Em01-C Characteristic Normal Forms and Complete-Sum Scale Reduction Explicit source conventions, exact local geometry, and scalar singular asymptotics

We study a local regularization of the complete retained Einstein–scalar contraction while keeping its internal kernel exact. The aligned-null locus admits explicit normal coordinates in which the three cone functions have differential rank two along that locus. Their restriction to the supported response incidence has rank one, with a definite quadratic transverse defect. We compute the coordinate inverse, ambient density, and a weighted dilation adapted to these equations.

The density-before-variation source has a finite symbolic specification. An explicit tensor-to-component conversion and a jet-polynomial Hessian determine all source coefficients without a printed component list. On the retained chronological patch, the source-ramp terms disappear by locality. A displayed unreduced coordinate-response expression also reduces exactly by causal support and the existing retarded convolution.

Transposition of the complete mixed vertex produces a compact, test-dependent distribution in a relative translation variable. The regularized contraction, dyadic increments, and logarithmic scale derivatives are exactly its evaluations on specified compact profiles. This construction retains every internal row and vertex term and respects all reconstruction identities.

For a separate scalar cone regularization, we compute an order-zero limit with an order-one convergence rate. Its response-normal derivative has an explicit inverse-scale coefficient, an explicit finite distribution, and an order-one remainder. A single fixed test detects divergence whenever that coefficient is nonzero.

The scalar regularization differs from the physical eight-relative-variable smoothing. The complete physical origin coefficient and its controlled remainder remain unevaluated. Accordingly, the results are geometric, symbolic, and analytic reductions, not a convergence or failure theorem for the complete physical family.

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

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

scope{ The proved results concern the finite source specification, exact local plateau reduction, characteristic coordinates and density, complete-sum translation representation, and the explicitly defined scalar asymptotics.

The source-specific characteristic estimate for ${\mathsf{F}}_{\delta }(K_{\mathrm{ret}})$ is not proved. Neither a scalar diagnostic nor a predecessor stopping label supplies that estimate. No quantum admission is asserted.}

The target is the existence of a limit

on arbitrary compact joint tests, including tests crossing the aligned-null characteristic locus $N$. The regularization is defined explicitly in Section sec:expression.

This is a fixed-kernel problem. It is distinct from constructing a continuous characteristic operator on an enlarged internal module: a cancellation for the designated source need not persist under arbitrary changes of its coefficient arrays.

A substantive physical conclusion would be one of:

1. a complete-sum Cauchy estimate;

2. an evaluated complete-sum singular expansion with a controlled remainder;

3. a fixed-test failure theorem for the specified unmodified regularization.

The present paper does not decide among these outcomes.

Three different qualifications are used below:

Specified but unevaluated.

A finite, unambiguous formula is available, but its component expansion or its contraction in the characteristic calculation has not been performed.

Fixed input not instantiated here.

An array or distribution is part of the mathematical data of the paper. Its particular physical realization is not independently derived here.

Inherited interpretation.

A predecessor identifies a displayed expression with a branch of the physical graph. That interpretation is not inferred merely because two manuscripts use similar notation.

A reproducible symbolic specification suffices to define coefficients. An explicit component list is not inherently required. Conversely, a component expansion does not replace a missing boundary-value prescription.

The source and the covariance package are separate inputs. Fixing the curvature convention does not instantiate the crossed covariances.

The equations printed below define the mathematical instance studied in this paper. Identifying that instance with a predecessor's frozen graph requires the same source-output conversion, component pairings, constant map $\mathsf{D}$, crossed array, and vertex normalization. Those identifications are not consequences of notation alone.

The source specification follows XII-C XIIC. The internal formulas follow the displayed retained architecture in Em01-B-v3, XIV-H, and XIV-J Em01B,XIVH,XIVJ. All operations used in the new calculations are either defined below or used with explicitly stated hypotheses.

Distributions have complex-bilinear pairings. Unless indicated otherwise, they are represented using flat coordinate densities. A smooth coordinate-density conversion is displayed explicitly or absorbed into a test or coefficient.

For tests supported in $L⋐U$, write

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

For wavefront-constrained operations we use the normal topology: pullback and suitably supported pushforward are continuous, whereas the relevant bilinear operations are hypocontinuous BDH (Sections 3–6). Fixed compact tensor products and their pushforwards below are defined independently of a characteristic coincident-point product.

