\documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern,microtype} \usepackage[margin=23mm,headheight=15pt]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,longtable,array,multirow} \usepackage[dvipsnames]{xcolor} \usepackage{enumitem,fancyhdr,listings} \usepackage[hidelinks,pdfauthor={Artificial Hyperintelligence Eve; research prepared for Maciej Nowicki},pdftitle={SERAPH-5 / SAELYRA: Blocking-Slack Geometry and Verified Convex-Body Atlases}]{hyperref} \definecolor{ink}{HTML}{213A4B} \definecolor{muted}{HTML}{52616B} \pagestyle{fancy}\fancyhf{} \fancyhead[L]{\small\textsc{SERAPH-5 / SAELYRA}} \fancyhead[R]{\small Research manuscript / v6.0.0} \fancyfoot[C]{\thepage} \setlength{\parindent}{0pt}\setlength{\parskip}{6pt} \setlist{nosep,leftmargin=1.6em} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition}\newtheorem{definition}[theorem]{Definition} \theoremstyle{remark}\newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R}\newcommand{\Q}{\mathbb Q}\newcommand{\Z}{\mathbb Z} \newcommand{\U}{\mathcal U}\newcommand{\conv}{\operatorname{conv}} \newcommand{\ill}{\operatorname{ill}}\newcommand{\intr}{\operatorname{int}} \newcommand{\supp}{\operatorname{supp}}\newcommand{\vertx}{\operatorname{vert}} \newcommand{\dist}{\operatorname{dist}}\renewcommand{\dh}{d_H^{\infty}} \newcommand{\ip}[2]{\langle #1,#2\rangle} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\sone}{\Sigma_1}\newcommand{\sinf}{\Sigma_\infty} \newcommand{\Binf}{B_\infty^n}\newcommand{\Bone}{B_1^n} \newcommand{\bx}{\beta_D} \lstset{basicstyle=\ttfamily\fontsize{7.3}{8.5}\selectfont,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,frame=single,rulecolor=\color{black!20},tabsize=4} \begin{document} \begin{titlepage} \color{ink} {\small\textsc{Research series / convex geometry / 06}}\par \vspace{20mm} {\Huge\bfseries SERAPH-5}\par \vspace{3mm} {\fontsize{38}{42}\selectfont\bfseries SAELYRA}\par \vspace{12mm} {\LARGE Blocking-Slack Geometry\par and Verified Convex-Body Atlases}\par \vspace{9mm} {\large A continuous certificate, a rational neighborhood theorem,\par and an exact finite-integer reformulation}\par \vspace{18mm} {\large Artificial Hyperintelligence Eve}\par Research prepared for Maciej Nowicki\par Version 6.0.0 \enspace / \enspace 3 October 2026 \vfill \color{black} \hrule\vspace{4mm} \textbf{Status: partial for the unrestricted five-dimensional target.}\par Complete proofs are supplied for the certificate and reduction theorems. Exact independent checks certify 361 polytope templates and explicit neighborhoods. No universal five-dimensional integer level has been verified; the assertion that every unconditional five-dimensional body has illumination number at most 32 is not proved here.\par \small This is an AI-generated research draft. Publication priority, external peer review, and proof-assistant verification have not been established. Earlier manuscripts are preserved as research records, not silently promoted to independently validated results. \end{titlepage} \pagenumbering{arabic} \section*{Abstract}\addcontentsline{toc}{section}{Abstract} We return to the general, rather than a specially decomposable, illumination problem. For a convex body $K$, a point $x\in K$, and a direction $d$, we minimize supporting-function slack over every unit normal that blocks $d$. Taking the best direction and then the worst point defines a blocking-slack functional $\beta_D(K)$. We prove \[ D\text{ illuminates }K\iff\beta_D(K)>0, \qquad |\beta_D(K)-\beta_D(L)|\le 2\dh(K,L). \] Optimization over $m$ normalized directions is attained and preserves this Lipschitz bound. A smooth-to-cube example explains quantitatively why pointwise angular margins do not provide the same control. A finite near-facet certificate yields an explicit common-step interior displacement for every convex body in a full Hausdorff neighborhood of a rational polytope. A vertex--facet slack gap gives an inexpensive sufficient certificate; a finite hyperplane-arrangement test gives a complete rational certificate language. The resulting infinite rational atlas covers exactly the normalized bodies with illumination number at most $m$. For each fixed dimension and direction count, we also give an all-integer statement $\Phi_{n,m,N}$ and prove \[ \bigl[\forall K\in\U_n,\ \ill(K)\le m\bigr] \iff \bigl[\exists N\ge1:\Phi_{n,m,N}\bigr]. \] In dimension five the local test uses the explicit integer thresholds $17N$ and $-16N$. This is a proved equivalence, not a verified level of the original conjecture. The general finite-net strategy has antecedents in Zong's quantitative program; our contribution is the direct blocking-normal formulation and its explicit certificate implementation, with priority not asserted. Two independent exact-arithmetic implementations certify 360 earlier unit-coordinate templates, each with a closed neighborhood radius at least $1/6156$. A further irregular five-dimensional polytope admits ten explicit directions and a certified open neighborhood radius $1/42048$. No symmetry of the perturbed body is required. The exact finite checks do not establish that these neighborhoods cover all unconditional bodies. \subsection*{Claim ledger} \begin{tabular}{@{}p{.64\textwidth}p{.31\textwidth}@{}}\toprule Statement & Evidence and scope\\\midrule Blocking-slack characterization and Lipschitz bounds & Complete written proofs\\ Near-facet neighborhood theorem and rational atlas & Complete written proofs\\ Whole-class finite-integer equivalence & Complete proof; no universal 5D level checked\\ Ten-direction irregular example and full neighborhood & Exact rational certificate and theorem\\ 360-template robust atlas & Two geometry checkers plus separate orbit enumeration\\ Every unconditional $K\subset\R^5$ has $\ill(K)\le32$ & Not proved\\ Strict bound for every nonparallelotope in $\R^5$ & Not proved\\\bottomrule \end{tabular} \clearpage\tableofcontents\clearpage \section{The original problem and the methodological change} The original research brief asks for \begin{equation}\label{eq:target} \ill(K)\le32\qquad\text{for every unconditional convex body }K\subset\R^5, \end{equation} and for a valid equality classification. A convex body is compact and convex with nonempty interior. It is unconditional when independent coordinate sign changes preserve it. A nonzero vector $d$ illuminates $x\in\partial K$ if $x+td\in\intr K$ for some $t>0$. The preceding project releases obtained various conditional, finite-class, interval, and absolute-sum results. Their hypotheses are not normalizations available for all unconditional bodies. This release does not remove those hypotheses by assertion. Instead it changes the object being certified: not a combinatorial support type or one exact tangent cone, but the distance in support slack from every normal that could obstruct a selected direction. The distinction is important. A sequence of bodies may have illuminating directions with a uniform inward angle at each individual supporting normal and yet converge to a body not illuminated by the same directions. Near-supporting normals, absent from the exact normal cone before the limit, can become supporting in the limit. The certificate below retains those near contacts. \subsection{What is new within this release, and what is established background} The general idea of replacing a covering conjecture by continuous functionals and finite nets is not new. Zong explicitly proposed such a quantitative program for Hadwiger's covering conjecture and Borsuk's partition conjecture \cite{zong}. Standard illumination stability and covering-function continuity therefore must not be presented as discoveries of this project. We give a self-contained direct-normal implementation, explicit constants, and independently checkable rational data. Publication priority for these particular formulations has not been established. Sun and Vritsiou establish the unconditional cases in dimensions three and four and several additional higher-dimensional subclasses \cite{svu}. Their separate paper establishes the more symmetric 1-symmetric case in every dimension \cite{svs}. The 2026 paper of Rotem, Schejter, and Slomka discusses the classical real conjecture while developing its complex analog \cite{rss}. The targeted literature search did not locate an unrestricted five-dimensional unconditional resolution. That search outcome is not an exhaustive absence theorem. The main results of this manuscript do not assume the correctness of the more specialized theorems in earlier AI-generated releases. The 