Em02-D, revision 4 Active Retarded Rows and Intersecting Crossed Cones Signed Internal Convolution and Complete Seven-Row Leading-Order Assembly

We study a fixed two-component retarded distribution model with a compact product smoothing profile, a nonconstant advanced scalar root, and an ambient first-order source jet. The localization includes aligned crossed cones, internal coincidence, and separated internal endpoints.

The response rows and two-line adjoints are assembled in one distributional convention. For the negative placement $5z$, we integrate the internal cones exactly before taking smoothing asymptotics. This replaces a radius–angle absolute-value estimate by a boundary/interior decomposition. A derivative-controlled first correction to the weighted master convolution determines the previously missing third temporal derivative of the integrated vertex.

The complete seven-row family has an inverse-cubic leading distribution on arbitrary localized tests, with a compact-uniform $O({\delta }^{-2})$ remainder and an independently controlled logarithmic-scale derivative. The nominal inverse-quartic contribution of $5z$ vanishes. The advanced cone fundamental solution identity also gives a genuine cancellation between the inner-tip part of the positive placement $5a$ and the response leading coefficient.

The collected inverse-cubic distribution nevertheless survives: a separated support-isolated test detects it. The raw family therefore does not converge, and corrections supported on $z=a$ cannot restore convergence. The available remainder estimate does not establish convergence after leading subtraction. No complete lower-order expansion, finite part, profile universality, or physical quantum admission is claimed.

All weights, component data, smoothing profiles, localizations, and correction supports are fixed before the smoothing limit. The analysis concerns the mathematical model specified below; it does not import an inherited physical observable.

For the fixed model of Sections sec:model and sec:rows, there is a continuous distribution $C_{\mathrm{seven}}$, given explicitly in (eq:Cseven), such that

For every fixed compact joint-test support $H$,

for sufficiently small $\delta >0$.

The leading coefficient is nonzero. One fixed compact test supported away from $z=a$ satisfies

Consequently the complete raw family has no weak distributional limit, and no correction family supported on $z=a$ restores convergence.

The coefficient calculation is proved in Sections sec:known–sec:seven. The separated witness and correction theorem are proved independently in Section sec:obstruction.

boundary. The result is a leading smoothing expansion for one specified distribution model. It is not an interacting Ward identity, a physical renormalization theorem, or an extension prescription for another manuscript's graph.

boundary. An $O({\delta }^{-2})$ upper bound does not prove that the remainder diverges. It also does not prove convergence or integrability after leading subtraction. These questions require additional lower-order analysis.

Let

Pairings are complex bilinear and use coordinate volume. Component indices range over $\{1,2\}$.

For $\mathrm{supp}\Phi \subset H$, where $H\subset W$ is compact, write

For a scalar weight,

When weights depend on spectators, their seminorms include the indicated mixed derivatives. Constants may depend on the fixed compact supports and charts, but not on the test or on small $\delta$.

Define

Sphere area measure has total mass $4\pi$. Away from the vertex,

Set

The identity $R\ast R=B$ will be proved below.

Set

Thus $\vartheta =1$ for $v\le 1$ and $\vartheta =0$ for $v\ge 4$.

Fix

The source switch is identically one.

The function $F$ is the advanced zero-future solution of $({\square }_{\eta }+1)F=f$. On $|x{|}_{\mathrm{E}}<4$,

Proof. The equation holds by definition. Compact support gives zero Cauchy data on every sufficiently late slice. The homogeneous difference of two zero-future solutions vanishes by Cauchy uniqueness. This uniqueness follows locally from the energy identity on domains of dependence, so no spatial decay assumption on an arbitrary smooth difference is required. The plateau identities are immediate.

The sparse component data are

Every undisplayed $\mathsf{A}$ or $\mathsf{D}$ entry vanishes.

For $j=1,2$, define the source incidence

The products are tensor-product pullbacks in independent incidence coordinates.

For $E=R$ or $B$,

The derivative is transferred before incidence restriction. The identity can be checked in independent propagation and incidence coordinates; their pushforward is proper on compact joint tests.

Introduce

Then

Near ${\mathbf{n}}_{\ast }=(1,0,0)$, choose a smooth orthonormal spatial tangent frame ${\mathbf{e}}_A(\mathbf{n})$, and put

Use

The inverse includes

The other variables are the longitudinal and transverse projections of $y-a$. The exact cone equations are

Define

Fix $h=1/100$ and

Extend it by zero outside the chart. Its support is compact in a region with positive lower bounds on $\alpha ,\beta -\alpha ,\beta$. On a neighborhood of its support,

The outer propagation $S=b-a$ never reaches its vertex.

Let

The closure of $N$ in $W$ is contained in $\mathcal{S}$.