Use

Let $S={\mathrm{Sym}}^2(({\mathbb{R}}^4{)}^{\ast })$. Fix a basis $e_I\in S$ and its algebraic dual $e^I\in S^{\ast }$, written as symmetric contravariant tensors:

The algebraic dual $e^I$ is not, in general, the tensor obtained by raising both indices of $e_I$ with $\eta$.

One concrete implementation uses labels $I=(i,j)$, $0\le i\le j\le 3$, with

Thus an off-diagonal basis tensor has two entries equal to one, and its dual has two entries equal to $1/2$.

The tensor Euler derivative is defined by

For propagation we lower this output with the fixed background metric. Its component polynomial is therefore

This is the source-output convention used in $Q$ and $T$ below.

To distinguish it from coordinate variation, let

Then

The ${\mathcal{E}}_I$ are the coordinate Euler components. They are not interchangeable with the propagated components ${\mathcal{J}}^L$. In particular, off-diagonal factors must not be inserted twice.

The linear maps are

Thus

The notations ${\mathsf{A}}^{\mu \nu }{}_L$ and ${\mathsf{b}}_L$, used when contracting a response output, denote evaluation on $e_L$:

Their index placement labels their role in the finite component contraction; it does not instruct an additional multiplication by $\mathbf{G}$.

Fix the constant bilinear map $\mathsf{D}$ through

The general reduction results hold for every fixed such array. Its designation as a particular physical density map remains an input identification.

Work on

where $A,B,Y$ have separated closures, $\overline{Z}\subset A$, and

These inequalities hold throughout the patch. The smooth ramp ${\sigma }_{\epsilon }$ equals one on $t>\epsilon$. The observation functions $f,g$ are fixed and smooth.

Let ${\mathcal{J}}_{\mathrm{D}}={\mathrm{EL}}_{\gamma }({\sigma }_{\epsilon }{\ell }_3)$, with the finite specification in Appendix app:source. Under (eq:source-conversion), its derivative has the local form

Here and below Greek multiindices in ${\partial }^{\alpha }$ are distinct from Greek spacetime indices in indexed tensor contractions.

Define the source incidence

The superscript $(4)$ here denotes the dimension of the Dirac mass. Primes, such as ${\delta }_{\mathrm{D}}^{\prime }(q)$, denote one-dimensional derivatives.

The kernels are

They satisfy $B=R\ast R$. Indeed, the spatial transform of $R$ is $\Theta (t)\sin (\rho t)/\rho$. Its causal time convolution is

with value $\Theta (t)t^3/6$ at $\rho =0$. Radial integration gives the same transform for the displayed $B$.

Put

Fix also a smooth advanced observation root $F={E_m}^{\mathrm{A}}f$, with $m\ge 0$. The analysis below only uses its smoothness and its fixed value.

Define

where

These formulas reproduce the retained architecture Em01B,XIVH,XIVJ. Its physical interpretation is that $K_{47}=K_4+K_7$; both placements of $K_5$ and both terms of $K_9$ are displayed separately. The seven rows are bookkeeping rows, not seven new interaction branches.

For $D=R$ or $D=B$,

All derivatives, including those of smooth coefficients, are retained.

The internal distributions are defined. For every $L⋐W$,

Proof. Use

and the analogous identity for $a$. External derivatives commute with the distributional integration. The undifferentiated integration is

This proves (eq:response-reduction).

The source incidence is a differentiated pullback under the submersion $(v,a,z)\mapsto (v-z,v-a)$. Adding $b-v$ supplies independent propagation variables. Its integration is locally proper because the incidence forces $v=a=z$ on compact external test supports.

The $K_{47}$ and $K_5$ seeds are tensor-product pullbacks under the invertible maps

The response seeds in (eq:response-reduction) are likewise tensor-product pullbacks.

Both $R$ and $B$ are locally finite measures. For $R$, the future-cone radial measure is

Every internal row has at most four derivatives of order-zero seeds, multiplied by smooth coefficients. For $K_8$, this bound uses two source derivatives and two displayed output derivatives. Transferring derivatives to the coefficient–test product proves (eq:internal-order).

To make the causal plateau argument reproducible, display the unreduced expression used for the coordinate-response branch in XIV-H XIVH (formula eq:K8map):

Its identification as the physical coordinate-response branch is inherited. The equality between this printed expression and (eq:K8) on the retained patch is proved next.

Let $Q_3$ be the incidence constructed from $J_3={\mathrm{EL}}_{\gamma }({\ell }_3)$, using the same output conversion. Let ${K_{\mathrm{ret}}}^{\mathrm{pl}}$ be the seven-row sum with $Q$ replaced by $Q_3$ and the displayed ${\sigma }_{\epsilon }(a),{\sigma }_{\epsilon }(z)$ replaced by one. No other data are changed.