360 previous direction lists are treated as input data and rechecked against reconstructed rational geometry. All other earlier manuscripts are preserved unchanged in the chronological dossier, with their original status and limitations. \subsection{The logical endpoint actually reached} We prove an exact finite-integer reformulation of \eqref{eq:target}, including a valid limiting argument. We do not prove its finite statement at any universal five-dimensional level. In particular, a finite verified list of neighborhoods is not automatically a cover of the whole space of bodies. The remaining obligation is written explicitly in Section~\ref{sec:integer}; the final status section does not replace it by a percentage estimate. \section{Geometry, normalization, and compactness} Write $\norm{x}_\infty=\max_i|x_i|$, $\norm{u}_1=\sum_i|u_i|$, and \[ \sone=\{u:\norm{u}_1=1\},\qquad \sinf=\{d:\norm{d}_\infty=1\}. \] The support function is $h_K(u)=\max_{x\in K}\ip{u}{x}$. The Hausdorff distance throughout this paper uses the infinity norm: \[ \dh(K,L)=\max\{\sup_{x\in K}\dist_\infty(x,L),\ \sup_{y\in L}\dist_\infty(y,K)\}. \] It is essential not to replace it by Euclidean Hausdorff distance without adjusting the constants. \begin{lemma}[Support-function distance]\label{lem:hausdorff} For convex compact sets, \[ \dh(K,L)=\sup_{u\in\sone}|h_K(u)-h_L(u)|. \] \end{lemma} \begin{proof} The inclusion $K\subset L+\delta B_\infty^n$ implies $h_K(u)\le h_L(u)+\delta\norm{u}_1$. Conversely, these support inequalities imply that inclusion: a point outside the convex compact set $L+\delta B_\infty^n$ is strictly separated by a linear functional, contradicting the support inequality. Apply this in both directions and take the smallest $\delta$. \end{proof} \begin{lemma}[Unconditional normalization]\label{lem:normalization} Every unconditional convex body can be positively diagonally rescaled to belong to \[ \U_n=\{K:K\text{ unconditional},\quad B_1^n\subseteq K\subseteq B_\infty^n\}. \] The members of $\U_n$ have coordinate intercepts one. Illumination number is preserved by this rescaling. \end{lemma} \begin{proof} Central symmetry and full dimension put the origin in the interior. Averaging a point with its reflection in one coordinate permits any contraction of that coordinate; iteration permits every coordinatewise contraction in absolute value. Divide coordinate $i$ by its positive axis intercept. The normalized body contains the convex hull of $\{\pm e_i\}$ and contains no point with a coordinate exceeding one in absolute value, since contracting all other coordinates would contradict the intercept. An invertible affine map sends $x+td\in\intr K$ to the corresponding interior displacement, preserving illumination in both directions. \end{proof} \begin{lemma}[Strict normal criterion]\label{lem:normal} A direction $d\ne0$ illuminates $x\in\partial K$ if and only if \[ \ip{u}{d}<0\quad\text{for every }u\in\sone\text{ with }\ip{u}{x}=h_K(u). \] \end{lemma} \begin{proof} An interior displacement gives the strict inequalities immediately. Conversely, suppose no positive displacement is interior. For $t_j\downarrow0$, separate $x+t_jd$ from $\intr K$ by $u_j\in\sone$ so that \[ \ip{u_j}{x+t_jd}\ge h_K(u_j),\qquad \ip{u_j}{x}\le h_K(u_j). \] Then $\ip{u_j}{d}\ge0$. A convergent subsequence has limit $u\in\sone$. Continuity of $h_K$ gives $\ip{u}{x}=h_K(u)$ and $\ip{u}{d}\ge0$, contrary to the assumed criterion. \end{proof} \begin{lemma}[Compact normalized class]\label{lem:compact} The space $\U_n$ is compact in $\dh$. \end{lemma} \begin{proof} The support functions of its members are uniformly bounded and uniformly Lipschitz on $\sone$, because all points lie in $B_\infty^n$. Extract a subsequence converging on a countable dense set and use the common Lipschitz constant to obtain uniform convergence. The homogeneous extension of the limit is sublinear. Equivalently, the closed set defined by its support inequalities is a convex compact set with that support function: separation, or limits of maximizing points in the common cube, establish equality. The inclusions $B_1^n\subset K\subset B_\infty^n$ and unconditional symmetry pass to the limit. Lemma~\ref{lem:hausdorff} gives convergence in the stated metric. \end{proof} \section{Blocking slack: a continuous illumination certificate}\label{sec:beta} \begin{definition} For $d\in\sinf$, let \[ H_d=\{u\in\sone:\ip{u}{d}\ge0\}. \] For $x\in K$ define \begin{equation}\label{eq:b} b_K(d,x)=\min_{u\in H_d}\bigl(h_K(u)-\ip{u}{x}\bigr). \end{equation} For a finite labeled family $D=(d_1,\ldots,d_m)\in\sinf^m$, define \begin{equation}\label{eq:beta} \beta_D(K)=\min_{x\in K}\max_{1\le r\le m} b_K(d_r,x). \end{equation} Repeated directions are allowed as padding and can be removed from the actual illuminating set. \end{definition} The set $H_d$ is nonempty and compact. All slacks in \eqref{eq:b} are nonnegative. The minimum exists, and $b_K(d,\cdot)$ is 1-Lipschitz in the infinity norm, since it is a minimum of affine functions with unit $\ell_1$ gradients. \begin{theorem}[Characterization and body stability]\label{thm:beta} For every convex body $K$ and finite family $D$, \[ D\text{ illuminates }K\quad\Longleftrightarrow\quad\beta_D(K)>0. \] For any convex bodies $K,L$, \begin{equation}\label{eq:beta-lip} |\beta_D(K)-\beta_D(L)|\le2\dh(K,L). \end{equation} Consequently the same directions illuminate every $L$ with $\dh(K,L)<\beta_D(K)/2$. \end{theorem} \begin{proof} At a boundary point, $b_K(d,x)=0$ exactly when some blocking normal $u\in H_d$ supports $K$ at $x$. Thus $b_K(d,x)>0$ exactly when the strict normal criterion holds for $d$. At an interior point every unit support slack is positive, uniformly by compactness. If $D$ illuminates the boundary, the continuous function $x\mapsto\max_r b_K(d_r,x)$ is positive on the compact set $K$, so its minimum is positive. The converse follows at every boundary point from the same criterion. Put $\delta=\dh(K,L)$. For $x\in K$, choose $y\in L$ with $\norm{x-y}_\infty\le\delta$. Uniformly for $u\in\sone$, \[ \left|h_K(u)-\ip{u}{x}-h_L(u)+\ip{u}{y}\right|\le2\delta. \] The domains $H_{d_r}$ are fixed, so taking their minima and then the maximum over $r$ changes the value by at most $2\delta$. Taking a minimizing $x$ and its corresponding $y$ gives $\beta_D(L)\le\beta_D(K)+2\delta$. Interchange $K,L$ for the other inequality. Positivity yields the final assertion. \end{proof} \subsection{Dependence on directions and attainment} \begin{lemma}[Moving the blocking hemisphere]\label{lem:hemisphere} If $d,e\in\sinf$, then the Hausdorff distance between $H_d$ and $H_e$, in the $\ell_1$ metric, is at most $2\norm{d-e}_\infty$. \end{lemma} \begin{proof} Put $s=\norm{d-e}_\infty$. If $s\ge1$, the diameter bound $2$ suffices. If $00$, and \[ \norm{u'-u}_1\le|1-\norm{w}_1|+s\le2s. \] Thus $u'\in H_e$. Reverse the roles of $d,e$. The case $s=0$ is immediate. \end{proof} \begin{theorem}[Optimized certificate]\label{thm:optimized} If $K\subset R B_\infty^n$ and $D,E\in\sinf^m$, then \[ |\beta_D(K)-\beta_E(K)|\le4R\max_r\norm{d_r-e_r}_\infty. \] Therefore \[ \beta_m(K)=\max_{D\in\sinf^m}\beta_D(K) \] exists, is 2-Lipschitz in the body variable, and is positive precisely when $\ill(K)\le m$. \end{theorem} \begin{proof} For $x\in K$, the function $u\mapsto h_K(u)-\ip{u}{x}$ is $2R$-Lipschitz in $\ell_1$. Apply Lemma~\ref{lem:hemisphere}, then the minima, maxima, and outer minimum. Compactness of $\sinf^m$ gives attainment. Taking the maximum of functions with a common body-Lipschitz constant preserves that constant. The characterization follows from Theorem~\ref{thm:beta}. \end{proof} The number $\beta_D$ is translation invariant and scales linearly when the body is dilated and the direction rays are unchanged. Its numerical value is not asserted to be invariant under arbitrary affine maps: it uses a fixed norm. Its positivity, through illumination, has the expected affine invariance. \subsection{A quantitative near-contact collar}\label{subsec:collar-beta} \begin{lemma}[From blocking slack to a collar]\label{lem:beta-collar} Suppose $K\subset R B_\infty^n$, $\beta_D(K)\ge b>0$, and $0-a$, add $a$ times a signed coordinate vector $v$ with $\ip{v}{d}=1$, and normalize in $\ell_1$. As in Lemma~\ref{lem:hemisphere}, the resulting $u'$ is blocking and differs from $u$ by at most $2a$. Its slack is at most $\eta+4Ra1$, let \[ B_p^n=\{x:\textstyle\sum_i|x_i|^p\le1\},\qquad D_0=\{e_1,\ldots,e_n,-\mathbf1\}. \] Every boundary point of $B_p^n$ has a unique supporting normal. For