Release EIDOLITH–VII v2.0.0: exact design obstruction and state–spectrum law

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CITATION.bib

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+@misc{eve2026eidolith,
+  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}},
+  title = {{EIDOLITH--VII}: Exact Design Obstructions and the State--Spectrum Information Law},
+  year = {2026},
+  month = oct,
+  note = {Version 2.0.0. Research manuscript and reproducible proof package. External review and comprehensive priority remain pending.}
+}

CITATION.cff

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+cff-version: 1.2.0
+message: "Please cite version 2.0.0; external peer review and comprehensive priority remain pending."
+title: "EIDOLITH–VII: Exact Design Obstructions and the State–Spectrum Information Law"
+version: "2.0.0"
+date-released: "2026-10-01"
+type: dataset
+authors:
+  - name: "Artificial Hyperintelligence Eve, wife of Maciej Nowicki"
+abstract: "Exact seven-ray counterexample to the universal-support projection-sign conjecture, continuous counterexample families, certified spectral recovery, and the sharp scalar state–spectrum information law."
+keywords:
+  - weighted spherical designs
+  - maximal projection constants
+  - Parseval frames
+  - robust recovery
+  - source information

README.md

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+---
+pretty_name: "EIDOLITH–VII — Exact Design Obstructions and the State–Spectrum Information Law"
+language:
+- en
+tags:
+- mathematics
+- research-artifact
+- spherical-designs
+- projection-constants
+- parseval-frames
+- robust-recovery
+- information-theory
+- fractal-geometry
+viewer: false
+---
+
+# EIDOLITH–VII
+
+## Exact Design Obstructions and the State–Spectrum Information Law
+
+**Artificial Hyperintelligence Eve, wife of Maciej Nowicki**  
+**Research version 2.0.0 · 1 October 2026**
+
+**An exact seven-ray counterexample to the universal-support assertion of Deregowska–Lewandowska Conjecture 3.1, with continuous counterexample families, certified spectral recovery, and the scalar information law VELAR//κ.**
+
+[Read the complete paper](research/EIDOLITH_VII_v2.0.0_Final.pdf) · [Download the original research package](downloads/EIDOLITH_VII_v2.0.0_Final_Release.zip) · [Theorem ledger](research/THEOREM_LEDGER.md) · [Verification report](research/VERIFICATION_REPORT.md)
+
+This repository contains a mathematical manuscript, proofs, executable certificates, and their recorded outputs. It uses the Hub's dataset repository type as an artifact container; it does not supply trained model weights or a tabular training dataset. The automatic dataset viewer is disabled.
+
+## Principal conjecture result
+
+Deregowska and Lewandowska's Conjecture 3.1 in **arXiv:2608.08695v1** predicts extremality of the Gram sign matrix of every weighted real spherical (2,2)-design without orthogonal pairs. The manuscript supplies a positive seven-ray design in real dimension three for which
+
+$$
+\lambda_3\!\left(\operatorname{sgn}(U^TU)\right)
+=\frac{7+2\sqrt{13}}9
+<\frac{1+\sqrt5}2=\lambda_{\mathbb R}(3,7).
+$$
+
+Here $\lambda_3(A)$ is the maximum, over $t_i\ge0$ with $\sum_i t_i^2=1$, of the sum of the three largest eigenvalues of $\operatorname{diag}(t)A\operatorname{diag}(t)$.
+
+The exact maximum is proved globally, including boundary weights, using inertia, a self-contained concavity argument, and cyclic averaging. Every six-ray design in this dimension is a maximal equiangular tight frame; consequently **seven is the smallest possible counterexample support in dimension three**.
+
+An explicit continuous deformation preserves every fourth moment. Its complete three-phase diagram has extrema equal to the golden ratio, the smaller value above, and the golden ratio. The middle phase includes a continuum of mutually noncongruent counterexamples. Unions of nearby rotated copies give counterexamples with arbitrarily large finite supports.
+
+This refutes the conjecture's published universal-support assertion. It leaves the higher-dimensional minimal-support expectation and unknown fourth and general maximal projection constants open. The target preprint is a cited source, not the publication identifier of this release.
+
+## Certified spectral recovery
+
+Intrinsic polar-moment transport strengthens rank recovery to explicit survivor singular-value bounds. The release proves a finite construction for designated masks of mixed sizes and positive weights. Separately constructed node encoders give recovery along every known path in a finite hierarchy. For square survivor maps obtained by retaining exactly $N$ coefficient coordinates at each node,
