% Em02-D-v4.tex % Standalone pdfLaTeX source; no external files are required. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error Em02-D-v4.tex % pdflatex -interaction=nonstopmode -halt-on-error Em02-D-v4.tex % % Revision 4: % Exact signed internal convolution for 5z. % Derivative-controlled first correction to the smoothing layer. % Explicit inverse-cubic 5z coefficient. % Complete seven-row leading-order assembly. % Distributional 5a--response inner-tip cancellation. % Independent separated obstruction and correction-support theorem. % % Boundary: % No complete lower-order singular expansion or finite part is proved. % No physical identification, compilation, or machine verification % is claimed. \documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\,\mathrm d} \newcommand{\pair}[2]{\langle #1,#2\rangle} \newcommand{\deltar}{\delta^{(4)}(r)} \newcommand{\Hr}{H_r} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\CS}{\mathfrak C_S} \newcommand{\CN}{\mathfrak C_N} \newcommand{\CL}{\mathfrak C_L} \newcommand{\Qcal}{\mathcal Q} \newcommand{\Gcal}{\mathcal G} \newcommand{\Acal}{\mathcal A} \newcommand{\Bcal}{\mathcal B} \newcommand{\Ccal}{\mathcal C} \newcommand{\Tcal}{\mathcal T} \newcommand{\Vcal}{\mathcal V} \newcommand{\Ecal}{\mathsf E} \newcommand{\scaleder}{\delta\partial_\delta} \newcommand{\dirac}{\delta_{\mathrm D}} \DeclareMathOperator{\supp}{supp} \title{\textbf{Em02-D, revision 4}\\[0.35em] Active Retarded Rows and Intersecting Crossed Cones\\[0.35em] \large Signed Internal Convolution and\\ Complete Seven-Row Leading-Order Assembly} \author{} \date{} \begin{document} \maketitle \begin{abstract} 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. \end{abstract} \tableofcontents %===================================================================== \section{Main result and scope} \label{sec:main} %===================================================================== 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. \begin{theorem}[Complete leading-order result] \label{thm:main} For the fixed model of Sections~\ref{sec:model} and \ref{sec:rows}, there is a continuous distribution \(C_{\mathrm{seven}}\), given explicitly in \eqref{eq:Cseven}, such that \begin{equation} t_\delta^\zeta = \delta^{-3}C_{\mathrm{seven}}+\mathcal R_\delta. \label{eq:main-expansion} \end{equation} For every fixed compact joint-test support \(H\), \begin{equation} |\mathcal R_\delta(\Phi)| +|(\scaleder\mathcal R_\delta)(\Phi)| \le C_H\delta^{-2}\|\Phi\|_{17,H} \label{eq:main-remainder} \end{equation} for sufficiently small \(\delta>0\). The leading coefficient is nonzero. One fixed compact test supported away from \(z=a\) satisfies \[ t_\delta^\zeta(\Phi_A) = c_A\delta^{-3}+O(\delta^{-2}), \qquad c_A\ne0. \] Consequently the complete raw family has no weak distributional limit, and no correction family supported on \(z=a\) restores convergence. \end{theorem} The coefficient calculation is proved in Sections~\ref{sec:known}--\ref{sec:seven}. The separated witness and correction theorem are proved independently in Section~\ref{sec:obstruction}. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.36\linewidth}Y@{}} \toprule Result & Status in this revision\\ \midrule Exact response and seed adjoints & Proved, with line and incidence derivatives kept separate.\\ Uniform leading master estimates & Proved through the derivative orders used below.\\ First correction to the smoothing layer & Proved through four scaled-normal derivatives and one tangential derivative, with uniform spectators.\\ Signed internal \(5z\) convolution & Exact boundary/interior identity, including the inner vertex.\\ \(5z\) leading coefficient & Explicit inverse-cubic distribution; nominal inverse-quartic term vanishes.\\ Seven-row leading assembly & Proved on arbitrary localized tests.\\ \(5a\)--response tip cancellation & Proved by the cone fundamental solution identity.\\ Separated raw obstruction & Proved without relying on the \(5z\) asymptotics.\\ Complete lower-order expansion & Not established.\\ Subtracted finite part & Not established.\\ Physical identification and quantum admission & Not asserted.\\ \bottomrule \end{tabularx} \end{center} \paragraph{Interpretation 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. \paragraph{Remainder 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. %===================================================================== \section{Fixed model, root, and localization} \label{sec:model} %===================================================================== \subsection{Conventions} Let \[ M=\R^{1+3},\qquad W=M^4_{(a,b,z,y)}, \qquad \eta=\operatorname{diag}(1,-1,-1,-1), \qquad \Box_\eta=\partial_0^2-\Delta_{\mathbf x}. \] Pairings are complex bilinear and use coordinate volume. Component indices range over \(\{1,2\}\). For \(\supp\Phi\subset H\), where \(H\subset W\) is compact, write \[ \|\Phi\|_{j,H} = \max_{J,|\gamma|\le j}\sup_W|\partial^\gamma\Phi_J|. \] For a scalar weight, \[ \|w\|_j=\max_{|\gamma|\le j}\sup_M|\partial^\gamma w|. \] 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 \(\delta\). \subsection{Retarded cone measure} Define \begin{equation} \pair R\psi = \frac1{4\pi}\int_0^\infty \beta \int_{S^2}\psi(\beta,\beta\mathbf n) \dd\mathbf n\,\dd\beta. \label{eq:R} \end{equation} Sphere area measure has total mass \(4\pi\). Away from the vertex, \[ R(x)=\frac{\delta(x^0-|\mathbf x|)} {4\pi|\mathbf x|}. \] Set \begin{equation} B(x)=\frac1{8\pi}\Theta(x^0-|\mathbf x|). \label{eq:B} \end{equation} The identity \(R*R=B\) will be proved below. \subsection{Advanced root and plateaux} Set \[ e(t)= \begin{cases} \exp(-1/t),&t>0,\\ 0,&t\le0, \end{cases} \qquad \vartheta(v)= \frac{e(4-v)}{e(4-v)+e(v-1)}. \] Thus \(\vartheta=1\) for \(v\le1\) and \(\vartheta=0\) for \(v\ge4\). Fix \begin{equation} F(x)=(1+x^0)\vartheta(|x|_{\mathrm E}^2/16), \qquad f=(\Box_\eta+1)F, \qquad g(x)=\vartheta(|x|_{\mathrm E}^2/36). \label{eq:weights} \end{equation} The source switch is identically one. \begin{proposition}[Root prescription] \label{prop:root} The function \(F\) is the advanced zero-future solution of \((\Box_\eta+1)F=f\). On \(|x|_{\mathrm E}<4\), \[ F=f=1+x^0,\qquad \partial_0F=\partial_0f=1. \] \end{proposition} \begin{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. \end{proof} \subsection{Ambient source jet} The sparse component data are \[ \mathsf A^{00,2}=\mathsf A^{00}{}_2 =\mathsf D^{00,2,2}=1,\qquad \mathcal P^L{}_M=\delta^L_M,\qquad \mathsf b_1=\mathsf b_2=1. \] Every undisplayed \(\mathsf A\) or \(\mathsf D\) entry vanishes. For \(j=1,2\), define the source incidence \[ Q^{j,j,j}(v;a,z) = \delta^{(4)}(v-z)\delta^{(4)}(v-a) + \partial_{v^1}\delta^{(4)}(v-z)\delta^{(4)}(v-a). \] The products are tensor-product pullbacks in independent incidence coordinates. For \(E=R\) or \(B\), \begin{align} \int E(b-v)Q^{j,j,j}(v;a,z)\dd^4v ={}& E(b-a)\delta^{(4)}(a-z)\notag\\ &-\partial_{z^1} [E(b-a)\delta^{(4)}(a-z)]. \label{eq:source-reduction} \end{align} 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 \begin{equation} S=b-a,\qquad r=z-a,\qquad x=y-a, \qquad \Hr=\deltar-\partial_{r^1}\deltar. \label{eq:relative} \end{equation} Then \[ \pair{\Hr}{\psi} =\psi(0)+\partial_{r^1}\psi(0). \] \subsection{Characteristic chart} Near \(\mathbf n_*=(1,0,0)\), choose a smooth orthonormal spatial tangent frame \(\mathbf e_A(\mathbf n)\), and put \[ \ell_{\mathbf n}=(1,\mathbf n),\qquad k_{\mathbf n}=(1,-\mathbf n),\qquad E_A=(0,\mathbf e_A). \] Use \begin{equation} \begin{split} z&=a+r,\\ b&=a+\beta\ell_{\mathbf n}+u k_{\mathbf n},\\ y&=a+\alpha\ell_{\mathbf n} +v k_{\mathbf n}+\xi^AE_A. \end{split} \label{eq:chart} \end{equation} The inverse includes \[ \mathbf n=\frac{\mathbf b-\mathbf a}{|\mathbf b-\mathbf a|}, \quad \beta=\frac{b^0-a^0+|\mathbf b-\mathbf a|}{2}, \quad u=\frac{b^0-a^0-|\mathbf b-\mathbf a|}{2}. \] The other variables are the longitudinal and transverse projections of \(y-a\). The exact cone equations are \[ S^2=4\beta u,\qquad x^2=4\alpha v-|\xi|^2,\qquad (S-x)^2=4(\beta-\alpha)(u-v)-|\xi|^2. \] Define \[ N=\{r=0,\ u=v=0,\ \xi=0\}. \] Fix \(h=1/100\) and \begin{align} \zeta={}& \vartheta(|a|_{\mathrm E}^2/h^2) \vartheta(|\mathbf n-\mathbf n_*|^2/h^2) \vartheta((\alpha-1)^2/h^2) \vartheta((\beta-2)^2/h^2)\notag\\ &\times \vartheta((u^2+v^2+|\xi|^2)/h^2) \vartheta(|r|_{\mathrm E}^2/h^2). \label{eq:zeta} \end{align} 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, \begin{equation} f(b)=F(b)=1+b^0,\quad \partial_0f(b)=\partial_0F(b)=1,\quad g(b)=g(y)=1. \label{eq:local-data} \end{equation} The outer propagation \(S=b-a\) never reaches its vertex. \subsection{Prescribed correction supports} Let \[ \mathcal S=\{z=a\},\qquad \mathcal L=\{a=b=z=y\}. \] The closure of \(N\) in \(W\) is contained in \(\mathcal S\). The class \(\CS\) consists of all distribution-valued families \(c_\delta\) with \(\supp c_\delta\subset\mathcal S\) at every scale, without order or scale restrictions. Define \(\CN\) and \(\CL\) similarly using \(\overline N\) and \(\mathcal L\). These supports are fixed before coefficient collection. %===================================================================== \section{Smoothing, vertex, and exact rows} \label{sec:rows} %===================================================================== \subsection{Fixed smoothing profile} Let \[ b_0(v)= \begin{cases} \exp(-1/(1-v)),&0\le v<1,\\ 0,&v\ge1, \end{cases} \quad p(X)=N_p b_0(|X|_{\mathrm E}^2), \quad N_p=\left(\pi^2\int_0^1v b_0(v)\dd v\right)^{-1}. \] Then \(p\) is smooth, even, radial, nonnegative, compactly supported, and normalized. Put \[ p_\delta(X)=\delta^{-4}p(X/\delta),\qquad \rho(P,Q)=p(P)p(Q). \] The crossed array is \[ C_{\mu\nu|I}(U,V) = \delta_{\mu0}\delta_{\nu0}R(U)R(-V), \qquad I=1,2. \] Choose \(\chi\in C_c^\infty(\R^8)\) equal to one near the relative image of \(\supp\zeta\), and smooth \(\chi C\) by \(\rho_\delta\). For sufficiently small scales, on a neighborhood of the retained support, \begin{equation} D_\delta(S,x) = A_\delta(x)A_\delta(S-x), \qquad A_\delta=p_\delta*R. \label{eq:smoothing} \end{equation} This identity and all its derivatives hold there because translated profile supports stay in the plateau of \(\chi\). \subsection{Vertex and integrated test} The trace-adjusted vector for the specified crossed tensor is \[ Z^\rho_\delta = -\frac12 A_\delta(x) \partial_\rho^{\mathrm E}A_\delta(S-x). \] The Euclidean coordinate gradient occurs because the trace-adjusted \(00\) coefficient matrix is one half of the Euclidean identity. The family is \begin{equation} t_\delta^\zeta(\Phi) = -\ii\sum_{I,J} \pair{K_{\mathrm{ret}}^{I,J}} {g(y)\left[ Z^\rho_\delta\partial_{y^\rho}(\zeta\Phi_J) +\frac12D_\delta\zeta\Phi_J \right]}. \label{eq:target} \end{equation} The internal distributions are tensored with coordinate volume in \(y\). Define \[ \phi_j(a,S,r,x) = (\zeta\Phi_j)(a,a+S,a+r,a+x), \] and, for a frozen weight \(w\), \begin{equation} J_\delta(S;w) = \int A_\delta(x)A_\delta(S-x)w(x)\dd^4x. \label{eq:Jdelta} \end{equation} The integrated vertex is \begin{align} M_{\delta,j}(a,S,r) ={}& -\frac12\sum_{\rho=0}^3 \left. \partial_{\widetilde S^\rho} J_\delta(\widetilde S; \partial_{x^\rho}\phi_j(a,S,r,\cdot)) \right|_{\widetilde S=S}\notag\\ &+\frac12J_\delta(S;\phi_j(a,S,r,\cdot)). \label{eq:M} \end{align} Propagation derivatives in \(\widetilde S\) freeze the weight. Total derivatives in \(S\) also differentiate its parameter dependence. Integration by parts in \(y\) alternatively gives the multiplier \begin{align} w_\delta(S,x)={}& \frac12A_\delta(x)A_\delta(S-x) +\frac12\nabla A_\delta(x)\cdot\nabla A_\delta(S-x)\notag\\ &-\frac12A_\delta(x)\Delta_{\mathrm E}A_\delta(S-x). \label{eq:multiplier} \end{align} \subsection{Rows in independent seed variables} Temporal primes on \(R\) mean derivatives of its propagation argument. On the retained neighborhood, the two-line kernels are equivalently \begin{align} K_{47}^{22} &= (\partial_{s^0}+\partial_{t^0})[R'(s)R'(t)], &s=b-a,\quad t=b-z, \label{eq:K47}\\ K_{5a}^{22} &= \partial_{s^0}[f(a+s)R'(s)R''(t)], &s=b-a,\quad t=a-z, \label{eq:K5a}\\ K_{5z}^{22} &= \partial_{s^0}[f(c+s)R'(s)R''(t)], &c=z,\quad s=b-z,\quad t=z-a. \label{eq:K5z} \end{align} 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 \[ \partial_{b^0} [f(b)\partial_{a^0}R(b-a)\partial_{a^0}L^{0,2}(a,z)], \] \[ \partial_{b^0} [f(b)\partial_{z^0}R(b-z)\partial_{z^0}L^{0,2}(z,a)]. \] For \(47\), the exact bracket cancellation gives \[ K_{47}^{22} = \partial_{b^0} [(\partial_0f)(b)\partial_{z^0}R(b-z) \partial_{a^0}R(b-a)]. \] The plateau value \(\partial_0f=1\) gives \eqref{eq:K47}. In relative variables, put \(D=\partial_{S^0}\). The response rows are \begin{align} K_6^{22}&=-D(fDR)\deltar, \label{eq:K6}\\ K_8^{22}&=-D[f(DB)\Hr], \label{eq:K8}\\ K_{9d}^{22}&=-DR\,\Hr, \label{eq:K9d}\\ K_{9m}^{jj}&=-fR\,\Hr,\qquad j=1,2. \label{eq:K9m} \end{align} Here \(f=f(a+S)\). The endpoint derivative in row \(6\) is line-only. It is not redistributed onto \(\deltar\). All undisplayed components vanish, and \[ \Kret=K_{47}+K_{5a}+K_{5z}+K_6+K_8+K_{9d}+K_{9m}. \] %===================================================================== \section{Exact response and seed assembly} \label{sec:assembly} %===================================================================== \subsection{Response assembly} On the retained non-tip region, \[ DB(S)=c_SR(S),\qquad c_S=S^0/2. \] Indeed both sides equal \(\delta(S^0-|\mathbf S|)/(8\pi)\). Put \(A_S=fc_S+1\). \begin{proposition}[Response distribution and adjoint] \label{prop:response} The second-component response sum is \[ K_{\mathrm{resp}}^{22} = -D(fDR)\deltar-[D(A_SR)+fR]\Hr. \] Its exact action is \begin{align} \pair{K_{\mathrm{resp}}^{22}}M = \int\dd^4a\,\dd R(S) \bigl[ -fD^2M+fc_SDM-fM +A_SD\partial_{r^1}M-f\partial_{r^1}M \bigr]_{r=0}. \label{eq:response-adjoint} \end{align} \end{proposition} \begin{proof} The separate actions are \begin{align*} \pair{K_6^{22}}M &=-\int\dd^4a\,\dd R(S)[fD^2M+(Df)DM]_{r=0},\\ \pair{K_8^{22}}M &=\int\dd^4a\,\dd R(S)[fc_SD(M+\partial_{r^1}M)]_{r=0},\\ \pair{K_{9d}^{22}}M &=\int\dd^4a\,\dd R(S)[D(M+\partial_{r^1}M)]_{r=0},\\ \pair{K_{9m}^{22}}M &=-\int\dd^4a\,\dd R(S)[f(M+\partial_{r^1}M)]_{r=0}. \end{align*} For row \(6\), transfer the outer derivative first and the line derivative second: \[ \pair{-D(fDR)}M =\pair{DR}{fDM} =-\pair R{fD^2M+(Df)DM}. \] Here \(Df=1\). Summation proves \eqref{eq:response-adjoint}. In particular, the \(-DM\) term from row \(6\) cancels the \(+DM\) term from row \(9d\). \end{proof} If expanded, row \(8\) contains all coefficient derivatives: \[ K_8^{22} = -[fc_SDR+(c_S+f/2)R]\Hr. \] Thus divergence-form adjoints do not omit derivatives of \(f\). \subsection{Seed adjoints} The substitutions for the integrated vertex are \[ \widetilde M_{47}=M_{\delta,2}(a,s,s-t),\qquad \widetilde M_{5a}=M_{\delta,2}(a,s,-t), \] \[ \widetilde M_{5z}=M_{\delta,2}(c-t,s+t,t). \] Their exact actions are \begin{align} \pair{K_{47}^{22}}{M_{\delta,2}} &= -\int\dd^4a\,\dd R(s)\,\dd R(t)\, \partial_{s^0}\partial_{t^0} (\partial_{s^0}+\partial_{t^0})\widetilde M_{47}, \label{eq:47-adjoint}\\ \pair{K_{5a}^{22}}{M_{\delta,2}} &= \int\dd^4a\,\dd R(s)\,\dd R(t)\, \partial_{s^0}\partial_{t^0}^2 [f(a+s)\partial_{s^0}\widetilde M_{5a}], \label{eq:5a-adjoint}\\ \pair{K_{5z}^{22}}{M_{\delta,2}} &= \int\dd^4c\,\dd R(s)\,\dd R(t)\, \partial_{s^0}\partial_{t^0}^2 [f(c+s)\partial_{s^0}\widetilde M_{5z}]. \label{eq:5z-adjoint} \end{align} For \(47\), the outer transfer gives a minus sign and the two line transfers give positive combined sign. For each \(5\) row, \[ \pair{\partial_{s^0}[fR'(s)R''(t)]}{\widetilde M} = -\pair{R'(s)R''(t)}{f\partial_{s^0}\widetilde M} = \pair{R(s)R(t)} {\partial_{s^0}\partial_{t^0}^2 (f\partial_{s^0}\widetilde M)}. \] These are identities before cone restriction. The derivative conversions are \begin{center} \small \begin{tabularx}{\linewidth}{@{}lYY@{}} \toprule Seed & Substitution & Temporal derivatives on \(M\)\\ \midrule \(47\) & \(M(a,s,s-t)\) & \(\partial_{s^0}=D+\partial_{r^0}\), \(\partial_{t^0}=-\partial_{r^0}\)\\ \(5a\) & \(M(a,s,-t)\) & \(\partial_{s^0}=D\), \(\partial_{t^0}=-\partial_{r^0}\)\\ \(5z\) & \(M(c-t,s+t,t)\) & \(\partial_{s^0}=D\), \(\partial_{t^0}=-\partial_{a^0}+D+\partial_{r^0}\)\\ \bottomrule \end{tabularx} \end{center} The first component contains only row \(9m\): \begin{equation} \pair{K_{\mathrm{ret}}^{11}}{M_{\delta,1}} = -\int\dd^4a\,\dd R(S) [f(M_{\delta,1}+\partial_{r^1}M_{\delta,1})]_{r=0}. \label{eq:first-action} \end{equation} Multiplication by \(-\ii\) gives the corresponding contribution to \(t_\delta^\zeta\). %===================================================================== \section{Weighted convolution and leading master estimates} \label{sec:master} %===================================================================== \subsection{Exact two-cone convolution} For a frozen weight \(w\), define \[ J(S;w)=\int R(x)R(S-x)w(x)\dd^4x. \] \begin{lemma}[Weighted cone convolution] \label{lem:weighted} For \(S=(T,q\mathbf n)\), \(T>q>0\), \begin{equation} J(S;w)=\Theta(T-q)\Gamma(S;w), \label{eq:J-unsmoothed} \end{equation} where on the timelike side \begin{equation} \Gamma(S;w) = \frac1{8\pi}\int_0^1\int_0^{2\pi} w(X(S,\theta,\varphi)) \frac{\dd\varphi}{2\pi}\dd\theta, \label{eq:Gamma} \end{equation} and \begin{align} X^0&=\frac{T+q(2\theta-1)}2,\notag\\ \mathbf X &=\frac{q+T(2\theta-1)}2\mathbf n +\sqrt{T^2-q^2}\sqrt{\theta(1-\theta)} (\cos\varphi\,\mathbf e_1+\sin\varphi\,\mathbf e_2). \label{eq:fiber} \end{align} The convolution vanishes outside the future causal cone. There is no additional null-boundary measure. In particular, \(R*R=B\). \end{lemma} \begin{proof} In the rest frame \(S=(\sqrt{S^2},0)\), temporal and radial cone integration gives \[ \frac1{32\pi^2}\int_{S^2} w\left(\frac{\sqrt{S^2}}2, \frac{\sqrt{S^2}}2\boldsymbol\omega\right) \dd\boldsymbol\omega. \] 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 \(R*R=B\). \end{proof} Let \[ d(S)=S^0-|\mathbf S|,\qquad \mathbf n(S)=\mathbf S/|\mathbf S|,\qquad k(S)=(1,-\mathbf n(S)), \] and \(\overline S=(|\mathbf S|,\mathbf S)\). At \(s=q\ell_{\mathbf n}\), define \begin{align} \Gcal[w] &= \frac1{8\pi}\int_0^1w(\theta s)\dd\theta, \label{eq:G}\\ \Acal[w] &= \frac1{8\pi}\int_0^1 \left[ \frac12\partial_0w+ \left(\theta-\frac12\right)\mathbf n\cdot\nabla_{\mathbf x}w +\frac q2\theta(1-\theta)\Delta_\perp w \right](\theta s)\dd\theta, \label{eq:A}\\ \Tcal[w] &= \frac1{8\pi}\int_0^1 \theta\,\ell_{\mathbf n}\cdot\nabla_xw(\theta s) \dd\theta. \label{eq:T} \end{align} Here \(\Delta_\perp=\Delta_{\mathbf x} -(\mathbf n\cdot\nabla_{\mathbf x})^2\). Formula \eqref{eq:fiber} gives \[ \Gamma(s;w)=\Gcal[w],\qquad (\partial_{S^0}\Gamma)(s;w)=\Acal[w]. \] Every propagation derivative of \(\Gamma\) freezes \(w\). \subsection{Boundary regularity} Fix a compact non-tip domain \[ \mathcal U= \{S=(q+d,q\mathbf n): q_0\le q\le q_1,\ \mathbf n\in\Omega,\ |d|\le d_0\}, \] where \(q_0>0\), \(d_00, \quad \kappa=H_*''(0) =-\int_\R|P_*'(t)|^2\dd t<0. \label{eq:moments} \end{equation} The density \(H_*\) is even, so \(H_*'(0)=H_*'''(0)=0\). Product convolution gives the exact translation formula \begin{equation} J_\delta(S;w) = \int\rho(P,Q) J(S-\delta(P+Q);w(\,\cdot+\delta P)) \dd^4P\,\dd^4Q. \label{eq:translation} \end{equation} \begin{lemma}[Leading boundary-layer estimates] \label{lem:master} For \(1\le m=|\gamma|\le5\), uniformly on \(|d(S)|\le\Lambda\delta\) in the fixed non-tip domain, \begin{align} \partial_S^\gamma J_\delta(S;w) ={}& \delta^{-m}H_*^{(m-1)}(d(S)/\delta) k(S)^\gamma\Gcal_{\overline S}[w] +E_{\delta,\gamma}(S;w),\notag\\ |E_{\delta,\gamma}|+ |(\scaleder E_{\delta,\gamma})| \le{}& C_{\Lambda,m}\delta^{1-m}\|w\|_{2m}. \label{eq:leading-master} \end{align} The scale derivative here is at fixed \(S\). Also \(J_\delta\) and its scale derivative are bounded by \(C\|w\|_0\). For \(d(S)>4\delta\), propagation derivatives and their scale derivatives are bounded by \(C_m\|w\|_{2m}\). For \(d(S)<-4\delta\), they vanish. \end{lemma} \begin{proof} Differentiate \(J=\Theta(d)\Gamma\). The unique highest boundary term of order \(m\) is \[ \dirac^{(m-1)}(d) \prod_\mu(\partial_\mu d)^{\gamma_\mu}\Gamma. \] The other boundary terms contain \(\dirac^{(j)}(d)\), \(j\le m-2\), and the interior term is bounded by Lemma~\ref{lem:regularity}. Put \(H=P+Q\), and solve the translated boundary in \(H^0\): \[ H^0= \frac{S^0-|\mathbf S-\delta\mathbf H|}{\delta} = \frac{d(S)}{\delta} +\mathbf n(S)\cdot\mathbf H+O(\delta). \] Temporal coarea contributes \(\delta^{-m}\) for the highest term. The profile density gives \(H_*^{(m-1)}\), the normal gives \(k(S)^\gamma\), and the coefficient gives \(\Gcal_{\overline S}[w]\). Their changes are \(O(\delta)\) in the required temporal derivatives. Lemma~\ref{lem:regularity}, together with the shifted-weight Taylor estimate, bounds these changes by \(C\delta\|w\|_{2m}\). Lower boundary terms are at most \(O(\delta^{1-m})\); interior terms are bounded. For the scale derivative, replace \(\rho\) by \[ \rho^{[1]}=-8\rho-(P,Q)\cdot\nabla\rho. \label{eq:profile-scale} \] Its normal marginal is \[ H_*^{[1]}(z)=-H_*(z)-zH_*'(z), \] and \[ (H_*^{[1]})^{(m-1)}(z) =-mH_*^{(m-1)}(z)-zH_*^{(m)}(z). \] 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, \[ |d(S-\delta H)-d(S)|\le2\sqrt2\,\delta \] 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. \end{proof} 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 \(\kappa\). %===================================================================== \section{Derivative-controlled first correction} \label{sec:first-correction} %===================================================================== Define \[ \mathscr F(z)=\int_{-\infty}^zH_*(h)\dd h,\qquad \mathscr L(z)=\int_{-\infty}^zhH_*(h)\dd h. \] For the boundary operators of the preceding section, set \begin{equation} \Bcal[w] = -\Acal[w]+\frac12\Tcal[w] +\frac14\Gcal[k_{\mathbf n}\cdot\nabla_xw] +\frac1{2q}\Gcal[w]. \label{eq:Boperator} \end{equation} \begin{theorem}[First-correction layer expansion] \label{thm:first-correction} Let \(S_\delta=(q+\delta z,q\mathbf n)\). Uniformly for bounded \(z\) and compact non-tip \((q,\mathbf n)\), \begin{align} J_\delta(S_\delta;w) ={}& \Gcal[w]\mathscr F(z)\notag\\ &+\delta\left[ z\Acal[w]\mathscr F(z)+\mathscr L(z)\Bcal[w] \right]+\mathcal E_\delta(z,q,\mathbf n;w). \label{eq:first-correction} \end{align} For up to four \(z\) derivatives, one tangential derivative in \((q,\mathbf n)\), and \(l\) additional smooth spectator derivatives, \begin{equation} |\partial_z^j\partial_{\mathrm{tan}}^\alpha \partial_{\mathrm{sp}}^\beta\mathcal E_\delta| \le C\delta^2\|w\|_{14+l}, \quad j\le4,\quad|\alpha|\le1,\quad|\beta|\le l. \label{eq:first-error} \end{equation} The same estimate holds for \[ \mathscr S\mathcal E_\delta,\qquad \mathscr S=\delta\partial_\delta\big|_{z,q,\mathbf n} -z\partial_z. \] This operator is the logarithmic-scale derivative at fixed unscaled \(S\). \end{theorem} \begin{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 \[ c_\delta(q,\mathbf n,\mathbf H) = \frac{|\;q\mathbf n-\delta\mathbf H\;|-q +\delta\mathbf n\cdot\mathbf H}{\delta^2}. \] This extends smoothly to \(\delta=0\) on the fixed compact profile support, with \[ c_0=\frac{|\mathbf H_\perp|^2}{2q}. \] Exactly, \[ d(S_\delta-\delta H) =\delta(z-h-\delta c_\delta). \] Make the change \(u=h+\delta c_\delta\). Its temporal Jacobian is one. The retarded indicator becomes \(\Theta(z-u)\), independent of \(\delta\). Thus \eqref{eq:translation} becomes an integral over \(u\le z\) of the smooth integrand \[ \rho(P,H_\delta-P)\, \Gamma(S_\delta-\delta H_\delta; w(\,\cdot+\delta P)), \] where \[ H_\delta= (u-\delta c_\delta+\mathbf n\cdot\mathbf H,\mathbf H). \] 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~\ref{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~\ref{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 \eqref{eq:first-error}. It remains to calculate the first derivative. At \(\delta=0\), the zeroth term is \(\Gcal[w]\mathscr F(z)\). The coefficient/weight derivative contributes \[ \int_{h\le z}\rho(P,Q) \left[(ze_0-H)\cdot\nabla_S\Gamma(s;w) +\Gcal[P\cdot\nabla_xw]\right]\dd P\dd Q. \label{eq:first-integrand} \] The density derivative from \(h=u-\delta c_\delta\) contributes \[ -\frac{\Gcal[w]}{2q} \int\rho(P,Q)|\mathbf H_\perp|^2 \dirac(z-h)\dd P\dd Q. \label{eq:curvature-term} \] The sum density of \(H\) is radial in four Euclidean dimensions. Rotation symmetry and interchange of \(P,Q\) give the density identities \begin{align} \int H\rho\,\dirac(z-h)\dd P\dd Q &=\frac z2 k_{\mathbf n}H_*(z),\notag\\ \int P\rho\,\dirac(z-h)\dd P\dd Q &=\frac z4 k_{\mathbf n}H_*(z). \label{eq:first-moments} \end{align} No division by a marginal density is needed. For \[ B_\perp(z)= \int\rho|\mathbf H_\perp|^2\dirac(z-h)\dd P\dd Q, \] radial integration by parts gives \[ B_\perp'(z)=-zH_*(z),\qquad B_\perp(z)=-\mathscr L(z). \label{eq:transverse-moment} \] For completeness, choose the coordinate \(t=h/\sqrt2\) and three perpendicular Euclidean coordinates \(Y\). If the sum density is \(a(t^2+|Y|^2)\), then \[ B_\perp'(z) = \frac z{\sqrt2}\int(Y_1^2+Y_2^2) a'(z^2/2+|Y|^2)\dd^3Y =-zH_*(z). \] Compact support fixes the integration constant. Along the null boundary, \[ \ell_{\mathbf n}\cdot\nabla_S\Gamma(s;w)=\Tcal[w], \qquad k_{\mathbf n}\cdot\nabla_S\Gamma(s;w) =2\Acal[w]-\Tcal[w]. \] Substitute these identities and \eqref{eq:first-moments} into \eqref{eq:first-integrand}. Then add \eqref{eq:curvature-term} using \eqref{eq:transverse-moment}. The result is precisely the first-order term in \eqref{eq:first-correction}. For direct scale control, repeat the same fixed-domain Taylor argument with the signed profile \(\rho^{[1]}\). It retains rotation and interchange symmetries, so the same calculation applies. Its cumulative moments are \[ \mathscr F^{[1]}(z)=-zH_*(z),\qquad \mathscr L^{[1]}(z)=\mathscr L(z)-z^2H_*(z). \] These are exactly the \(\mathscr S\)-derivatives of the zeroth and first-order terms in \eqref{eq:first-correction}. The remainder for this signed profile has the same bound, proving the scale estimate without differentiating \eqref{eq:first-error}. \end{proof} \begin{remark}[Meaning of the derivative budget] The order fourteen arises from \(2(4+1+2)\): four scaled-normal derivatives, one tangential derivative, and two Taylor derivatives in the scale. It is a sufficient count for the proof, not a claim of optimality. \end{remark} %===================================================================== \section{Integrated-vertex jets and the known six rows} \label{sec:known} %===================================================================== \subsection{Second and third temporal derivatives} At \(S=s=q\ell_{\mathbf n}\), write \[ g_\phi=k_{\mathbf n}\cdot\nabla_x\phi, \qquad D_{\mathrm w}\phi=\partial_{S^0}\phi \quad\text{with }a,r,x\text{ fixed}. \] The notation \(g_\phi\) here is unrelated to the observation cutoff \(g\). Define \begin{equation} \Qcal\phi(a,s,r) = \int_0^1g_\phi(a,s,r,\theta s)\dd\theta, \qquad P_2=-\frac{\kappa}{2}\Gcal[g_\phi] =-\frac{\kappa}{16\pi}\Qcal\phi. \label{eq:P2} \end{equation} Also set \begin{align} \Ccal[w] &=4\Acal[w]+3\Bcal[w]\notag\\ &=\Acal[w]+\frac32\Tcal[w] +\frac34\Gcal[k_{\mathbf n}\cdot\nabla_xw] +\frac{3}{2q}\Gcal[w], \label{eq:Coperator}\\ \Vcal[\phi] &= \frac1{8\pi}\int_0^1 \theta\left[ (\mathbf n\cdot\nabla_{\mathbf x})\partial_{x^0}\phi +\Delta_{\mathbf x}\phi \right](\theta s)\dd\theta, \label{eq:Voperator}\\ P_3 &= \frac{\kappa}{2} \left\{ \Gcal[\phi]-\Vcal[\phi] -\Ccal[g_\phi]-3\Gcal[D_{\mathrm w}g_\phi] \right\}. \label{eq:P3} \end{align} The operations in these formulas act on the \(x\)-weight, with \(a,s,r\) as displayed spectators. \begin{proposition}[Integrated-vertex asymptotics] \label{prop:vertex-jets} On the compact outer null cone, \begin{align} M_{\delta,j} &=-\frac{\nu}{16\pi\delta}\Qcal\phi_j+O_H(1), \label{eq:M-leading}\\ DM_{\delta,j}&=O_H(\delta^{-1}), \label{eq:DM-bound}\\ D^2M_{\delta,j} &=\delta^{-3}P_2[\phi_j]+O_H(\delta^{-2}), \label{eq:D2M}\\ D^3M_{\delta,j} &=\delta^{-3}P_3[\phi_j]+O_H(\delta^{-2}). \label{eq:D3M} \end{align} For the uses below, the remainders and their direct scale derivatives are controlled by \(\|\Phi\|_{17,H}\). The \(D^2M\) estimate remains valid after two \(r\) derivatives. In particular, \(D^3M\) has no inverse-quartic term. \end{proposition} \begin{proof} The first three statements follow from Lemma~\ref{lem:master}. Total \(S\) differentiation splits into propagation and weight differentiation. The only inverse-cubic terms in \(D^2M\) are \[ -\frac12\sum_\rho \partial_{\widetilde S^\rho} \partial_{\widetilde S^0}^2 J_\delta(\widetilde S;\partial_{x^\rho}\phi) \big|_{\widetilde S=s}. \] They give \(P_2\). Parity removes the nominal second-order master coefficient, so \(DM=O(\delta^{-1})\). For \(D^3M\), use Theorem~\ref{thm:first-correction}. At zero, \[ \mathscr F^{(4)}(0)=H_*'''(0)=0,\quad (z\mathscr F)^{(4)}(0)=4\kappa,\quad \mathscr L^{(4)}(0)=3\kappa. \] Thus, for frozen weights, \[ \partial_{\widetilde S^0}^4J_\delta(s;w) =\kappa\delta^{-3}\Ccal[w]+O(\delta^{-2}). \label{eq:fourth-time} \] For a spatial propagation derivative, define \[ \mathscr T_i = n_i\partial_q+ q^{-1}(\delta_{ij}-n_in_j)\partial_{n_j}. \] In layer coordinates, at fixed unscaled time, \[ \partial_{\widetilde S^i} =\mathscr T_i-\frac{n_i}{\delta}\partial_z, \qquad \partial_{\widetilde S^0}=\delta^{-1}\partial_z. \] Therefore \[ \partial_{\widetilde S^0}^3 \partial_{\widetilde S^i}J_\delta(s;w) = \kappa\delta^{-3} \{\mathscr T_i\Gcal[w]-n_i\Ccal[w]\} +O(\delta^{-2}). \label{eq:mixed-fourth} \] For frozen \(w\), \[ \mathscr T_i\Gcal[w] = \frac1{8\pi}\int_0^1 \theta(n_i\partial_0w+\partial_iw)(\theta s)\dd\theta. \] Summing \eqref{eq:fourth-time} and \eqref{eq:mixed-fourth} with weights \(\partial_{x^\rho}\phi\) gives \[ \kappa\delta^{-3} \{\Ccal[g_\phi]+\Vcal[\phi]\}. \] The kinetic vertex contributes minus one half of this. When one of the three total time derivatives falls on the weight, the coefficient is \(-3\kappa\Gcal[D_{\mathrm w}g_\phi]/2\). Terms with two or three weight derivatives are lower order. The mass vertex contributes \(\kappa\Gcal[\phi]/2\) at inverse-cubic order. These are exactly \eqref{eq:P3}. The first-correction estimate uses fourteen weight derivatives; the output derivative raises this to fifteen. Allowing two incidence spectators gives seventeen. The remaining leading-master operations require no more: the interior estimates below use at most eleven. Repeating the calculation with \(\rho^{[1]}\) gives the scale-leading coefficients \(-3P_2\) and \(-3P_3\), and the same remainder bounds. \end{proof} \subsection{Affine checks of the third-derivative coefficient} These checks are local: weights may be cut off outside a fixed neighborhood of all relevant convolution fibers. For \(\phi=1\), \[ P_3=\frac{\kappa}{16\pi}. \] This agrees with \(M_\delta=J_\delta(S;1)/2\). For \(\phi=x^0\), symmetry gives the exact identity \[ J_\delta(S;x^0)=\frac{S^0}{2}J_\delta(S;1), \] hence \[ M_\delta=-\frac12D J_\delta(S;1) +\frac{S^0}{4}J_\delta(S;1). \] Formula \eqref{eq:P3} gives \[ P_3=\frac{\kappa}{32\pi}(q-3/q), \] which agrees with direct differentiation. For \(\phi=x^j\), \[ J_\delta(S;x^j)=\frac{S^j}{2}J_\delta(S;1), \] and \eqref{eq:P3} gives \[ P_3=\frac{\kappa n_j}{32\pi}(q+3/q). \] These identities audit signs and factors; the arbitrary-test proof is Proposition~\ref{prop:vertex-jets}. \subsection{Six-row coefficients} Define \begin{align} C_{47}(\Phi) &= \frac{\ii\kappa}{16\pi} \int\dd^4a\,\dd R(s)\,\dd R(t) [\partial_{r^0}\Qcal\phi_2(a,s,r)]_{r=s-t}, \label{eq:C47}\\ C_{5a}(\Phi) &= \frac{\ii\kappa}{16\pi} \int\dd^4a\,\dd R(s)\,\dd R(t)\, f(a+s)[\partial_{r^0}^2\Qcal\phi_2(a,s,r)]_{r=-t}, \label{eq:C5a}\\ C_{\mathrm{resp}}(\Phi) &= -\frac{\ii\kappa}{16\pi} \int\dd^4a\,\dd R(s)\, f(a+s)\Qcal\phi_2(a,s,0). \label{eq:Cresp} \end{align} \begin{proposition}[Known partial assembly] \label{prop:six} The six-row family with \(5z\) removed satisfies \[ t_\delta^\zeta-t_{\delta,5z}^\zeta = \delta^{-3}(C_{47}+C_{5a}+C_{\mathrm{resp}}) +O_H(\delta^{-2}), \] with the same \(O_H(\delta^{-2})\) bound for the scale derivative of the remainder. \end{proposition} \begin{proof} The \(47\) adjoint becomes \[ \int\dd^4a\,\dd R(s)\,\dd R(t) [(D+\partial_{r^0})\partial_{r^0}D M_{\delta,2}]_{r=s-t}. \] Only \(D^2\partial_{r^0}M\) contributes at inverse-cubic order. Multiplication by \(-\ii\) gives \(C_{47}\). The \(5a\) adjoint becomes \[ \int\dd^4a\,\dd R(s)\,\dd R(t) [fD^2\partial_{r^0}^2M+D\partial_{r^0}^2M]_{r=-t}. \] Its first term gives \(C_{5a}\). No estimate divides by the inner radius: after the adjoint, inner derivatives act on the smooth \(r\)-weight. The cone measure is finite through its vertex. In \eqref{eq:response-adjoint}, only \(-fD^2M\) contributes at inverse-cubic order, giving \(C_{\mathrm{resp}}\). Every other response term is \(O(\delta^{-1})\). The first-component action \eqref{eq:first-action} is also \(O(\delta^{-1})\). Uniform master estimates and finite compact seed measures give the stated remainder bounds, including their direct scale derivatives. \end{proof} %===================================================================== \section{Exact signed internal convolution for \(5z\)} \label{sec:signed} %===================================================================== In \((a,S,r)\), the negative placement is \[ K_{5z}^{22} = D[f(a+S)R'(S-r)R''(r)]. \] Transferring only the outer derivative in the target gives \begin{equation} t_{\delta,5z}^\zeta = \ii\int R'(S-r)R''(r)\, U_\delta(a,S,r)\dd^4a\,\dd^4S\,\dd^4r, \qquad U_\delta=f(a+S)DM_{\delta,2}. \label{eq:5z-signed-start} \end{equation} The auxiliary amplitude \(U_\delta\) is not the smoothed propagator \(A_\delta\). \subsection{Frozen-weight identity} For a frozen internal weight \(v(r)\), define \[ J_{\mathrm{int}}(S;v)=\int R(r)R(S-r)v(r)\dd^4r. \] By Lemma~\ref{lem:weighted}, \[ J_{\mathrm{int}}(S;v)=\Theta(d(S))\Gamma_{\mathrm{int}}(S;v). \] \begin{lemma}[Signed internal integration] \label{lem:internal-identity} With propagation derivatives freezing the weight, \begin{align} \int R'(S-r)R''(r)v(r)\dd^4r ={}& D_{\mathrm p}^3J_{\mathrm{int}}(S;v) -2D_{\mathrm p}^2J_{\mathrm{int}}(S;\partial_{r^0}v)\notag\\ &+D_{\mathrm p}J_{\mathrm{int}}(S;\partial_{r^0}^2v). \label{eq:internal-identity} \end{align} The identity includes the internal cone vertex. \end{lemma} \begin{proof} Transfer the two derivatives on \(R''(r)\): \[ \partial_{r^0}^2[R'(S-r)v(r)] = R'''(S-r)v-2R''(S-r)\partial_{r^0}v +R'(S-r)\partial_{r^0}^2v. \] Pair with \(R(r)\). Each resulting term is a frozen propagation derivative of \(J_{\mathrm{int}}\). The computation is an identity for the proper pushforward of an independent two-cone distribution and remains valid at the vertex. \end{proof} \subsection{Boundary/interior decomposition} For \(s=(q,q\mathbf n)\), let \[ \Gcal_j^{\,r}[v] = \left.