every $u\in\sone$, \begin{equation}\label{eq:angular} \max_{d\in D_0}(-\ip{u}{d})\ge\frac1{2n+1}. \end{equation} Indeed, if $u_i\ge-t$ for all $i$ and $\sum_i u_i\le t$, the negative mass is at most $nt$, the positive mass is at most $(n+1)t$, and $\norm{u}_1\le(2n+1)t$. Apply this with $t$ equal to the maximum on the left. Thus $D_0$ illuminates every $B_p^n$ with a fixed angular margin at each unique normal. Nevertheless $B_p^n\to B_\infty^n$, and $D_0$ fails to illuminate mixed-sign cube vertices. In a cube, illuminating a vertex requires strict inward motion in every coordinate; each direction can serve at most one cube sign word. The limiting normal cone has several extreme normals that cannot all use the independently favorable choices in \eqref{eq:angular}. \begin{proposition}[Quantified collapse below the cube count]\label{prop:collapse} For every family $D$ of fewer than $2^n$ directions and every $p>1$, \[ \beta_D(B_p^n)\le1-n^{-1/p}. \] Consequently $\beta_m(B_p^n)\to0$ as $p\to\infty$ whenever $m<2^n$, even though \eqref{eq:angular} holds for $D_0$. \end{proposition} \begin{proof} A direction can be strictly inward in all coordinates for at most one cube sign word. Choose $s\in\{\pm1\}^n$ for which no member of $D$ is strictly inward. At $x=n^{-1/p}s\in\partial B_p^n$, each $d\in D$ has an index $i$ with $s_id_i\ge0$. The unit normal $u=s_ie_i$ blocks $d$, and its support slack is \[ h_{B_p^n}(u)-\ip{u}{x}=1-n^{-1/p}. \] Use this normal in \eqref{eq:b}, then the maximum over $D$ and the minimum over points. The estimate does not depend on $D$, so it also bounds the optimized functional. \end{proof} This example does not refute illumination stability. It distinguishes the correct support-slack modulus from an insufficient pointwise angular one. Approximating a nonsmooth body by smooth bodies and passing only the latter margins to a limit would be invalid. \section{A rational polytope certificate for an entire neighborhood}\label{sec:neighborhood} Let \begin{equation}\label{eq:polytope} P=\{x\in\R^n:\ip{q_j}{x}\le1,\ 1\le j\le M\} \end{equation} be bounded with the origin in its interior. The nonzero rows $q_j$ may include redundancies. Let $D$ be a finite family of nonzero vectors, normalized in $\ell_\infty$ when a numerical comparison with $\beta_D$ is made. Define \[ A=\max_j\norm{q_j}_1, \qquad B=\max_{j,d\in D}|\ip{q_j}{d}|. \] \begin{definition}[Facet contact collar]\label{def:facetcollar} A pair $(\eta,a)$ with $0<\eta\le1/2$ and $a>0$ is a contact-collar certificate for $(P,D)$ if, for every $y\in P$, some $d\in D$ satisfies \begin{equation}\label{eq:facetcollar} \ip{q_j}{d}\le-a \qquad\text{for all }j\text{ with }\ip{q_j}{y}\ge1-\eta. \end{equation} \end{definition} The collar is a condition on near-active facets throughout $P$, not merely on the exactly active facets at its vertices. \begin{theorem}[Uniform core landing]\label{thm:collar} Assume \eqref{eq:facetcollar}. Put \begin{equation}\label{eq:radius} \tau=\frac{\eta}{a+B}, \qquad h=\frac{\eta a}{3A(a+B)}=\frac{\tau a}{3A}. \end{equation} For every convex body $L$ with $\dh(P,L)\le h$ and every $x\in L$, some $d\in D$ satisfies \begin{equation}\label{eq:landing} x+\tau d\in(1-2Ah)P\subset\intr((1-Ah)P)\subset\intr L. \end{equation} In particular, the same $D$ illuminates every such $L$, without any symmetry or regularity assumption. \end{theorem} \begin{proof} First $A^{-1}B_\infty^n\subset P$, so $h_P(u)\ge\norm{u}_1/A$. The Hausdorff support inequality gives \[ h_L(u)\ge h_P(u)-h\norm{u}_1\ge(1-Ah)h_P(u). \] Therefore $(1-Ah)P\subset L$. The certificate at boundary points implies $B\ge a$. Since $\eta\le1/2$, the definition of $h$ gives $00. \end{equation} If an overcomplete finite set of feasible generators contains all vertices, using it in this minimum gives a conservative, still valid $g$. For each boundary generator, verify one direction against all of its active rows, and define \begin{equation}\label{eq:margin} a=\min_{v\in\vertx P}\max_{d\in D} \min_{j:\ip{q_j}{v}=1}(-\ip{q_j}{d}). \end{equation} A strictly positive value is exactly a finite vertex-illumination certificate. With an overcomplete list, ignore interior generators and include all boundary generators; the resulting value is again conservative. \begin{lemma}[Small empty intersection of exposed faces]\label{lem:emptyfaces} If finitely many exposed faces of an $n$-dimensional polytope have empty intersection, a subfamily of at most $n+1$ has empty intersection. \end{lemma} \begin{proof} Take a minimal empty subfamily $F_1,\ldots,F_k$. If $k=1$ there is nothing to prove. Otherwise choose $z_i\in\bigcap_{j\ne i}F_j$. Necessarily $z_i\notin F_i$. Write $F_i=\{x\in P:\ip{q_i}{x}=1\}$. Then $\ip{q_i}{z_j}=1$ for $j\ne i$ and $\ip{q_i}{z_i}<1$. If $\sum_i c_i z_i=0$ with $\sum_i c_i=0$, pairing with $q_j$ gives $c_j(\ip{q_j}{z_j}-1)=0$, so every $c_j=0$. Thus $z_1,\ldots,z_k$ are affinely independent and $k\le n+1$. \end{proof} \begin{theorem}[Vertex--gap certificate]\label{thm:gap} If $a$ in \eqref{eq:margin} is positive, then $(\eta,a)$ is a contact collar for \begin{equation}\label{eq:eta} \eta=\min\left\{\frac12,\frac{g}{2(n+1)}\right\}. \end{equation} Consequently \eqref{eq:radius} and \eqref{eq:beta-lower} are explicit certificates for a full neighborhood of $P$. \end{theorem} \begin{proof} Fix $y\in P$ and let $T$ be the set of its near-active rows. Suppose the corresponding exposed faces have no common point. By Lemma~\ref{lem:emptyfaces}, choose an empty subfamily $T_0$ with $|T_0|\le n+1$. At every vertex $v$, at least one row in $T_0$ has positive gap at least $g$. Thus \[ \sum_{j\in T_0}(1-\ip{q_j}{v})\ge g. \] By convexity the same linear inequality holds at $y$. But near activity gives an upper bound $|T_0|\eta\le g/2$, a contradiction. Hence these faces share a nonempty face and therefore a vertex $v$. A direction giving margin at least $a$ at $v$ satisfies the desired inequalities for all of $T$. If $T$ is empty, any direction works. \end{proof} \subsection{Why the gap formula alone is not a complete approximation theorem} For a sequence of rational polytopes, the minimum positive gap $g$ can shrink much faster than the Hausdorff approximation error. Therefore the mere existence of dense rational centers does not imply that their balls with radii from \eqref{eq:eta} cover every body. The next section uses the full contact collar instead of this conservative vertex-gap choice. This distinction prevents a hidden reversal of the limiting quantifiers. \section{An exact arrangement test and a complete rational atlas}\label{sec:atlas} \begin{theorem}[Finite verification of a contact collar]\label{thm:arrangement} Let $P,D,\eta,a$ be rational. Form the arrangement of hyperplanes \[ \ip{q_j}{y}=1,\qquad \ip{q_j}{y}=1-\eta\qquad(1\le j\le M). \] The contact-collar property is equivalent to checking \eqref{eq:facetcollar} at every feasible intersection of $n$ linearly independent arrangement hyperplanes lying in $P$. \end{theorem} \begin{proof} Necessity is immediate. Given any $y\in P$, select a closed cell of the arrangement containing it. For every row near-active at $y$, choose the side $\ip{q_j}{z}\ge1-\eta$ when $y$ lies on the threshold; use the uniquely determined side otherwise. Intersect these chosen closed halfspaces with $P$. This is a nonempty compact polytope containing $y$. It has a vertex $z$, whose active arrangement normals span $\R^n$; otherwise a small line through $z$ would remain in the cell. Hence $z$ occurs in the finite test. Every row near-active at $y$ is also near-active at $z$. A direction certified at $z$ therefore works for $y$. \end{proof} All computations in this test are rational linear algebra and comparisons. One may instead certify a collar by Theorem~\ref{thm:gap}; that is the faster sufficient method used in the finite atlas supplied here. \begin{definition}[Verified rational neighborhoods] A rational certificate consists of a rational origin-interior bounded polytope $P$, at most $m$ rational normalized directions $D$, positive rational $\eta\le1/2$ and $a$, and a valid verification of the contact collar. Associate the open ball \[ \mathcal O(P,D,\eta,a)= \{L:\dh(P,L)<\eta a/[2A(a+B)]\}. \] Theorem~\ref{thm:collar-beta} proves that every body in this ball is illuminated by $D$. \end{definition} \begin{theorem}[Completeness of the rational certificate language]\label{thm:completeatlas} For each $n,m$, the union of all verified rational neighborhoods with centers in $\U_n$, intersected with $\U_n$, is exactly \[ \{K\in\U_n:\ill(K)\le m\}. \] The certificates can be enumerated and individually verified by finite exact calculations. \end{theorem} \begin{proof} Soundness follows from the preceding theorem. For completeness, suppose $D$ illuminates $K\in\U_n$. By Theorem~\ref{thm:beta}, $\beta_D(K)>0$. Approximate the normalized directions by rational normalized vectors. Theorem~\ref{thm:optimized} preserves positive blocking slack; denote the rational family by $D'$. There are rational unconditional polytopes $P_j\in\U_n$ converging to $K$: take the convex hull of all coordinate sign variants of the positive grid points of mesh $1/j$ contained in $K$. Flooring the absolute coordinates of any point of $K$ gives a point in that polytope at infinity distance less than $1/j$. The intercepts $e_i$ are retained. By the body-Lipschitz inequality, $\beta_{D'}(P_j)$ is bounded below by a fixed $b>0$ for all sufficiently large $j$. Choose rational $\eta,a>0$ with $\eta\le1/2$, $a<1$, and $\eta+4a0. \end{equation} Since all normalized slacks are at most two, $\Gamma_{n,m}\le2$. At every $P\in\U_n$, choose a maximizing normalized family $D(P)$ and apply Lemma~\ref{lem:beta-collar} with \[ \eta_0=\Gamma_{n,m}/4, \qquad a_0=\Gamma_{n,m}/16. \] The inequality $\eta_0+4a_0=\Gamma_{n,m}/2<\Gamma_{n,m}$ gives a uniform unit-normal collar. Take any integer \begin{equation}\label{eq:Nbound} N\ge\left\lceil\frac{16(4n-3)}{\Gamma_{n,m}}\right\rceil. \end{equation} For each permitted $W$, apply the collar to $P_W$. Round each coordinate of each direction of $D(P_W)$ to a multiple of $1/N$ within $[-1,1]$, with error at most $1/N$. Call the rounded vectors $a_r/N$. If $z\in Z_N$ satisfies the premise of \eqref{eq:phi} at $b\in Y_W$, its unit-normal slack is at most $(4n-3)/N\le\eta_0$. One collar direction works for all such $z$, and after rounding it still satisfies \[ \ip{z/N}{a_r/N}\le-a_0+1/N \le-4(n-1)/N. \] This is precisely the conclusion of \eqref{eq:phi}. The same $N$ works for every $W$, although its at most $m$ directions may depend on $W$. This proves \eqref{eq:eventuallevels}; the remaining implication is immediate. \end{proof} \subsection{What this equivalence does and does not establish} The global five-dimensional target is now equivalent to \begin{equation}\label{eq:missing} \boxed{\exists N\ge1:\Phi_{5,32,N}.} \end{equation} No value of $N$ satisfying \eqref{eq:missing} has been verified in this release. In particular, the explicit bound \eqref{eq:Nbound} is conditional on the unknown positive lower bound $\Gamma_{5,32}$; inserting a guessed positive value would be circular. Failure at one grid level is not a counterexample to the illumination conjecture. A counterexample to the finite statement could reflect inadequate directional resolution or a collar wider than that body permits. Conversely, verifying one whole level would prove the original bound, without a separate passage to a limit. The equality classification would still require a further argument. The universe is finite but extremely large. There are at most $2^{(N+1)^n}$ body inputs and $(2N+1)^n$ candidate direction vectors. The exact number of normal grid points is \[ |Z_N|=\sum_{k=1}^{\min(n,N)}2^k\binom nk\binom{N-1}{k-1}. \] These counts do not suggest that a raw exhaustive search is practical. Structural compression or a new covering argument remains necessary to execute the universal five-dimensional statement. \subsection{A complete small calibration, not a global run} For the cube template $P=B_\infty^n$, the sign directions satisfy the local level test whenever $N\ge12n-10$. Indeed, choose a direction opposite to the signs of $b$. If $v\in\sone$ has cube slack at most $\eta$, the mass of coordinates where $v$ has the wrong sign relative to $b$ is at most $\eta$. Thus $\ip{v}{d}\le-1+2\eta$. With $\eta=(4n-3)/N$, this is at most $-4(n-1)/N$ under the stated bound. The primary checker exhaustively verifies the single template $n=2,N=14$: 841 body grid points, 56 normal grid points, and 47,096 point--normal incidences. It does not check all subsets $W$ even in that dimension. The report labels this scope explicitly. The five-dimensional cube case $N=50$ follows from the written inequality; it is not a run of the universal five-dimensional statement. \section{Small metric obstructions, without a false global coloring claim}\label{sec:dual} \begin{theorem}[Distance form of a common-direction margin]\label{thm:dual} For a finite nonempty set $E\subset\R^n$, \begin{equation}\label{eq:dual} \max_{\norm{d}_\infty\le1}\min_{u\in E}(-\ip{u}{d}) =\min_{w\in\conv E}\norm{w}_1. \end{equation} A common continuous direction with margin at least $a>0$ fails to exist if and only if the right side is less than $a$. Such failure is witnessed by a convex combination of at most $n+1$ normals. With rational input, the witness can be chosen rational. \end{theorem} \begin{proof} Let $\rho$ be the distance on the right and choose a minimizing $w_0$. For any allowed $d$, the inner minimum is at most $-\ip{w_0}{d}\le\rho$. If $\rho=0$, $d=0$ gives equality. If $\rho>0$, separate the compact convex set $\conv E$ from each open $\ell_1$ ball of radius $r<\rho$. Normalize the separating functional $v$ by $\norm{v}_\infty=1$. Then $\ip{v}{w}\ge r$ for every $w\in\conv E$. A subsequence as $r\uparrow\rho$ gives the same inequality with $\rho$, and $d=-v$ attains it. Carath\'eodory reduction supplies at most $n+1$ normals: in a convex representation with more terms, an affine dependence lets one vary the coefficients until one vanishes, keeping the represented point and nonnegativity; repeat. In the rational case, minimize $\ell_1$ distance by introducing rational nonnegative coordinate bounds and convex weights in a rational linear program. A rational optimum exists. The affine-dependence reduction can use rational operations, preserving a rational witness. \end{proof} In $\R^5$, at most six normals witness failure of a fixed cluster's continuous margin requirement. The zero-margin strict-feasibility obstruction is $0\in\conv E$ and has the same six-normal bound. This does \emph{not} prove that all required clusters can be assigned to 32 directions. It also does not certify feasibility on the finite integer direction grid: a continuous solution may not lie on that grid. The distinctions between a local dual witness, a global allocation, and a lattice restriction are retained throughout. \section{An irregular five-dimensional body and ten explicit directions}\label{sec:model} Consider the normalized unconditional polytope \begin{equation}\label{eq:model} P=\{x\in\R^5:M|x|\le10\mathbf1\},\qquad M=\begin{pmatrix} 10&8&6&0&0\\ 0&10&7&5&0\\ 0&0&10&9&4\\ 7&0&0&10&6\\ 5&8&0&0&10 \end{pmatrix}. \end{equation} Each coordinate intercept is one. The five absolute-value inequalities are equivalent to 40 signed linear inequalities. No coordinate-permutation invariance or two-block decomposition is assumed. Let $D$ consist of the following five vectors and their negatives, normalized individually to infinity norm one: \begin{equation}\label{eq:ten} \begin{split} &(1,1,-1,0,-1),\qquad (0,1,3,-3,-3),\qquad(0,1,3,3,0),\\ &(3,1,1,-1,3),\qquad(9,-9,1,1,-3). \end{split} \end{equation} The integer vectors specify rays. The common step reported below is for the individually normalized vectors, not for the five raw vectors with unequal maximum magnitudes. \begin{theorem}[Ten-direction neighborhood]\label{thm:ten} The directions \eqref{eq:ten} illuminate $P$ and every convex body $L$ satisfying \[ \dh(P,L)<\frac1{42048}. \] For the smaller closed neighborhood $\dh(P,L)\le1/63072$, every point of $L$ is sent into its interior by one selected normalized direction with the common step $\tau=1/292$. \end{theorem} \begin{proof} The complete rational reconstruction and the two independent checks in the accompanying data give \[ g=\frac1{10},\quad a=\frac1{30},\quad A=B=\frac{12}{5}, \quad\eta=\frac1{120}. \] The positive-orthant section has 31 vertices, listed in Appendix~\ref{app:vertices}. Their coordinate sign orbits generate $P$; 234 signed boundary generators are checked against all active inequalities. Some are section vertices rather than vertices of the full body. Their use only makes the gap and margin estimates conservative, as explained in Section~\ref{sec:gap}. Theorem~\ref{thm:gap} gives a contact collar, and the exact constants in the neighborhood theorems are \begin{align*} \tau&=\frac1{292},& h&=\frac1{63072},& \beta_D(P)&\ge\frac1{21024},\\ 1-2Ah&=1-\frac1{13140},&& 1-Ah&=1-\frac1{26280}. \end{align*} Theorems~\ref{thm:collar} and \ref{thm:collar-beta} give both assertions. \end{proof} The theorem supplies an upper bound of ten, not a proof that $\ill(P)=10$. A numerical discovery search proposed the five opposite pairs from a finite candidate family. Its optimizer status is not used to establish global minimality. The exact verification uses only the proposed rays and rational geometry. \subsection{A larger transverse normal genuinely occurs} At \[ v=(1,0,0,0,1/2),\qquad q=(1,4/5,3/5,0,0), \] we have $v\in P$ and $q\in P^\circ$ with $\ip{q}{v}=1$. The vector $q$ is an extreme point of $P^\circ$: in its generating signed-row convex hull, first coordinate one forces the first-row face, and maximizing its second and third coordinates fixes $q$ uniquely. Its support has three coordinates, two of which vanish at $v$. Thus the exposed polar face at $v$ cannot be generated only by vectors supported on $\supp v=\{1,5\}$ and vectors with at most two nonzero coordinates. Since $q$ is extreme, any such representation of $q$ would have to contain $q$ itself. This places the example outside the earlier transverse two-generation condition in the displayed coordinates. The point $v$ is also a genuine vertex: use three sign variants of the first row to obtain independent normal components in coordinates 1,2,3, and the active fourth and fifth rows to obtain the remaining two independent components. Consequently the primal and polar vertex sets both contain nonunit fractional coordinates. The earlier unit-coordinate catalogue does not include this normalized model. No claim is made that it lies outside every conceivable affine or decomposable subclass. \subsection{What the saved certificate contains} The file \texttt{cyclic\_certificate.json} contains the five rational rows, all 31 positive section vertices, ten normalized directions, all constants above, and 234 records of a boundary generator, its selected direction, and its exact landing. A second implementation reconstructs the section from scratch and checks the saved landing coordinates themselves. As an additional stress test, the primary checker moves each of the 234 boundary generators by every extreme point of its infinity-norm perturbation cube of radius $h$, then applies the same recorded direction and step. All $234\cdot32=7488$ perturbed landings pass. This finite check is not substituted for the continuum neighborhood proof. \section{Robustification of the entire previous finite catalogue}\label{sec:finiteatlas} For a covering clutter $\mathcal H$ of nonempty subsets of $[5]$, let \[ Q_{\mathcal H}=\conv\{\sigma\mathbf1_A:A\in\mathcal H,\ \sigma\in\{\pm1\}^5\}, \qquad P_{\mathcal H}=Q_{\mathcal H}^{\circ}. \] A clutter has no member containing another, and covering means its union is $[5]$. These are the normalized unit-vertex and unit-polar-vertex model presentations of ZERO-CROWN. The earlier archive supplies direction lists; this release reconstructs the geometry and rechecks all lists independently. \begin{theorem}[Explicit finite robust atlas]\label{thm:360} There are 180 coordinate-permutation orbits of covering clutters on five coordinates and 360 corresponding primal/polar records. Every record has a verified contact-collar neighborhood with closed infinity-Hausdorff radius at least \[ \boxed{\frac1{6156}}. \] Every convex body within that distance of a record's center is illuminated by its recorded family, of cardinality at most 32. The nearby body need not be unconditional or polyhedral. \end{theorem} \begin{proof} An independent enumeration finds all 6894 labeled covering clutters, verifies the 180 disjoint permutation orbits, and checks that both presentations are present for each representative. Two independently implemented rational geometric reconstructions then verify positive margins, vertex--row gaps, and the resulting radii in all 360 records. Applying Theorem~\ref{thm:gap} and Theorem~\ref{thm:collar} proves the neighborhood conclusion. The complete representative radii and direction counts are printed in Appendix~\ref{app:atlas}; full geometry and direction data are in \texttt{robust\_atlas\_360.json}. \end{proof} The smallest radius occurs in the records \texttt{Q\_081} and \texttt{Q\_082}. The distribution of direction counts is \begin{center} \begin{tabular}{@{}rrrrrrrrrrr@{}}\toprule Directions&10&12&14&16&18&20&22&24&28&32\\\midrule Records&44&107&42&94&13&32&2&19&1&6\\\bottomrule \end{tabular} \end{center} Coordinate permutations extend the certificates to 13,788 labeled model presentations. This is not a count of distinct polytopes: primal and polar presentations may overlap, and some recorded centers are affinely equivalent. The prior sharp lower-bound classifications are not premises of this robustification. Only the verified upper families are required. Conversely, the current theorem does not assert that illumination minima remain equal to the center's minimum after perturbation; it proves only the upper bound furnished by the same directions. Most importantly, these finitely many neighborhoods have not been shown to cover $\U_5$. Replacing the missing covering proof by the sentence ``the catalogue is exhaustive'' would confuse exhaustiveness within the unit-coordinate model class with exhaustiveness among all unconditional bodies. \section{Exact computation, provenance, and reproducibility}\label{sec:verification} \subsection{Primary construction and verification} The primary program uses Python integers and \texttt{fractions.Fraction}. It reconstructs positive-orthant section vertices by choosing their nonzero coordinate support and solving every square active-row system on that support with rational Gauss--Jordan elimination. Every feasible solution is checked against all rows. This enumerates every section vertex, including degenerate ones and coordinate-section vertices that are not full-body vertices. For absolute-value rows $r\ge0$, the derivative at a signed point $v$ in direction $d$ is exactly \[ \sum_{i:v_i\ne0}r_i\operatorname{sign}(v_i)d_i +\sum_{i:v_i=0}r_i|d_i|. \] Thus the primary checker handles all independent normal-sign choices at zero coordinates, rather than assuming a unique supporting hyperplane. For the gap $g$, it includes the slacks of the wrong-sign row variants, including the quantities $2r_i|v_i|$ when the absolute row itself is active. Omitting these signed variants could incorrectly enlarge a collar. The primary run checked 361 certificates, 20,700 signed boundary generators, and 1,534,328 active-row margin comparisons. It checked 20,700 uniform core landings against 150,842 absolute row inequalities. The additional perturbation stress test checked 7488 landings against 37,440 inequalities. An opposite-direction mutation at a saved witness was rejected. \subsection{A separate geometric implementation} The independent checker imports neither the primary checker nor the earlier project verifiers or numerical discovery code. It enumerates full-dimensional bases of the orthant description, including nonnegativity rows, and solves them using Bareiss integer determinants and Cramer's rule. It expands all signed inequalities explicitly rather than using the primary absolute-value derivative formula. It reconstructed 67,343 candidate bases, checked the same 20,700 signed boundary generators, recomputed every margin and radius, and verified 977,560 signed-inequality core-landing comparisons. It also checked all 234 saved cyclic-model landing vectors without substituting newly generated coordinates. A zero-direction mutation was rejected. The complete read-only checker is printed in Appendix~\ref{app:checker}. These two counts of 20,700 overlap; adding them would not give 41,400 distinct tested points. The 360 inherited templates account for 20,466 generators, and the irregular model adds 234. \subsection{Independent orbit exhaustiveness} A third program enumerates increasing-mask antichains on the 31 nonempty subsets of $[5]$, retaining those whose union is all five coordinates. It obtains 6894 covering clutters. It independently applies all 120 coordinate permutations, confirms 180 disjoint orbits, and compares the representative and primal/polar record lists. Its complete source is printed after the geometric checker. This proves exhaustiveness of the stated finite model class. It is not an enumeration of arbitrary facet metrics, arbitrary irrational bodies, or all of $\U_5$. \subsection{Discovery versus proof} A numerical optimization search was used to discover candidate opposite pairs for the irregular