{f}(\overline{U})$. Then (eq:target) is defined at every positive scale.

Let

The aligned-null locus is

where ${\ell }_{\mathbf{n}}=(1,\mathbf{n})$, $\mathbf{n}\in S^2$.

On a sphere chart choose an orthonormal transverse frame $E_A=(0,{\mathbf{e}}_A)$, and put $k_{\mathbf{n}}=(1,-\mathbf{n})$. Use

These are coordinates near a retained point of $N$, with

Writing $d=\beta -\alpha >0$, one has

The ambient flat density is

Along $N$, the three cone differentials have rank two. Throughout the chart,

Proof. For $s=b-a$, the inverse is

For $x=y-a$,

Also $r=z-a$. These formulas prove invertibility.

The null-frame identities give (eq:cones). The $(\beta ,u)$ Jacobian contributes $2(\beta -u{)}^2$, and the $(\alpha ,v,\xi )$ determinant contributes two, proving (eq:density). Differentiating (eq:cones) at $N$ gives the rank statement. Their divided sum gives (eq:defect).

The convolution is

Its displacement belongs to ${\mathbb{R}}^4\times {\mathbb{R}}^4$, not to a space of independently regularized cone functions.

Write a displacement as $\delta (H_1,H_2)$, where

With $x_1=y-a$, $x_2=y-b$, the shifted equations are

On the bounded rescaled core

their divided forms are

The coordinate density in $(u,v,\xi )$ contributes ${\delta }^3$, with its full smooth factor $4(\beta -\delta U{)}^2$.

All these algebraic formulas and their required derivatives are uniform on compact positive tangential parameter ranges and bounded rescaled cores.

Proof. In (eq:coordinates),

Subtracting $\delta H_2$ and taking the square gives (eq:shift2); the first equation is analogous. Substitution gives the divided equations.

The two longitudinal normal variables contribute ${\delta }^2$, and the two transverse variables contribute $\delta$. Uniformity follows directly from the polynomial expressions and the bounded coordinate transformations.

In invariant notation the second sign is

It must not be replaced by the first-slot sign.

The bounded core above does not establish a finite sector cover for the physical boundary values. In particular:

• supported rows impose $r=0$;

• $K_{47}$ also contains the cone of $b-z$;

• the two $K_5$ rows contain the cone-tip factors $R(a-z)$ and $R(z-a)$.

No common physical weight for $r$, no complete sector decomposition, and no physical scaling degree follows solely from Proposition prop:shifted.

Nor does the convolution permit pairing $K$ with every individual translated singular crossed factor. The exact distributional representation used next avoids that additional product assumption.

For $K=K_{\mathrm{ret}}$ and a compact joint test $\Phi$, define

The pre-pushforward distributions have compact support, so these pushforwards are defined by transposition.

Set

Let $\mathfrak{d}(x,v)=x-v$, and define

The compact distribution ${\Omega }_{\Phi }$ is well defined, and

All seven rows and all three vertex types are retained.

If $\chi C$ has order at most $p$, then locally

on a fixed compact neighborhood of its support.

Every equality of actual internal distributions is respected. In particular, a reconstruction-null presentation gives ${\Omega }_{\Phi }=0$ for every test.

Proof. Transposing (eq:vertextest) for a smooth symmetric array $H$ gives

The kinetic test sign $-i$ becomes $+i$ on transposition. The trace-kinetic sign $+i/2$ becomes $-i/2$. The mass sign remains $-i{m}^2/2$. Symmetry of $H$ gives the first-line symmetrization in (eq:Lambda).

Insert $H={\rho }_{\delta }\ast (\chi C)$. The resulting tensor pairing is

The factors are compact, so a compact cutoff can be inserted near their product support. Pushforward by $\mathfrak{d}$ gives (eq:exacttranslation).