(D_{\mathrm p}^j\Gamma_{\mathrm{int}})(S;v) \right|_{S=s}. \] Thus \[ \Gcal_0^{\,r}[v] =\frac1{8\pi}\int_0^1v(\lambda s)\dd\lambda, \qquad \Gcal_1^{\,r}[v]=\Acal^{\,r}[v], \] where \(\Acal^{\,r}\) is \eqref{eq:A} with \(x\) replaced by \(r\). Write \(\Gamma_j^r(S;v)=D_{\mathrm p}^j \Gamma_{\mathrm{int}}(S;v)\). \begin{proposition}[Exact \(5z\) decomposition] \label{prop:5z-decomposition} At every sufficiently small positive scale, \[ t_{\delta,5z}^\zeta = t_{\delta,5z}^{\mathrm{bdry}} +t_{\delta,5z}^{\mathrm{int}}, \] where \begin{align} t_{\delta,5z}^{\mathrm{bdry}} = \ii\int\dd^4a\,\dd^3\mathbf s\, \bigl\{& \Gcal_0^{\,r}[D^2U_\delta] -\Gcal_1^{\,r}[DU_\delta] +2\Gcal_0^{\,r}[\partial_{r^0}DU_\delta]\notag\\ &+\Gcal_2^{\,r}[U_\delta] -2\Gcal_1^{\,r}[\partial_{r^0}U_\delta] +\Gcal_0^{\,r}[\partial_{r^0}^2U_\delta] \bigr\}_{S=s}, \label{eq:5z-boundary} \end{align} and \begin{align} t_{\delta,5z}^{\mathrm{int}} = \ii\int_{d(S)>0}\dd^4a\,\dd^4S\, \bigl\{& \Gamma_3^r(S;U_\delta) -2\Gamma_2^r(S;\partial_{r^0}U_\delta)\notag\\ &+\Gamma_1^r(S;\partial_{r^0}^2U_\delta) \bigr\}. \label{eq:5z-interior} \end{align} Every internal coefficient derivative freezes its \(r\)-weight. The \(D\) derivatives on \(U_\delta\) are total temporal derivatives at fixed \(a,r\). \end{proposition} \begin{proof} Apply Lemma~\ref{lem:internal-identity} at each fixed \(a,S\), with its actual parameter-dependent weight \(U_\delta(a,S,\cdot)\). For temporal derivatives, \(Dd=1\), \(D^2d=0\), so \begin{align*} D_{\mathrm p}^3(\Theta(d)\Gamma) &=\Theta(d)\Gamma_3 +3\dirac(d)\Gamma_2 +3\dirac'(d)\Gamma_1+\dirac''(d)\Gamma_0,\\ D_{\mathrm p}^2(\Theta(d)\Gamma) &=\Theta(d)\Gamma_2 +2\dirac(d)\Gamma_1+\dirac'(d)\Gamma_0,\\ D_{\mathrm p}(\Theta(d)\Gamma) &=\Theta(d)\Gamma_1+\dirac(d)\Gamma_0. \end{align*} Integrate \(S^0\), with \(d=S^0-|\mathbf S|\). For example, \[ \int\dirac''(d)\Gamma_0(S;U_\delta(S))\dd S^0 = \Gcal_2^r[U_\delta] +2\Gcal_1^r[DU_\delta] +\Gcal_0^r[D^2U_\delta], \] whereas \[ 3\int\dirac'(d)\Gamma_1(S;U_\delta(S))\dd S^0 = -3\Gcal_2^r[U_\delta]-3\Gcal_1^r[DU_\delta]. \] Adding the \(3\dirac(d)\Gamma_2\) term leaves \[ \Gcal_2^r[U_\delta]-\Gcal_1^r[DU_\delta] +\Gcal_0^r[D^2U_\delta]. \] The \(-2D_{\mathrm p}^2\) term similarly leaves \[ -2\Gcal_1^r[\partial_{r^0}U_\delta] +2\Gcal_0^r[\partial_{r^0}DU_\delta]. \] The remaining boundary term is \(\Gcal_0^r[\partial_{r^0}^2U_\delta]\). The undifferentiated Heaviside terms give \eqref{eq:5z-interior}. \end{proof} This signed identity performs the internal-tip matching exactly. It does not assign an internal radius to the smoothing scale. %===================================================================== \section{The \(5z\) coefficient and complete leading assembly} \label{sec:seven} %===================================================================== \subsection{Explicit \(5z\) coefficient} In the following formulas, \(P_2,P_3\) are computed from \(\phi_2\), with their \(r\) dependence retained. \begin{theorem}[\(5z\) leading expansion] \label{thm:5z} Define \begin{align} C_{5z}(\Phi) = \ii\int\dd^4a\,\dd^3\mathbf s\, \bigl\{& f(a+s)\Gcal_0^r[P_3]+2\Gcal_0^r[P_2]\notag\\ &-f(a+s)\Gcal_1^r[P_2] +2f(a+s)\Gcal_0^r[\partial_{r^0}P_2] \bigr\}. \label{eq:C5z} \end{align} Then \[ t_{\delta,5z}^\zeta = \delta^{-3}C_{5z}+\mathcal R_{\delta,5z}, \] and \[ |\mathcal R_{\delta,5z}(\Phi)| +|(\scaleder\mathcal R_{\delta,5z})(\Phi)| \le C_H\delta^{-2}\|\Phi\|_{17,H}. \] There is no inverse-quartic leading term. \end{theorem} \begin{proof} Since \(Df=1\) and \(D^2f=0\) on the retained patch, \[ D^2U_\delta=fD^3M_\delta+2D^2M_\delta, \qquad DU_\delta=fD^2M_\delta+DM_\delta. \] Substitute Proposition~\ref{prop:vertex-jets} into \eqref{eq:5z-boundary}. Its inverse-cubic terms are exactly \eqref{eq:C5z}. The remaining boundary terms use \(U_\delta=fDM_\delta\), which is \(O(\delta^{-1})\) on the null boundary, including the required \(r\) derivatives. The coefficient \(\Gcal_2^r\) needs at most four \(r\)-weight derivatives; the other terms need fewer. They are controlled by the stated common seminorm. For the interior term, Lemma~\ref{lem:regularity} needs at most six \(r\)-weight derivatives. On \(04\delta\), the master and internal coefficient derivatives are uniformly bounded and all integration ranges are compact. This part is \(O_H(1)\). Therefore the interior contribution is lower order than \(\delta^{-3}\), with ample room in the claimed remainder. For scale control, repeat the boundary and interior estimates with \(\rho^{[1]}\). The leading \(P_2,P_3\) coefficients are multiplied by \(-3\), so the boundary leading coefficient is \(-3C_{5z}\), as required. Finally, the exact decomposition contains no boundary term more singular than \(D^3M\), and its nominal inverse-quartic coefficient vanishes by \(\mathscr F^{(4)}(0)=H_*'''(0)=0\). The interior term is lower order. This proves the signed inverse-quartic cancellation. \end{proof} \begin{remark}[Support and endpoints] The support of \(C_{5z}\) is contained in the closure of \[ S=s\text{ future null},\qquad r=\lambda s,\qquad x=\theta s,\qquad 0\le\lambda,\theta\le1, \] intersected with the retained localization. The \(x\)-localization removes its segment endpoints, but \(r=0\) remains. No restriction of a separated distribution to \(r=0\) is used. \end{remark} \subsection{Distributional inner-tip cancellation} \begin{lemma}[Cone fundamental solution] \label{lem:fundamental} With normalization \eqref{eq:R}, \[ \Box_\eta R=\delta^{(4)}. \] The reflected advanced cone measure satisfies the same identity. \end{lemma} \begin{proof} For a compact smooth test \(\psi\), let \[ h(t,\rho)=\frac1{4\pi}\int_{S^2} \psi(t,\rho\mathbf m)\dd\mathbf m. \] Spherical averaging gives \[ \pair R{\Box_\eta\psi} = \int_0^\infty [\rho(h_{tt}-h_{\rho\rho})-2h_\rho]_{t=\rho}\dd\rho. \] The integrand is \[ \frac{\dd}{\dd\rho} [\rho(h_t-h_\rho)-h]_{t=\rho}. \] The compact-support boundary at infinity is zero. At \(\rho=0\), the bracket is \(-\psi(0)\). Therefore the integral is \(\psi(0)\). Reflection preserves \(\Box_\eta\) and \(\delta^{(4)}\), proving the advanced assertion. \end{proof} \begin{theorem}[\(5a\)--response coincidence cancellation] \label{thm:tip-cancellation} The leading coefficients satisfy \begin{align} C_{5a}+C_{\mathrm{resp}} = \frac{\ii\kappa}{16\pi} \int\dd^4a\,\dd