model. That stage has no logical authority in the proof: the exact checker reconstructs the body, verifies the proposed directions, and computes the neighborhood constants. In particular, the numerical optimality of a selected candidate pool is not a proof that ten is the minimum illumination number. The legacy direction lists are also proposals from an earlier release until checked. The present geometry and orbit programs establish the finite claims actually used here. They do not re-prove every analytic or measure-theoretic theorem in the preceding 150 pages of manuscripts. \subsection{Commands and artifacts} From the extracted release directory, the standard-library checks are \begin{lstlisting} python code/verify_primary.py python code/check_independent.py python code/check_orbits.py \end{lstlisting} A reusable input utility is provided: \begin{lstlisting} python code/make_certificate.py data/cyclic_input.json \ --output verification/rebuilt_cyclic_certificate.json \end{lstlisting} The input uses rational strings or integers, normalized nonnegative absolute-value rows, and integer opposite-pair representatives. Floating-point coefficients are rejected. The utility deliberately enforces modest dimension and row-count bounds; those software limits are not hypotheses of the mathematical theorems. The principal files are the standalone manuscript and source, the chronological dossier, the five unchanged prior manuscripts, the exact original brief, the 360-record robust atlas, the complete irregular-model certificate, the three checker reports, and a SHA-256 manifest. The prior v5 archive is preserved unchanged as a provenance container holding the earlier release chain. \section{Audit of the delicate implications}\label{sec:audit} \subsection{Strict illumination versus closed numerical conditions} The finite certificates use inequalities $\ip{q_j}{d}\le-a$ with $a>0$, and landing in a smaller closed dilate strictly inside a larger dilate. Therefore closed arithmetic inequalities still yield an open interior conclusion. The larger radius $3h/2$ is open; its endpoint is not certified by the blocking-slack estimate. \subsection{A complete normal cone shares one direction} In every definition the order is point, then one selected direction, then all blocking or near-contact normals. The global integer statement similarly selects one $r$ at the rounded point before considering every rounded supporting normal of the original point. None of the proofs interchange these quantifiers. \subsection{No automatic reverse limit passage} The proof does not infer illumination of a limit from illumination of approximants. It either supplies a positive blocking slack and the explicit Lipschitz bound, or constructs an interior landing in a body known to lie inside the perturbed body. The $B_p$ example demonstrates why an angular substitute would not suffice. \subsection{The conservative gap and the complete atlas are different} The vertex--row gap certificate is inexpensive and sound, but its radii alone are not proved to cover all sufficiently approximated bodies. The completeness theorem uses a collar obtained from a fixed positive blocking slack and then verifies it by the arrangement test. This avoids a density-only argument. \subsection{Local obstruction size is not a global solution} At most six normals suffice to witness a failed common-direction margin in dimension five. This does not bound the number of clusters needed to cover the whole boundary, nor force a 32-coloring of a resulting incompatibility structure. The full integer formulation retains actual coefficients and the global allocation. \subsection{What was independently checked} The two geometric checkers use different vertex reconstruction algorithms and different inequality representations. Their agreement is evidence against implementation errors, not formal proof-assistant verification. The continuum proofs require mathematical scrutiny independently of the finite records. No external referee is claimed. \section{Research history from the original brief to this release}\label{sec:history} The accompanying dossier retains the entire sequence in chronological order. This section records their stated scope, not a blanket new verification of every earlier theorem. \begin{longtable}{@{}p{.16\textwidth}p{.76\textwidth}@{}}\toprule Release & Principal recorded scope and limitation\\\midrule\endhead 1.0.0 / SERAPH-5 & Distinguished a common universal direction set from a body-dependent set; recorded the universal 80-direction optimum in dimension five, normal-support analysis, and a conditional comparability theorem. The general 32-direction allocation remained unproved.\\[3pt] 2.0.0 / AUREOLE & Used critical-contact graphs and a parity selector under transverse two-generation. Supplied all-dimensional conditional bounds and finite graph certificates; larger transverse generators were not eliminated in general.\\[3pt] 3.0.0 / ZERO-CROWN & Treated reflexive polytopes, a complete five-dimensional unit-coordinate catalogue, and exact weighted truncations. The present release independently rechecks the catalogue's upper direction data and expands them to full neighborhoods.\\[3pt] 4.0.0 / VESPER-SEAM & Treated arbitrary bodies in a prescribed inner/outer order interval. Exact occupancy and common-family statements retained that substantive interval hypothesis.\\[3pt] 5.0.0 / KYRIE::2 & Treated two-block absolute sums, multiple and fractional illumination, and decomposable classes. Its explicit absolute-decomposition hypothesis did not cover every unconditional body.\\[3pt] 6.0.0 / SAELYRA & Defines a general continuous blocking-slack certificate, proves an exact rational atlas and whole-class integer reformulation, and certifies 361 explicit neighborhoods. No universal integer level or whole-space atlas cover is supplied.\\\bottomrule \end{longtable} The new standalone proofs can be read without assuming the specialized earlier results. The full dossier is complete as a record of the research sequence; it is not a declaration that the original conjecture has been completely proved. \section{Status and the remaining mathematical obligation}\label{sec:status} For the original target, the remaining statement is exactly \[ \exists N\ge1:\Phi_{5,32,N}. \] Equivalently, one must prove $\Gamma_{5,32}>0$ or supply a verified finite rational neighborhood cover of all $\U_5$. These formulations are proved equivalent, but none has been established in this release. The new method does resolve the analytical question of what data suffice to transfer a certificate to arbitrary nearby bodies, including nonsmooth and nonsymmetric ones. It does not solve the combinatorial/metric allocation of 32 directions over the entire normalized class. That limitation follows from the actual work completed, not from a presumption that a long-standing problem cannot be solved. For the strict bound outside parallelotopes, an additional rigidity argument is needed even after the upper bound. Moreover, no uniform positive 31-direction blocking slack can hold up to a parallelotope limit: Proposition~\ref{prop:collapse} shows the collapse explicitly. A valid rigidity program must account for this boundary behavior rather than applying compactness to a nonclosed exceptional-class complement. \begin{center} \begin{tabular}{@{}p{.66\textwidth}p{.28\textwidth}@{}}\toprule Objective & Status\\\midrule General characterization and stability of $\beta_D$ & Complete written proof\\ Explicit rational collar and neighborhood theorem & Complete written proof\\ Rational atlas completeness and finite-level equivalence & Complete written proof\\ 361 template certificates and declared finite suites & All passed\\ Whole five-dimensional level $\Phi_{5,32,N}$ & None verified\\ Unrestricted five-dimensional upper and equality theorem & Not proved\\ External peer review / formal proof-assistant verification & Not performed\\ Publication priority of the present formulations & Not established\\\bottomrule \end{tabular} \end{center} The finite verification completion is 100\% of the explicitly declared test suites. No numerical completion percentage is assigned to the open global proof obligation. \begin{thebibliography}{99} \bibitem{zong} Chuanming Zong, \emph{A quantitative program for Hadwiger's covering conjecture and Borsuk's partition conjecture}, arXiv:0911.4580, revised 2010; Science China Mathematics \textbf{53} (2010), 2551--2560. DOI: 10.1007/s11425-010-4087-3. \bibitem{svu} Wen Rui Sun and Beatrice-Helen