The class ${\mathfrak{C}}_S$ consists of all distribution-valued families $c_{\delta }$ with $\mathrm{supp}{c}_{\delta }\subset \mathcal{S}$ at every scale, without order or scale restrictions. Define ${\mathfrak{C}}_N$ and ${\mathfrak{C}}_L$ similarly using $\overline{N}$ and $\mathcal{L}$. These supports are fixed before coefficient collection.

Let

Then $p$ is smooth, even, radial, nonnegative, compactly supported, and normalized. Put

The crossed array is

Choose $\chi \in {C_c}^{\infty }({\mathbb{R}}^8)$ equal to one near the relative image of $\mathrm{supp}\zeta$, and smooth $\chi C$ by ${\rho }_{\delta }$.

For sufficiently small scales, on a neighborhood of the retained support,

This identity and all its derivatives hold there because translated profile supports stay in the plateau of $\chi$.

The trace-adjusted vector for the specified crossed tensor is

The Euclidean coordinate gradient occurs because the trace-adjusted $00$ coefficient matrix is one half of the Euclidean identity.

The family is

The internal distributions are tensored with coordinate volume in $y$.

Define

and, for a frozen weight $w$,

The integrated vertex is

Propagation derivatives in $\tilde{S}$ freeze the weight. Total derivatives in $S$ also differentiate its parameter dependence.

Integration by parts in $y$ alternatively gives the multiplier

Temporal primes on $R$ mean derivatives of its propagation argument. On the retained neighborhood, the two-line kernels are equivalently

Every seed product is a tensor-product pullback, including at its inner vertex.

For comparison with endpoint notation, set $L^{0,2}(q,a)={\partial }_{a^0}R(q-a)$. Then $K_{5a}$ and $K_{5z}$ are the two terms

For $47$, the exact bracket cancellation gives

The plateau value ${\partial }_{0}f=1$ gives (eq:K47).

In relative variables, put $D={\partial }_{S^0}$. The response rows are

Here $f=f(a+S)$. The endpoint derivative in row $6$ is line-only. It is not redistributed onto ${\delta }^{(4)}(r)$.

All undisplayed components vanish, and

On the retained non-tip region,

Indeed both sides equal $\delta (S^0-|\mathbf{S}|)/(8\pi )$. Put $A_S=f{c}_S+1$.

The second-component response sum is

Its exact action is

Proof. The separate actions are

For row $6$, transfer the outer derivative first and the line derivative second:

Here $Df=1$. Summation proves (eq:response-adjoint). In particular, the $-DM$ term from row $6$ cancels the $+DM$ term from row $9d$.

If expanded, row $8$ contains all coefficient derivatives:

Thus divergence-form adjoints do not omit derivatives of $f$.

The substitutions for the integrated vertex are

Their exact actions are

For $47$, the outer transfer gives a minus sign and the two line transfers give positive combined sign. For each $5$ row,

These are identities before cone restriction.

The derivative conversions are

The first component contains only row $9m$:

Multiplication by $-\mathrm{i}$ gives the corresponding contribution to ${t_{\delta }}^{\zeta }$.

For a frozen weight $w$, define

For $S=(T,q\mathbf{n})$, $T>q>0$,

where on the timelike side

and

The convolution vanishes outside the future causal cone. There is no additional null-boundary measure. In particular, $R\ast R=B$.

Proof. In the rest frame $S=(\sqrt{S^2},0)$, temporal and radial cone integration gives

The cone measure $R=\Theta (x^0)\delta (x^2)/(2\pi )$ is Lorentz invariant. Boost in direction $\mathbf{n}$, and put ${\omega }_{\parallel }=2\theta -1$. This gives the formula.

On compact sets of $S$, retarded support bounds both propagation times, so the convolution is proper. The set of parallel angular pairs has zero product-measure mass. Therefore no extra measure is carried by the null boundary. For $w=1$, $\Gamma =1/(8\pi )$, proving $R\ast R=B$.

Let

and $\overline{S}=(|\mathbf{S}|,\mathbf{S})$.

At $s=q{\ell }_{\mathbf{n}}$, define

Here ${\Delta }_{\perp }={\Delta }_{\mathbf{x}}-(\mathbf{n}\cdot {\nabla }_{\mathbf{x}}{)}^2$. Formula (eq:fiber) gives

Every propagation derivative of $\Gamma$ freezes $w$.

Fix a compact non-tip domain

where $q_0>0$, $d_0<q_0/2$, and $\Omega$ is a compact sphere subchart.

For each finite $N$, $\Gamma$ has a one-sided $C^N$ extension to $d=0$, and a local $C^N$ extension across it, with

Adding $l$ spectator derivatives requires at most $2N+l$ weight derivatives.