R(s)\,\dd R(t)\, f(a+s)\, [\Delta_{\mathbf r}\Qcal\phi_2(a,s,r)]_{r=-t}. \label{eq:tip-cancellation} \end{align} This is a distributional identity including the inner vertex. \end{theorem} \begin{proof} Lemma~\ref{lem:fundamental} gives \[ \partial_{r^0}^2R(-r) = \Delta_{\mathbf r}R(-r)+\delta^{(4)}(r). \] Pair this identity with the smooth compact \(r\)-weight \(\Qcal\phi_2(a,s,r)\), freezing \(a,s\). Then \[ \int\dd R(t) [\partial_{r^0}^2\Qcal\phi_2]_{r=-t} = \Qcal\phi_2(a,s,0) +\int\dd R(t) [\Delta_{\mathbf r}\Qcal\phi_2]_{r=-t}. \] The first term, multiplied by the prefactor of \(C_{5a}\), is exactly \(-C_{\mathrm{resp}}\). This proves \eqref{eq:tip-cancellation}. \end{proof} \subsection{Seven-row theorem} Define \begin{align} C_{\mathrm{seven}} &= C_{47}+C_{5a}+C_{\mathrm{resp}}+C_{5z}\notag\\ &= C_{47}+C_{5z} +\frac{\ii\kappa}{16\pi} \int\dd^4a\,\dd R(s)\,\dd R(t)\, f(a+s) [\Delta_{\mathbf r}\Qcal\phi_2(a,s,r)]_{r=-t}. \label{eq:Cseven} \end{align} \begin{theorem}[Complete seven-row leading assembly] \label{thm:seven} The expansion \eqref{eq:main-expansion} and remainder bound \eqref{eq:main-remainder} hold with coefficient \eqref{eq:Cseven}. In particular, \[ \scaleder t_\delta^\zeta = -3\delta^{-3}C_{\mathrm{seven}} +\scaleder\mathcal R_\delta. \] The coefficient is a finite-order distribution on arbitrary localized joint tests, including tests crossing \(N\). \end{theorem} \begin{proof} Add Proposition~\ref{prop:six} and Theorem~\ref{thm:5z}. The first-component inverse-scale term is lower order and remains inside the \(O(\delta^{-2})\) remainder. Theorem~\ref{thm:tip-cancellation} gives the second form of the coefficient. Every formula uses compact segment or cone-measure integration of finite ambient test jets, so it defines a continuous distribution. \end{proof} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.44\linewidth}Y@{}} \toprule Mechanism & Verdict\\ \midrule Rows \(6\) and \(9d\), zeroth-incidence first derivative & Exact algebraic cancellation.\\ Nominal order-two master coefficient & Vanishes by even-profile parity.\\ Nominal inverse-quartic \(5z\) coefficient & Vanishes after exact signed internal reduction.\\ Positive-\(5a\) inner-tip delta and response cubic term & Cancel by the cone fundamental solution identity.\\ Collected inverse-cubic seven-row coefficient & Survives; detected in Section~\ref{sec:obstruction}.\\ Raw weak distributional limit & Does not exist.\\ Leading-subtracted finite part & Not established.\\ \bottomrule \end{tabularx} \end{center} The cancellation in \eqref{eq:tip-cancellation} removes an explicit incidence term from one representation. It does not make the surviving spatial-Laplacian distribution zero, and it does not amount to a canonical restriction of the complete coefficient to \(N\). %===================================================================== \section{Independent separated obstruction and necessary support} \label{sec:obstruction} %===================================================================== \subsection{Exact isolation} Choose \[ a=0,\qquad s_*=2\ell_{\mathbf n_*},\qquad t_*=T_*(1,-\mathbf n_*),\qquad T_*=h/4, \] with \(z=a-t_*\) and \(y-a=\ell_{\mathbf n_*}\). The point lies where \(\zeta=1\). On the positive seed, \(s=b-a\) and \(t=a-z\) are future null, nonzero, and nonparallel. Thus \[ (b-z)^2=(s+t)^2>0,\qquad r=-t\ne0. \] A fixed small neighborhood \(O_A\) excludes the \(47\) and \(5z\) supports, both of which require \(b-z\) null, and excludes all response supports, which require \(r=0\). Therefore, for every test supported in \(O_A\), \begin{equation} t_\delta^\zeta(\Phi)=t_{\delta,5a}^\zeta(\Phi) \label{eq:isolation} \end{equation} at every positive scale. Differentiation does not enlarge internal supports, and only the crossed coefficient is smoothed. \subsection{Fixed inverse-cubic witness} In \(O_A\), let \[ d_t=t^0-|\mathbf t|,\qquad d_y=(y^0-a^0)-\mathbf n\cdot(\mathbf y-\mathbf a). \] Take a nonnegative nonzero compact tangential bump \(\psi_A\) on the positive seed and aligned crossed segment, with fixed normal cutoffs. Set \[ \Phi_{A,2}=\frac{d_t^2}{2}d_y\psi_A,\qquad \Phi_{A,1}=0. \] Choose its support where \(\zeta=1\). At \(r=-t\), the two \(r^0\) derivatives in \(C_{5a}\) must hit \(d_t^2/2\). On the crossed segment, the output normal derivative must hit \(d_y\). Since \(k_{\mathbf n}\cdot\nabla_xd_y=2\), \begin{equation} c_A=C_{5a}(\Phi_A) = \frac{\ii\kappa}{8\pi} \int\dd^4a\,\dd R(s)\,\dd R(t)\, f(a+s)\int_0^1\psi_A(a,s,t,\theta)\dd\theta \ne0. \label{eq:witness} \end{equation} \begin{theorem}[Separated raw failure] \label{thm:raw} The complete seven-row family has no weak distributional limit. The fixed test above satisfies \[ t_\delta^\zeta(\Phi_A) = \delta^{-3}c_A+O(\delta^{-2}),\qquad c_A\ne0. \] Moreover, \[ C_{\mathrm{seven}}(\Phi_A)=C_{5a}(\Phi_A)\ne0. \] \end{theorem} \begin{proof} Equation \eqref{eq:isolation} and the \(5a\) calculation in Proposition~\ref{prop:six} prove the fixed-test expansion. A weak limit would give a finite limit on every fixed compact test, contradicting this expansion. Theorem~\ref{thm:seven} identifies its coefficient with \(C_{\mathrm{seven}}(\Phi_A)\). The failure proof itself needs no \(5z\) asymptotic theorem. \end{proof} \subsection{Correction-support obstruction} \begin{theorem}[No incidence-supported correction] \label{thm:correction} For every \(c_\delta\in\CS\), the family \(t_\delta^\zeta+c_\delta\) fails to converge weakly. The same conclusion holds for \(\CN,\CL\), unions of their permitted strata, and all order- or component-restricted subclasses. \end{theorem} \begin{proof} The fixed support of \(\Phi_A\) is disjoint from \(\mathcal S=\{z=a\}\). Therefore \(\pair{c_\delta}{\Phi_A}=0\) at every scale, independently of its order and scale dependence. The unbounded pairing of Theorem~\ref{thm:raw} remains. All other listed supports are contained in \(\mathcal S\). \end{proof} Define \[ \Sigma_{5a}^{\mathrm{iso}} = \left\{ \begin{array}{l} s=b-a,\ t=a-z\text{ future null and nonzero},\\ s,t\text{ nonparallel},\ y=a+\theta s,\ 0<\theta<1 \end{array} \right\}. \] \begin{theorem}[Necessary support for a larger correction rule] \label{thm:necessary} If \(Z\subset W\) is closed and a correction family supported in \(Z\) makes \(t_\delta^\zeta+c_\delta\) converge weakly, then \[ \Sigma_{5a}^{\mathrm{iso}}\cap\{\zeta>0\}\subset Z. \] \end{theorem} \begin{proof} A point of the displayed stratum outside \(Z\) has a small neighborhood disjoint from \(Z\). The same nonparallel support isolation applies there. Use the preceding normal-polynomial witness with a nonnegative tangential bump. All three needed normal derivatives must hit the vanishing polynomial factors. The localization contributes its positive value on the stratum; no cutoff derivative survives. Because \(f>0\) and \(\kappa<0\), the coefficient is nonzero. The correction