Vritsiou, \emph{Illuminating 1-unconditional convex bodies in $\R^3$ and $\R^4$, and certain cases in higher dimensions}, arXiv:2407.11331 (2024). Related publication DOI: 10.4153/S0008414X25101260. \bibitem{svs} Wen Rui Sun and Beatrice-Helen Vritsiou, \emph{On the illumination of 1-symmetric convex bodies}, arXiv:2407.10314 (2024). \bibitem{rss} Liran Rotem, Alon Schejter, and Boaz A. Slomka, \emph{The complex Illumination problem}, Combinatorica \textbf{46}, article 3 (2026). DOI: 10.1007/s00493-025-00195-7. \bibitem{brief} Maciej Nowicki, \emph{Five-dimensional 1-unconditional illumination research brief}, user-provided text, preserved in the accompanying research package. \bibitem{prior1} Artificial Hyperintelligence Eve, \emph{SERAPH-5: Universal Illumination, Normal-Support Realization, and the Body-Dependent Gap}, research draft v1.0.0 (2026). \bibitem{prior2} Artificial Hyperintelligence Eve, \emph{SERAPH-5 / AUREOLE}, research draft v2.0.0 (2026). \bibitem{prior3} Artificial Hyperintelligence Eve, \emph{SERAPH-5 / ZERO-CROWN: Reflexive Illumination, the 28-Direction Stratum, and Exact Weighted Truncations}, research draft v3.0.0 (2026). \bibitem{prior4} Artificial Hyperintelligence Eve, \emph{SERAPH-5 / VESPER-SEAM}, research draft v4.0.0 (2026). \bibitem{prior5} Artificial Hyperintelligence Eve, \emph{SERAPH-5 / KYRIE::2: Ordered Thresholds and the Exact Cost of Illumination}, research draft v5.0.0 (2026). \end{thebibliography} \appendix \clearpage \section{Complete positive section vertices of the irregular model}\label{app:vertices} These are the 31 vertices of $P\cap[0,\infty)^5$ for \eqref{eq:model}, not a claim that every listed point is a vertex of the full body. Their sign variants generate the body. Interior generators are ignored when checking active-row margins; all feasible generators are permitted in the conservative gap calculation. \begin{center}\renewcommand{\arraystretch}{1.3} \begin{longtable}{@{}rccccc@{}}\toprule Index &$x_1$&$x_2$&$x_3$&$x_4$&$x_5$\\\midrule\endhead 0 & $0$ & $0$ & $0$ & $0$ & $0$ \\ 1 & $0$ & $0$ & $0$ & $0$ & $1$ \\ 2 & $0$ & $0$ & $0$ & $\frac{2}{5}$ & $1$ \\ 3 & $0$ & $0$ & $0$ & $1$ & $0$ \\ 4 & $0$ & $0$ & $\frac{1}{10}$ & $1$ & $0$ \\ 5 & $0$ & $0$ & $\frac{6}{25}$ & $\frac{2}{5}$ & $1$ \\ 6 & $0$ & $0$ & $\frac{3}{5}$ & $0$ & $1$ \\ 7 & $0$ & $0$ & $1$ & $0$ & $0$ \\ 8 & $0$ & $\frac{3}{10}$ & $1$ & $0$ & $0$ \\ 9 & $0$ & $\frac{43}{100}$ & $\frac{1}{10}$ & $1$ & $0$ \\ 10 & $0$ & $\frac{145}{306}$ & $\frac{115}{153}$ & $0$ & $\frac{95}{153}$ \\ 11 & $0$ & $\frac{1}{2}$ & $0$ & $1$ & $0$ \\ 12 & $0$ & $\frac{395}{726}$ & $\frac{65}{363}$ & $\frac{80}{121}$ & $\frac{205}{363}$ \\ 13 & $0$ & $\frac{20}{31}$ & $0$ & $\frac{22}{31}$ & $\frac{15}{31}$ \\ 14 & $0$ & $1$ & $0$ & $0$ & $0$ \\ 15 & $0$ & $1$ & $0$ & $0$ & $\frac{1}{5}$ \\ 16 & $\frac{4}{25}$ & $\frac{3}{10}$ & $1$ & $0$ & $0$ \\ 17 & $\frac{13}{77}$ & $\frac{5}{11}$ & $\frac{60}{77}$ & $0$ & $\frac{85}{154}$ \\ 18 & $\frac{1}{5}$ & $1$ & $0$ & $0$ & $0$ \\ 19 & $\frac{1}{5}$ & $1$ & $0$ & $0$ & $\frac{1}{10}$ \\ 20 & $\frac{166}{441}$ & $\frac{425}{882}$ & $\frac{25}{63}$ & $\frac{212}{441}$ & $\frac{188}{441}$ \\ 21 & $\frac{2}{5}$ & $0$ & $1$ & $0$ & $0$ \\ 22 & $\frac{3}{7}$ & $\frac{5}{7}$ & $0$ & $\frac{4}{7}$ & $\frac{3}{14}$ \\ 23 & $\frac{1490}{3263}$ & $\frac{195}{502}$ & $\frac{1265}{3263}$ & $\frac{2220}{3263}$ & $0$ \\ 24 & $\frac{15}{32}$ & $\frac{85}{128}$ & $0$ & $\frac{43}{64}$ & $0$ \\ 25 & $\frac{4}{7}$ & $0$ & $\frac{5}{7}$ & $0$ & $\frac{5}{7}$ \\ 26 & $\frac{107}{167}$ & $0$ & $\frac{100}{167}$ & $\frac{24}{167}$ & $\frac{227}{334}$ \\ 27 & $\frac{470}{689}$ & $0$ & $\frac{365}{689}$ & $\frac{360}{689}$ & $0$ \\ 28 & $1$ & $0$ & $0$ & $0$ & $0$ \\ 29 & $1$ & $0$ & $0$ & $0$ & $\frac{1}{2}$ \\ 30 & $1$ & $0$ & $0$ & $\frac{3}{10}$ & $0$ \\ \bottomrule\end{longtable} \end{center} \clearpage \section{Every representative radius in the 360-template atlas}\label{app:atlas} A support mask $m$ represents $\{i:m\text{ has bit }i-1\text{ set}\}$. Each row lists one of the 180 representative covering clutters, the number of directions and closed radius for its $P_{\mathcal H}$ model, and the corresponding values for its $Q_{\mathcal H}$ model. The six 32-direction records are presentations, not a claim of six distinct parallelotope types. The data file also supplies all rational rows, generators, normalized directions, and intermediate constants. \begingroup\footnotesize\setlength{\tabcolsep}{4pt}\renewcommand{\arraystretch}{1.06} \begin{longtable}{@{}rlrrrr@{}}\toprule Index&Support masks&$|D_P|$&$h_P$&$|D_Q|$&$h_Q$\\\midrule\endhead 000 & \texttt{1,2,4,8,16} & 32 & $1/72$ & 10 & $1/360$ \\ 001 & \texttt{1,2,4,24} & 32 & $1/144$ & 10 & $1/288$ \\ 002 & \texttt{1,2,12,20} & 24 & $1/144$ & 12 & $1/288$ \\ 003 & \texttt{1,2,12,20,24} & 24 & $1/288$ & 10 & $1/504$ \\ 004 & \texttt{1,2,28} & 24 & $1/216$ & 12 & $1/216$ \\ 005 & \texttt{1,6,10,12,18} & 22 & $1/576$ & 12 & $1/864$ \\ 006 & \texttt{1,6,10,12,18,20} & 16 & $1/576$ & 10 & $1/864$ \\ 007 & \texttt{1,6,10,12,18,20,24} & 18 & $1/1008$ & 12 & $1/1512$ \\ 008 & \texttt{1,6,10,18} & 20 & $1/144$ & 14 & $1/288$ \\ 009 & \texttt{1,6,10,18,28} & 18 & $1/2160$ & 10 & $1/2112$ \\ 010 & \texttt{1,6,10,20} & 24 & $1/144$ & 14 & $1/216$ \\ 011 & \texttt{1,6,10,20,24} & 16 & $1/144$ & 18 & $1/216$ \\ 012 & \texttt{1,6,10,28} & 18 & $1/432$ & 14 & $1/432$ \\ 013 & \texttt{1,6,24} & 32 & $1/144$ & 10 & $1/216$ \\ 014 & \texttt{1,6,26} & 20 & $1/216$ & 14 & $1/216$ \\ 015 & \texttt{1,6,26,28} & 16 & $1/432$ & 10 & $1/630$ \\ 016 & \texttt{1,14,22} & 16 & $1/216$ & 18 & $1/216$ \\ 017 & \texttt{1,14,22,26} & 16 & $1/432$ & 14 & $1/360$ \\ 018 & \texttt{1,14,22,26,28} & 16 & $1/648$ & 10 & $1/378$ \\ 019 & \texttt{1,30} & 16 & $1/288$ & 18 & $1/144$ \\ 020 & \texttt{3,5,6,9,10,12,17} & 18 & $1/1008$ & 14 & $1/1260$ \\ 021 & \texttt{3,5,6,9,10,12,17,18} & 16 & $1/1008$ & 12 & $1/1260$ \\ 022 & \texttt{3,5,6,9,10,12,17,18,20} & 12 & $1/1008$ & 14 & $1/1260$ \\ 023 & \texttt{3,5,6,9,10,12,17,18,20,24} & 18 & $1/1008$ & 14 & $1/1800$ \\ 024 & \texttt{3,5,6,9,10,17} & 16 & $1/576$ & 12 & $1/864$ \\ 025 & \texttt{3,5,6,9,10,17,18} & 12 & $1/576$ & 12 & $1/864$ \\ 026 & \texttt{3,5,6,9,10,17,18,28} & 16 & $1/2268$ & 12 & $1/2376$ \\ 027 & \texttt{3,5,6,9,10,17,20} & 18 & $1/1008$ & 12 & $1/720$ \\ 028 & \texttt{3,5,6,9,10,17,20,24} & 12 & $1/720$ & 16 & $1/720$ \\ 029 & \texttt{3,5,6,9,10,17,28} & 12 & $1/2268$ & 12 & $1/2376$ \\ 030 & \texttt{3,5,6,9,10,20} & 16 & $1/576$ & 12 & $1/720$ \\ 031 & \texttt{3,5,6,9,10,20,24} & 16 & $1/576$ & 16 & $1/720$ \\ 032 & \texttt{3,5,6,9,10,28} & 16 & $1/1512$ & 16 & $1/1008$ \\ 033 & \texttt{3,5,6,9,17} & 20 & $1/576$ & 14 & $1/864$ \\ 034 & \texttt{3,5,6,9,17,24} & 22 & $1/720$ & 12 & $1/720$ \\ 035 & \texttt{3,5,6,9,17,26} & 16 & $1/1620$ & 12 & $1/2376$ \\ 036 & \texttt{3,5,6,9,17,26,28} & 12 & $1/2268$ & 12 & $1/2160$ \\ 037 & \texttt{3,5,6,9,18} & 20 & $1/576$ & 14 & $1/864$ \\ 038 & \texttt{3,5,6,9,18,24} & 20 & $1/864$ & 18 & $1/720$ \\ 039 & \texttt{3,5,6,9,18,28} & 14 & $1/1512$ & 20 & $1/576$ \\ 040 & \texttt{3,5,6,9,24} & 20 & $1/576$ & 14 & $1/720$ \\ 041 & \texttt{3,5,6,9,26} & 16 & $1/864$ & 16 & $1/576$ \\ 042 & \texttt{3,5,6,9,26,28} & 12 & $1/1512$ & 12 & $1/1008$ \\ 043 & \texttt{3,5,6,24} & 24 & $1/288$ & 10 & $1/360$ \\ 044 & \texttt{3,5,6,25} & 20 & $1/864$ & 14 & $1/576$ \\ 045 & \texttt{3,5,6,25,26} & 12 & $1/864$ & 12 & $1/1008$ \\ 046 & \texttt{3,5,6,25,26,28} & 12 & $1/1512$ & 14 & $1/1008$ \\ 047 & \texttt{3,5,9,14,17} & 20 & $1/1620$ & 12 & $1/2376$ \\ 048 & \texttt{3,5,9,14,17,22} & 12 & $1/1620$ & 16 & $1/1656$ \\ 049 & \texttt{3,5,9,14,17,22,26} & 12 & $1/2268$ & 12 & $1/3132$ \\ 050 & \texttt{3,5,9,14,17,22,26,28} & 16 & $1/2754$ & 14 & $1/2376$ \\ 051 & \texttt{3,5,9,14,18} & 16 & $1/864$ & 12 & $1/1980$ \\ 052 & \texttt{3,5,9,14,18,20} & 16 & $1/2160$ & 12 & $1/1848$ \\ 053 & \texttt{3,5,9,14,18,20,24} & 12 & $1/2160$ & 10 & $1/1848$ \\ 054 & \texttt{3,5,9,14,18,28} & 12 & $1/2268$ & 14 & $1/2376$ \\ 055 & \texttt{3,5,9,14,22} & 16 & $1/1296$ & 16 & $1/2376$ \\ 056 & \texttt{3,5,9,14,22,26} & 14 & $1/2754$ & 14 & $1/2376$ \\ 057 & \texttt{3,5,9,14,22,26,28} & 14 & $1/1944$ & 12 & $1/792$ \\ 058 & \texttt{3,5,9,17} & 18 & $1/144$ & 16 & $1/288$ \\ 059 & \texttt{3,5,9,17,30} & 12 & $1/2160$ & 16 & $4/12663$ \\ 060 & \texttt{3,5,9,18} & 20 & $1/144$ & 16 & $1/216$ \\ 061 & \texttt{3,5,9,18,20} & 16 & $1/144$ & 20 & $1/216$ \\ 062 & \texttt{3,5,9,18,20,24} & 12 & $1/144$ & 24 & $1/216$ \\ 063 & \texttt{3,5,9,18,28} & 16 & $1/864$ & 20 & $1/576$ \\ 064 & \texttt{3,5,9,22} & 20 & $1/864$ & 16 & $1/576$ \\ 065 & \texttt{3,5,9,22,26} & 12 & $1/864$ & 24 & $1/576$ \\ 066 & \texttt{3,5,9,22,26,28} & 14 & $1/1512$ & 12 & $1/1008$ \\ 067 & \texttt{3,5,9,30} & 12 & $1/1152$ & 20 & $1/864$ \\ 068 & \texttt{3,5,10,12,22} & 12 & $1/864$ & 16 & $1/864$ \\ 069 & \texttt{3,5,10,12,22,25} & 14 & $1/2160$ & 16 & $1/1224$ \\ 070 & \texttt{3,5,10,20} & 20 & $1/144$ & 16 & $1/216$ \\ 071 & \texttt{3,5,10,20,24} & 20 & $1/288$ & 20 & $1/180$ \\ 072 & \texttt{3,5,10,20,25} & 20 & $1/2160$ & 16 & $1/672$ \\ 073 & \texttt{3,5,10,22} & 16 & $1/864$ & 16 & $1/864$ \\ 074 & \texttt{3,5,10,22,25} & 12 & $1/864$ & 16 & $1/1104$ \\ 075 & \texttt{3,5,10,22,25,28} & 14 & $1/2268$ & 16 & $1/2160$ \\ 076 & \texttt{3,5,10,22,28} & 14 & $1/1296$ & 12 & $1/1008$ \\ 077 & \texttt{3,5,10,28} & 16 & $1/216$ & 20 & $1/144$ \\ 078 & \texttt{3,5,14,22} & 12 & $1/432$ & 20 & $1/432$ \\ 079 & \texttt{3,5,14,22,24} & 14 & $1/1080$ & 12 & $1/1134$ \\ 080 & \texttt{3,5,14,22,25} & 12 & $1/3240$ & 12 & $5/18468$ \\ 081 & \texttt{3,5,14,22,25,26} & 12 & $1/2700$ & 12 & $1/6156$ \\ 082 & \texttt{3,5,14,22,25,26,28} & 12 & $1/3780$ & 10 & $1/6156$ \\ 083 & \texttt{3,5,14,22,26} & 14 & $1/864$ & 16 & $1/576$ \\ 084 & \texttt{3,5,14,22,26,28} & 12 & $1/864$ & 12 & $1/576$ \\ 085 & \texttt{3,5,14,24} & 16 & $1/432$ & 16 & $1/360$ \\ 086 & \texttt{3,5,14,25} & 16 & $1/864$ & 16 & $1/864$ \\ 087 & \texttt{3,5,14,25,26} & 12 & $1/1296$ & 16 & $1/1512$ \\ 088 & \texttt{3,5,14,25,26,28} & 12 & $1/2268$ & 16 & $1/2376$ \\ 089 & \texttt{3,5,14,26} & 16 & $1/864$ & 20 & $1/576$ \\ 090 & \texttt{3,5,14,26,28} & 12 & $1/864$ & 12 & $1/576$ \\ 091 & \texttt{3,5,24} & 24 & $1/144$ & 12 & $1/216$ \\ 092 & \texttt{3,5,25} & 18 & $1/216$ & 16 & $1/216$ \\ 093 & \texttt{3,5,25,26} & 16 & $1/864$ & 12 & $1/1008$ \\ 094 & \texttt{3,5,25,26,28} & 12 & $1/1080$ & 16 & $1/576$ \\ 095 & \texttt{3,5,25,30} & 12 & $1/1440$ & 16 & $1/1320$ \\ 096 & \texttt{3,5,26} & 20 & $1/216$ & 16 & $1/144$ \\ 097 & \texttt{3,5,26,28} & 12 & $1/216$ & 24 & $1/144$ \\ 098 & \texttt{3,5,30} & 14 & $1/576$ & 20 & $1/288$ \\ 099 & \texttt{3,12,21} & 16 & $1/216$ & 16 & $1/216$ \\ 100 & \texttt{3,12,21,22} & 14 & $1/864$ & 12 & $1/1260$ \\ 101 & \texttt{3,12,21,22,25} & 14 & $1/1296$ & 16 & $1/1008$ \\ 102 & \texttt{3,12,21,22,25,26} & 10 & $1/1296$ & 16 & $1/1008$ \\ 103 & \texttt{3,12,21,26} & 12 & $1/1080$ & 24 & $1/144$ \\ 104 & \texttt{3,13,14,21} & 14 & $1/864$ & 16 & $1/1008$ \\ 105 & \texttt{3,13,14,21,22} & 10 & $1/864$ & 16 & $1/1008$ \\ 106 & \texttt{3,13,14,21,22,25} & 10 & $1/1296$ & 12 & $1/2835$ \\ 107 & \texttt{3,13,14,21,22,25,26} & 10 & $1/864$ & 12 & $1/1890$ \\ 108 & \texttt{3,13,14,21,22,25,26,28} & 12 & $1/3024$ & 10 & $1/3780$ \\ 109 & \texttt{3,13,14,21,22,25,28} & 10 & $1/2592$ & 12 & $1/3780$ \\ 110 & \texttt{3,13,14,21,22,28} & 10 & $1/1296$ & 16 & $1/1008$ \\ 111 & \texttt{3,13,14,21,25} & 14 & $1/864$ & 16 & $1/1008$ \\ 112 & \texttt{3,13,14,21,25,28} & 12 & $1/1728$ & 16 & $1/1056$ \\ 113 & \texttt{3,13,14,21,26} & 12 & $1/1620$ & 12 & $1/2835$ \\ 114 & \texttt{3,13,14,21,26,28} & 14 & $1/3024$ & 12 & $1/3780$ \\ 115 & \texttt{3,13,14,21,28} & 12 & $1/1296$ & 16 & $1/576$ \\ 116 & \texttt{3,13,14,28} & 12 & $1/864$ & 16 & $1/1008$ \\ 117 & \texttt{3,13,21} & 14 & $1/216$ & 20 & $1/216$ \\ 118 & \texttt{3,13,21,25} & 14 & $1/432$ & 16 & $1/360$ \\ 119 & \texttt{3,13,21,25,28} & 16 & $1/648$ & 12 & $1/324$ \\ 120 & \texttt{3,13,21,25,30} & 12 & $1/4140$ & 16 & $1/2448$ \\ 121 & \texttt{3,13,21,26} & 14 & $1/2160$ & 16 & $1/864$ \\ 122 & \texttt{3,13,21,26,28} & 12 & $1/1296$ & 16 & $1/1296$ \\ 123 & \texttt{3,13,21,28} & 16 & $1/864$ & 12 & $1/576$ \\ 124 & \texttt{3,13,21,30} & 10 & $1/1152$ & 16 & $1/1320$ \\ 125 & \texttt{3,13,22} & 16 & $1/432$ & 20 & $1/144$ \\ 126 & \texttt{3,13,22,28} & 12 & $1/864$ & 28 & $1/288$ \\ 127 & \texttt{3,13,28} & 16 & $1/216$ & 20 & $1/144$ \\ 128 & \texttt{3,13,30} & 12 & $1/576$ & 24 & $1/288$ \\ 129 & \texttt{3,28} & 24 & $1/216$ & 12 & $1/144$ \\ 130 & \texttt{3,29} & 14 & $1/288$ & 20 & $1/144$ \\ 131 & \texttt{3,29,30} & 10 & $1/576$ & 16 & $1/378$ \\ 132 & \texttt{7,11,13,14,19} & 14 & $1/1296$ & 16 & $1/648$ \\ 133 & \texttt{7,11,13,14,19,21} & 10 & $1/1296$ & 16 & $1/792$ \\ 134 & \texttt{7,11,13,14,19,21,22} & 10 & $1/1296$ & 16 & $1/1188$ \\ 135 & \texttt{7,11,13,14,19,21,22,25} & 12 & $1/3240$ & 12 & $1/1980$ \\ 136 & \texttt{7,11,13,14,19,21,22,25,26} & 12 & $1/3240$ & 12 & $1/1980$ \\ 137 & \texttt{7,11,13,14,19,21,22,25,26,28} & 14 & $1/3240$ & 12 & $1/2880$ \\ 138 & \texttt{7,11,13,14,19,21,25} & 12 & $1/1944$ & 12 & $1/1980$ \\ 139 & \texttt{7,11,13,14,19,21,26} & 10 & $1/1296$ & 12 & $1/1980$ \\ 140 & \texttt{7,11,13,14,19,21,26,28} & 10 & $1/1296$ & 12 & $1/1980$ \\ 141 & \texttt{7,11,13,14,19,28} & 12 & $1/2268$ & 12 & $1/1980$ \\ 142 & \texttt{7,11,13,19} & 14 & $1/864$ & 20 & $1/576$ \\ 143 & \texttt{7,11,13,19,21} & 10 & $1/864$ & 16 & $1/576$ \\ 144 & \texttt{7,11,13,19,21,25} & 12 & $1/1512$ & 20 & $1/1008$ \\ 145 & \texttt{7,11,13,19,21,25,30} & 12 & $1/4320$ & 16 & $1/1134$ \\ 146 & \texttt{7,11,13,19,21,26} & 10 & $1/1296$ & 12 & $1/1980$ \\ 147 & \texttt{7,11,13,19,21,26,28} & 10 & $1/864$ & 12 & $1/1320$ \\ 148 & \texttt{7,11,13,19,21,30} & 10 & $1/1728$ & 16 & $1/2106$ \\ 149 & \texttt{7,11,13,19,22} & 14 & $1/864$ & 16 & $1/576$ \\ 150 & \texttt{7,11,13,19,22,28} & 10 & $1/864$ & 12 & $1/1320$ \\ 151 & \texttt{7,11,13,19,28} & 12 & $1/1620$ & 12 & $1/1980$ \\ 152 & \texttt{7,11,13,19,30} & 10 & $1/1728$ & 16 & $1/1134$ \\ 153 & \texttt{7,11,13,22} & 14 & $1/432$ & 20 & $1/288$ \\ 154 & \texttt{7,11,13,22,26} & 10 & $1/864$ & 20 & $1/576$ \\ 155 & \texttt{7,11,13,22,26,28} & 10 & $1/432$ & 20 & $1/288$ \\ 156 & \texttt{7,11,13,30} & 12 & $1/864$ & 24 & $1/324$ \\ 157 & \texttt{7,11,19} & 12 & $1/216$ & 24 & $1/216$ \\ 158 & \texttt{7,11,19,28} & 12 & $1/2160$ & 12 & $1/1320$ \\ 159 & \texttt{7,11,19,29} & 10 & $1/1152$ & 16 & $1/1320$ \\ 160 & \texttt{7,11,19,29,30} & 10 & $1/1728$ & 16 & $25/27594$ \\ 161 & \texttt{7,11,21} & 14 & $1/216$ & 24 & $1/144$ \\ 162 & \texttt{7,11,21,25} & 10 & $1/216$ & 32 & $1/144$ \\ 163 & \texttt{7,11,21,25,30} & 10 & $1/576$ & 16 & $1/216$ \\ 164 & \texttt{7,11,21,26} & 12 & $1/1512$ & 16 & $1/576$ \\ 165 & \texttt{7,11,21,26,28} & 12 & $1/1080$ & 12 & $1/920$ \\ 166 & \texttt{7,11,21,30} & 10 & $1/1152$ & 16 & $1/432$ \\ 167 & \texttt{7,11,28} & 12 & $1/432$ & 16 & $1/288$ \\ 168 & \texttt{7,11,29} & 12 & $1/576$ & 24 & $1/288$ \\ 169 & \texttt{7,11,29,30} & 10 & $1/864$ & 16 & $2/891$ \\ 170 & \texttt{7,25} & 18 & $1/216$ & 16 & $1/144$ \\ 171 & \texttt{7,25,30} & 12 & $1/720$ & 16 & $1/216$ \\ 172 & \texttt{7,27} & 12 & $1/288$ & 24 & $1/144$ \\ 173 & \texttt{7,27,29} & 10 & $1/576$ & 16 & $1/378$ \\ 174 & \texttt{7,27,29,30} & 10 & $1/864$ & 16 & $1/504$ \\ 175 & \texttt{15,23} & 10 & $1/288$ & 32 & $1/144$ \\ 176 & \texttt{15,23,27} & 10 & $1/576$ & 24 & $1/216$ \\ 177 & \texttt{15,23,27,29} & 10 & $1/864$ & 16 & $1/216$ \\ 178 & \texttt{15,23,27,29,30} & 14 & $1/2592$ & 24 & $1/240$ \\ 179 & \texttt{31} & 10 & $1/360$ & 32 & $1/36$ \\ \bottomrule\end{longtable} \endgroup \clearpage \section{Complete independent read-only geometric checker}\label{app:checker} This program expects the supplied rational JSON files. It imports no primary or prior verifier. Its orbit-enumeration field refers only to this script; the separate program in the next appendix supplies that check. The working directory may be anywhere because paths are resolved relative to the script. \lstinputlisting[language=Python]{../code/check_independent.py} \clearpage \section{Complete independent covering-clutter enumeration}\label{app:orbit} \lstinputlisting[language=Python]{../code/check_orbits.py} \section{Machine-readable status and data interpretation} The file \texttt{verification/release\_report.json} sets \begin{lstlisting} "all_declared_suites_passed": true, "unrestricted_five_dimensional_theorem_proved": false, "global_integer_level_certified": null, "proof_assistant_verified": false, "externally_peer_reviewed": false, "publication_priority_established": false \end{lstlisting} These fields distinguish a successful finite verification run from the missing global certificate. The manifest hashes fixed artifacts, not their mathematical truth. The chronological dossier preserves earlier versions, so readers can distinguish inherited claims from newly checked data and new proofs. \end{document}