The internal order-four estimate gives order at most five for $\Lambda [\Phi ]$, with a factor $\Vert \Phi {\Vert }_5$. Combining at most five $x$-derivatives and $p$$v$-derivatives on $\psi (x-v)$ gives (eq:Omegaorder).

Every operation starts from the actual $K$ and is linear. A zero reconstructed distribution therefore remains zero.

Put

Then

and

The family is strongly smooth on compact positive-scale intervals.

Proof. Differentiate the compact profile in its test-function topology. On a compact positive-scale interval, all profiles and scale derivatives have common support and uniformly bounded seminorms. Estimate (eq:Omegaorder) justifies differentiation uniformly on bounded joint test sets.

This representation concerns the complete contraction before any conclusion about its origin behavior.

For $u\in {\mathcal{D}}^{\prime }({\mathbb{R}}^8)$ and $s>0$, define

In particular,

This convention fixes the signs and powers in the coefficient calculation.

A chain $H_0,\dots ,H_J\in {\mathcal{D}}^{\prime }({\mathbb{R}}^8)$ of degree $\lambda \in \mathbb{R}$ satisfies

The chain is defined as distributions, including at the origin. A merely off-origin homogeneous expression is not enough.

For the fixed complete translation map $\Phi \mapsto {\Omega }_{\Phi }$, assume:

1. A finite set $\mathcal{E}\subset (-\infty ,0]$ of distinct real degrees and integers $J_{\lambda }\ge 0$ is specified.

2. For each $\lambda$, linear maps

   $$H_{\lambda ,j}:\mathcal{D}(U;V_{\mathrm{out}}^{\ast })⟶{\mathcal{D}}^{\prime }({\mathbb{R}}^8),0\le j\le J_{\lambda },$$

   give chains satisfying (eq:chain). For every compact joint support $L$ and compact relative support $B$, there are finite orders $m,n$ and a constant such that

   $$|H_{\lambda ,j}[\Phi ](\psi )|\le C_{L,B}\Vert \Phi {\Vert }_m\Vert \psi {\Vert }_n,\mathrm{supp}\Phi \subset L,\mathrm{supp}\psi \subset B.$$

3. For every compact $L⋐U$, a radius $r_L>0$ and linear continuous-function representatives $W_{\Phi }$ on $B(0,r_L)$ are specified such that

   $${\Omega }_{\Phi }=\sum _{\lambda \in \mathcal{E}}{H}_{\lambda ,J_{\lambda }}[\Phi ]+W_{\Phi }\text{on }B(0,r_L),\mathrm{supp}\Phi \subset L.$$

4. For some $a_L\in (0,1]$, finite order $m_L$, and constant $C_L$,

   $$\begin{array}{rl}|W_{\Phi }(0)|& \le C_L\Vert \Phi {\Vert }_{m_L},\\ |W_{\Phi }(h)-W_{\Phi }(0)|& \le C_L|h{|}^{a_L}\Vert \Phi {\Vert }_{m_L},|h|<r_L.\end{array}$$

The certificate concerns the entire ${\Omega }_{\Phi }$, including all sectors and all joint tests. Its existence is not inferred from the cone rank, the source rule, or moderate growth.

The certificate is sufficient, not asserted to be necessary. It supplies one concrete route to a decision without assuming a power–logarithm expansion solely for convenience.

Its maps must be computed from the fixed package when an evaluated application is claimed. Formula (eq:maincoeff) is coefficient extraction from that computed germ; it is not itself a physical evaluation of the germ.

If a rowwise certificate has independently been proved, rows can be collected into one common distributional germ:

Missing chain entries are handled by first collecting the full distribution of each degree and forming its chain.

The conclusion depends on the resulting complete ${A_{\lambda ,k}}^{\rho }$, not on the nonvanishing of one $H_{\lambda ,j;b,v}$. In particular the two tensor products in (eq:C) remain part of the complete summation.

For a sufficiently small scale, $\delta \mathrm{supp}\rho \subset B(0,r_L)$. Using (eq:germ) and (eq:chain),