pairs to zero with that fixed test, a contradiction. \end{proof} This is a necessary support condition, not a sufficient subtraction or extension theorem. %===================================================================== \section{Response jets, activity, and release boundaries} \label{sec:release} %===================================================================== \subsection{Individual response and first-component jets} Let \[ \tau=(a,\mathbf n,\alpha,\beta),\qquad \nu_N=(r,u,v,\xi),\qquad \dd\tau=\dd^4a\,\dd\mathbf n\,\dd\alpha\,\dd\beta. \] Define insertion by its action: \[ (\Ecal_\gamma c)(\Phi) = \int c(\tau)[\partial_{\nu_N}^\gamma\Phi]_{\nu_N=0} \dd\tau. \] On the outer cone, \[ \dd R(s)\dd\theta = \frac1{4\pi}\dd\mathbf n\,\dd\beta\,\dd\alpha, \qquad \alpha=\beta\theta, \qquad k_{\mathbf n}\cdot\nabla_x=\partial_v. \] Put \(f_N=1+a^0+\beta\), \(\zeta_N=\zeta|_N\). Then the individual response coefficient is \[ C_{\mathrm{resp}} = \Ecal_v\left(-\frac{\ii\kappa}{64\pi^2}f_N\zeta_N\right) \] in the second component. The cutoff has \(\partial_v\zeta=0\) on \(N\), so there is no additional cutoff jet. Tests \(v\psi(\tau)\) detect this nonzero order-one jet. If \(\Phi_2=0\), the complete first component satisfies \[ t_\delta^\zeta(\Phi)=\delta^{-1}A_B(\Phi)+O_H(1), \] with the same bounded scale remainder, where \[ A_B(\Phi)= -\frac{\ii\nu}{16\pi} \int\dd^4a\,\dd R(s)\, f(a+s)\Qcal(\phi_1+\partial_{r^1}\phi_1)(a,s,0). \] Its normal-jet representation is \[ A_B= \Ecal_v\left(-\frac{\ii\nu}{64\pi^2}f_N\zeta_N\right) + \Ecal_{vr^1}\left(-\frac{\ii\nu}{64\pi^2}f_N\zeta_N\right). \] The corresponding first cutoff derivatives and mixed derivative vanish on \(N\). Tests \(v\psi\) and \(r^1v\psi\) independently detect the two jets. These are statements about individual coefficients. They do not override the cross-sector identity \eqref{eq:tip-cancellation} for the second component. \subsection{Positive-scale activity} For completeness, all seven retained row contractions remain nonzero at a common sufficiently small positive scale. At a non-tip null point \(x=\alpha\ell_{\mathbf n}\), direct temporal cone integration gives \[ \delta A_\delta(x)\longrightarrow\frac{P_*(0)}{4\pi\alpha}, \quad \delta^2\nabla A_\delta(x)\longrightarrow0, \quad \delta^3\Delta_{\mathrm E}A_\delta(x) \longrightarrow\frac{2P_*''(0)}{4\pi\alpha}. \] Here \(P_*''(0)<0\). Consequently, for positive \(\alpha,\lambda\), \[ \delta^4w_\delta(\,(\alpha+\lambda)\ell_{\mathbf n}, \alpha\ell_{\mathbf n}\,) \longrightarrow -\frac{P_*(0)P_*''(0)}{16\pi^2\alpha\lambda}>0. \] These limits follow by expanding the translated non-tip cone formula for \(A_\delta\); compact profile support permits differentiation and dominated convergence. At regular independent cone points, the highest normal polynomials of the two-line kernels are proportional to \[ \sigma_0\tau_0(\sigma_0+\tau_0), \qquad f\,\sigma_0^2\tau_0^2. \] The response rows also have nonzero highest line or incidence jets, as their separate adjoints show. The multiplier is nonvanishing near a central aligned point, and each regular seed stratum meets that neighborhood. Multiplication by a smooth nonvanishing function is locally invertible on distributions. Thus the nonzero internal row jets give nonzero contractions. A common scale exists because there are finitely many rows. A complete kinetic witness is obtained in the first component by choosing \[ \Phi_{B,1}=2v\psi(\tau),\qquad \Phi_{B,2}=0 \] with normal cutoffs constant near \(N\), where \(\zeta=1\). It has \[ A_B(\Phi_B)= -\frac{\ii\nu}{8\pi} \int\dd^4a\,\dd R(s)\, f(a+s)\int_0^1\psi(a,\mathbf n,\beta,\theta)\dd\theta\ne0. \] The mass-vertex contribution is bounded. Hence the complete kinetic contraction is nonzero for all sufficiently small scales. \subsection{Exact mathematical release} \begin{theorem}[Release] \label{thm:release} For the model fixed in this paper: \begin{enumerate}[label=\textup{(\roman*)}] \item all internal rows and positive-scale contractions exist, including the inner cone vertices; \item the advanced root and local weight identities hold; \item the response and seed adjoints are exact; \item the first-correction master expansion has the stated mixed-derivative and direct scale-remainder control; \item \(P_2,P_3\), and \(C_{5z}\) are explicitly calculated; \item the complete seven-row leading expansion is \eqref{eq:main-expansion}, with \eqref{eq:main-remainder}; \item the inner-tip cancellation is \eqref{eq:tip-cancellation}; \item the complete leading coefficient survives and the raw family fails to converge; \item incidence-supported corrections fail, and the necessary larger-support condition of Theorem~\ref{thm:necessary} holds. \end{enumerate} No complete lower-order singular expansion, subtracted finite part, profile-independent limit, physical identification, or admitted quantum coefficient is established. \end{theorem} \begin{proof} The construction and exact adjoints prove (i)--(iii). Theorem~\ref{thm:first-correction} proves (iv). Proposition~\ref{prop:vertex-jets} and Theorem~\ref{thm:5z} prove (v). Theorems~\ref{thm:seven} and \ref{thm:tip-cancellation} prove (vi)--(vii). The separated witness proves (viii), independently of the negative-placement asymptotics. Theorems~\ref{thm:correction} and \ref{thm:necessary} prove (ix). \end{proof} \paragraph{Remaining analytic problem.} A finite-part theorem must calculate every surviving lower singular coefficient and then prove a convergent remainder or an integrable logarithmic-scale derivative. The present leading-order bound does neither. Possible inverse-square, inverse-scale, or logarithmic terms are not determined here. \paragraph{Submission and verification boundary.} The source is standalone and intended for pdfLaTeX. No successful compilation, numerical execution, machine proof, referee approval, or verification of external manuscript claims is asserted. %===================================================================== \appendix \section{Circle-average derivatives and boundary regularity} \label{app:circle} %===================================================================== For a smooth function \(F\) on a transverse Euclidean plane, let \[ \mathcal M_F(p,h)= \int_0^{2\pi} F(p+\sqrt h\,(\cos\theta,\sin\theta)) \frac{\dd\theta}{2\pi}, \qquad h\ge0. \] \begin{lemma}[Exact squared-radius identity] \label{lem:circle} For \(j\ge1\), \[ \partial_h^j\mathcal M_F(p,h) = \frac1{4^j(j-1)!}\int_0^1 (1-v)^{j-1}\mathcal M_{\Delta^jF}(p,vh)\dd v. \] In particular, \[ |\partial_h^j\mathcal M_F| \le\frac1{4^jj!}\|\Delta^jF\|_0. \] Mixed derivatives with \(l\) moving-base or frame parameters require at most \(2j+l\) derivatives of \(F\). The Taylor expansion \[ \mathcal M_F(p,h) = \sum_{j=0}^{N-1} \frac{h^j}{4^j(j!)^2}\Delta^jF(p)+\mathcal R_N(p,h) \] has \[ |\partial_h^q\mathcal R_N| \le C_N h^{N-q}\|F\|_{2N}\quad(0\le q