Proof. Appendix app:circle proves that the $j$-th derivative of a circle average in its squared radius is controlled by $2j$ derivatives of its weight. In (eq:fiber), that squared radius is $(S^2)\theta (1-\theta )$; the longitudinal point and frame are smooth on the non-tip domain. A total propagation derivative of order $N$ therefore uses at most $2N$ weight derivatives. The estimate is uniform at $\theta =0,1$.

For a two-sided extension, choose $a_1,\dots ,a_{N+1}$ such that ${{\sum }_{j=1}}^{N+1}{a}_j(-j{)}^k=1$ for $0\le k\le N$. For negative small $d$, use

The normal and mixed jets match through order $N$, with the same type of bound.

The density of $P^0-\mathbf{n}\cdot \mathbf{P}$ under $p$ is

Set

The density $H_{\ast }$ is even, so $H_{\ast }^{\prime }(0)=H_{\ast }^{\prime \prime \prime }(0)=0$.

Product convolution gives the exact translation formula

For $1\le m=|\gamma |\le 5$, uniformly on $|d(S)|\le \Lambda \delta$ in the fixed non-tip domain,

The scale derivative here is at fixed $S$. Also $J_{\delta }$ and its scale derivative are bounded by $C\Vert w{\Vert }_0$.

For $d(S)>4\delta$, propagation derivatives and their scale derivatives are bounded by $C_m\Vert w{\Vert }_{2m}$. For $d(S)<-4\delta$, they vanish.

Proof. Differentiate $J=\Theta (d)\Gamma$. The unique highest boundary term of order $m$ is

The other boundary terms contain ${{\delta }_{\mathrm{D}}}^{(j)}(d)$, $j\le m-2$, and the interior term is bounded by Lemma lem:regularity.

Put $H=P+Q$, and solve the translated boundary in $H^0$:

Temporal coarea contributes ${\delta }^{-m}$ for the highest term. The profile density gives $H_{\ast }^{(m-1)}$, the normal gives $k(S{)}^{\gamma }$, and the coefficient gives ${\mathcal{G}}_{\overline{S}}[w]$. Their changes are $O(\delta )$ in the required temporal derivatives. Lemma lem:regularity, together with the shifted-weight Taylor estimate, bounds these changes by $C\delta \Vert w{\Vert }_{2m}$. Lower boundary terms are at most $O({\delta }^{1-m})$; interior terms are bounded.

For the scale derivative, replace $\rho$ by

Its normal marginal is

and

This is the fixed-$S$ scale derivative of the leading term. The same estimates apply to every lower boundary and interior term for this compact signed profile.

Finally,

on the profile support. For $d>4\delta$, only smooth timelike coefficient derivatives occur; for $d<-4\delta$, all translated arguments are outside causal support.

At a null point $s=q{\ell }_{\mathbf{n}}$, order-two propagation derivatives are $O({\delta }^{-1})$ by parity. Order-three derivatives have leading moment $\kappa$.

Define

For the boundary operators of the preceding section, set

Let $S_{\delta }=(q+\delta z,q\mathbf{n})$. Uniformly for bounded $z$ and compact non-tip $(q,\mathbf{n})$,

For up to four $z$ derivatives, one tangential derivative in $(q,\mathbf{n})$, and $l$ additional smooth spectator derivatives,

The same estimate holds for

This operator is the logarithmic-scale derivative at fixed unscaled $S$.

Proof. We give an exact fixed-domain reduction before Taylor expansion.

Put $H=P+Q$, ${\mathbf{H}}_{\perp }=\mathbf{H}-(\mathbf{n}\cdot \mathbf{H})\mathbf{n}$, and $h=H^0-\mathbf{n}\cdot \mathbf{H}$. Define

This extends smoothly to $\delta =0$ on the fixed compact profile support, with

Exactly,

Make the change $u=h+\delta c_{\delta }$. Its temporal Jacobian is one. The retarded indicator becomes $\Theta (z-u)$, independent of $\delta$.

Thus (eq:translation) becomes an integral over $u\le z$ of the smooth integrand

where

A fixed compact interval in $u$ contains every nonzero integrand for the parameter range under consideration. The coefficient may be extended across $d=0$ using Lemma lem:regularity; on the actual integration domain for $\delta >0$, its argument is causal.

Taylor expansion in $\delta$, with this fixed domain, is therefore legitimate in the stated mixed norms. Two $\delta$ derivatives, four $z$ derivatives, and one tangential derivative use at most seven coefficient derivatives. Lemma lem:regularity uses at most fourteen weight derivatives. Spectator derivatives add $l$. Endpoint terms from differentiating the upper limit in $u$ have no larger derivative count. The integral Taylor remainder gives (eq:first-error).

It remains to calculate the first derivative. At $\delta =0$, the zeroth term is $\mathcal{G}[w]\mathcal{F}(z)$. The coefficient/weight derivative contributes

The density derivative from $h=u-\delta c_{\delta }$ contributes