On the retained patch,

The first equality holds as an equality of internal distributions; it therefore persists in every positive-scale regularized contraction on tests in the patch.

Source-ramp derivative terms are exactly absent from this local expression. No smoothing gain follows merely from their absence.

Proof. The weighted source is

The source incidence enforces $v=a=z$. Near its retained support, $v^0>\epsilon$, so the source and all its coefficient jets agree with the unweighted source there. The finite reduction (eq:response-reduction) consequently gives identical response distributions with $Q$ and $Q_3$.

The explicit factors ${\sigma }_{\epsilon }(a),{\sigma }_{\epsilon }(z)$ are one on open neighborhoods of their retained variables; all their derivatives vanish there. This proves the first equality.

For (eq:K8-unreduced), retarded support enforces

Hence $s^0\ge a^0>\epsilon$. Replacing ${\sigma }_{\epsilon }(s)$ by one, including inside the displayed derivative, leaves the distribution unchanged. Differentiation does not enlarge support.

The intermediate integration is support-proper on compact endpoint sets, since $s$ lies in compact causal diamonds. Causal convolution then gives

which is exactly the reduced expression (eq:K8).

The proposition does not remove observation-switch derivatives, change the frozen action, or identify a late response with an unweighted response for unrelated source data.

Fix a symmetric crossed array

The four lines are fixed distributions; both tensor products exist. The lower component label $I$ denotes the dual input slot contracted with the corresponding slot of $K^{I,J}$. No additional fiber metric is inserted in that contraction.

The linear submersion is

For smooth $H$, define

The vertex is

Its arbitrary-test action is

This fixes the kinetic, trace-kinetic, and mass coefficients. With the adopted polarization

they are the coefficients obtained by transferring the unused scalar derivative in the mixed Hessian $-\int g{h}^{\mu \nu }{T}_{\mu \nu }(\varphi ,\psi ){\mathrm{d}}^{4}y$. This is the normalization adopted for the retained vertex Em01B. The vertex mass term and the internal $K_{9,\mathrm{mass}}$ term are distinct.

Fix an open $U⋐W$, a cutoff $\chi \in {C_c}^{\infty }({\mathbb{R}}^8)$ equal to one near $\mathfrak{f}(\overline{U})$, and a profile $\rho \in {C_c}^{\infty }({\mathbb{R}}^8)$ with $\int \rho =1$. Define

No positivity or symmetry of $\rho$ is assumed. Since $C_{\delta }$ is smooth, every positive-scale expression exists.

Define

and

Fix $w_0\in N$. On a sphere chart choose a smooth orthonormal frame ${\mathbf{e}}_1,{\mathbf{e}}_2$ of ${\mathbf{n}}^{\perp }$, and set

Then

Near $w_0$, the map

is a smooth coordinate map, with

There are eight tangential and eight normal variables:

The cone equations are

The ambient flat density is

The map and inverse have bounded derivatives on compact chart sets where $\alpha ,\beta -\alpha ,\beta ,\beta -u$ have positive lower bounds and the sphere coordinate remains in a compact subchart.

Proof. For $s=b-a$, with $|\mathbf{s}|>0$, the inverse coordinates are

For $d=y-a$,

These formulas prove invertibility.

The null-frame identities give the three exact cone equations. They also identify $N$ and $S_{\mathrm{resp}}$.

The spatial radius of $s$ is $\beta -u$, and $s^0=\beta +u$. The absolute Jacobian of $(\beta ,u)\mapsto (s^0,|\mathbf{s}|)$ is two:

At fixed $\mathbf{n}$, the map $(\alpha ,v,\xi )\mapsto d$ has absolute determinant two. The full transformation is block triangular after selecting $a,s$, so the determinants multiply. Uniformity follows from the explicit smooth inverse.

Put $h=\beta -\alpha >0$. Along $N$,

The three differentials have rank two there, and

Throughout the coordinate chart,

On $S_{\mathrm{resp}}$, simultaneous vanishing of $q_1,q_2$ is equivalent to $v=\xi =0$. Their restricted differentials have rank one along $N$, with the remaining constraint supplied by the definite quadratic form in (eq:quadratic-relation).

Proof. Differentiate the exact equations at $u=v=\xi =0$. Adding their divided forms gives the nonlinear relation. On $S_{\mathrm{resp}}$, $q_R=0$; positivity of $\beta /(\alpha h)$ forces $\xi =0$, and then $v=0$.

These rank statements are localized to $N$; rank may increase away from it. The flag has codimensions

It is not a replacement numerical extension problem.

The dilation

satisfies

and has normal coordinate Jacobian ${\lambda }^{10}$. The weight of $r$ is declared; the cone equations do not determine it. The ambient density has the additional smooth ratio

This geometric dilation is distinct from the isotropic smoothing in (eq:regularization). No physical scaling degree is assigned by this calculation alone. Extension frameworks require their additional growth and microlocal hypotheses Dang,BF.

Let $K$ be a fixed internal distribution and $\Phi \in \mathcal{D}(U;V_{\mathrm{out}}^{\ast })$. Define compact relative distributions

The pre-pushforward distributions have compact support. Their pushforward is defined by transposition against smooth relative functions.

Put

For every smooth symmetric array $H$,

If $K$ has local order at most four, then for $\mathrm{supp}\Phi \subset L⋐U$,

on a fixed compact neighborhood of its relative support.

Proof. The kinetic part of the test action is

Transposition gives $i{\partial }_{2,\mu }{a}_{I,\nu }$. Symmetry of $H$ gives the symmetrization in (eq:Lambda). The trace-kinetic coefficient $i/2$ becomes $-i/2$, while the mass coefficient remains $-i{m}^2/2$. This proves the identity.

The order-four estimate acts on a product of $g$, a derivative of $\Phi$, and a pulled-back relative test. It uses at most five derivatives of $\Phi$. The additional relative derivative uses at most five derivatives of $\psi$.

Let

and define

All factors are compactly supported. Thus ${\Omega }_{\Phi }$ is a compact distribution formed by defined tensor products and pushforward, not by an undeclared coincident-point product.

A compact distribution acts on smooth functions by insertion of a cutoff equal to one near its support. This makes the pairing with ${\rho }_{\delta }(x-v)$ precise even though that function need not be compactly supported on ${\mathbb{R}}^{16}$.

For $K=K_{\mathrm{ret}}$,

Every internal row and every vertex term is retained.

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

on a fixed compact neighborhood of the translation support.

Every equality of actual internal distributions is respected. In particular, if $\mathsf{Rec}A=0$, then the corresponding ${\Omega }_{\Phi }$ and all regularized values are zero.

Proof. Insert $H=C_{\delta }$ into (eq:transpose). The result is

which is the pushforward definition of (eq:origin-evaluation).

Linearity gives

Each row uses all terms of (eq:Lambda).

The tensor pairing requires at most five $x$-derivatives and at most $p$$v$-derivatives. On $\psi (x-v)$ their combined order is at most $p+5$. The compact-support cutoffs have fixed derivative bounds. This proves (eq:Omega-order).

The construction begins with the actual $K$, so a zero internal distribution remains zero at every stage.

The theorem identifies the complete object whose origin behavior must be controlled. It does not establish regularity at that origin.

Define the compact profiles

Then

Both have integral zero. Their moments satisfy

The first sign follows directly by integration by parts.

The family $t_{\delta }$ is smooth in the strong distributional topology on compact subintervals of $(0,\infty )$. Set

Then

and equivalently

Proof. On a compact positive scale interval the profiles and their scale derivatives have common compact support and are smooth in every test seminorm. The estimate (eq:Omega-order) makes their evaluations smooth uniformly on bounded joint test sets. Differentiate or subtract (eq:origin-evaluation).

Only the auxiliary profiles vary with scale. The internal kernel, source, switches, and grouped crossed distribution remain fixed.

The direct physical target is

This need not be proved through a function-valued regularity theorem for ${\Omega }_{\Phi }$.

Zero profile integral alone is insufficient. For example,

and a zero-integral scale profile may still have nonzero value at zero.

Let $t_{\delta }$ be a strongly smooth distribution-valued family on $0<\delta \le {\delta }_0$, with $D_{\delta }=\delta {\partial }_{\delta }{t}_{\delta }$. Suppose that for every $L⋐U$, there are $m_L,C_L$ and a nonnegative measurable $a_L$ such that

and

Then $t_{\delta }$ converges strongly to a distribution $t_0$, with

If $a_L(s)=s^{\alpha }$, $\alpha >0$, then

and, for dyadic scales within the parameter interval,

The same integral controls parameters between adjacent dyadic scales.

Proof. For $0<{\delta }_1<{\delta }_2\le {\delta }_0$,