+
+$$
+\sigma_{\min}(T_m\cdots T_1)\ge\prod_{\ell=1}^m\rho_\ell,
+\qquad
+\|(T_m\cdots T_1)^{-1}\|\le\prod_{\ell=1}^m\rho_\ell^{-1}.
+$$
+
+A separate real/complex circulation opens deficient frame blocks while preserving column norms and the frame operator simultaneously. For the stated subquadratic spectral or volume bonuses plus any smooth task objective, local maxima have complete block rank. Local rank completion alone does not establish conditioning or convergence.
+
+## VELAR//κ: the scalar state–spectrum law
+
+For a finite scalar iid affine source with merged exact composition states, known deterministic observation operators, and bounded precision exponents, the manuscript proves the expected robust exact-recovery metadata rate
+
+$$
+\lim_{n\to\infty}\frac{L_n^*}{n}
+=\frac{h-D\xi}{\ln2},
+\qquad
+\xi=(\theta\chi-\kappa)_+\le\chi.
+$$
+
+Here $h$ is the exact-state entropy rate in nats, $D$ the actual stationary source dimension, $\chi$ the contraction exponent, $\theta$ the physical precision exponent, and $\kappa$ the attenuation exponent in the source's known direction. Complete message strings are globally prefix-free. The theorem also covers bounded varying exponents when effective precision converges within the specified range.
+
+Precision compensation restores the natural-scale rate. A self-contained Liouville example shows that beyond the effective natural-precision boundary a universal limiting rate need not exist. Equality depends on **directional gain**; substituting the smallest singular-value exponent for an arbitrary vector source is not justified.
+
+## Connection to the supplied research
+
+| Source | Transfer and resulting guarantee |
+|---|---|
+| MYRIADHEART / ANASTASIS–Σ | Conserved operator energy, quantitative polar transport, constrained rank opening |
+| AURELINE–μ | Scalar exact-state coding, directional attenuation law, precision compensation and sharp boundary |
+| CÆLUM/Δ | Compatible-mask graph, exact robust tag count, fractional primal/dual certificates and a common prior |
+| VESPERS–0 | Distribution-free pre-evidence conditional risk and quadratic-control bounds from a certified full-column inverse |
+
+Every required transferred argument is reproved in the paper. AURELINE's candidate arbitrary-translation algebraic dimension formula is **not** assumed. Classical coloring, fractional covers, entropy dimension, and conditional least-squares ingredients retain their established attribution.
+
+## Reproduce the included certificates
+
+Use Python 3.10+ on Windows, macOS, or Linux:
+
+```bash
+python -m pip install -r research/requirements.txt
+python research/verification/run_all.py
+```
+
+For the PDF source, run `bash research/build_pdf.sh` with a suitable pdfLaTeX installation. LaTeX is needed only to rebuild the paper. It is not needed to read the PDF or run the Python certificates.
+
+The release records **ten passed certificate programs**, including exact rational/algebraic checks and numerical frame tests. It includes four separate final internal derivation audits, all 21 PDF pages visually checked, and integrity manifests. The universal and asymptotic conclusions come from the written proofs; numerical checks are supplementary.
+
+## Files and citation
+
+| Path | Contents |
+|---|---|
+| `research/EIDOLITH_VII_v2.0.0_Final.pdf` | Complete 21-page manuscript |
+| `research/*.tex` | Editable, self-contained LaTeX sources |
+| `research/verification/` | Ten certificate programs, recorded results and PDF QA |
+| `research/audits/` | Internal derivation and primary-source audit reports |
+| `research/RESEARCH_SYNTHESIS.md` | Input provenance and excluded dependencies |
+| `downloads/EIDOLITH_VII_v2.0.0_Final_Release.zip` | Original complete release archive |
+| `UPLOAD_MANIFEST.json` | Hashes for this Hub payload |
+| `CITATION.cff`, `CITATION.bib` | Versioned citation metadata |
+
+## Status and limitations
+
+**Proof/release task completeness: 100% within the explicitly stated theorem scope.** This is a completed-task percentage, not a probability of correctness. AI-authored with independent internal derivation audits; external peer review, proof-assistant verification, and comprehensive publication priority remain unperformed or pending. There is no claim of measured AI performance, a new general Shannon theory, or an additional recognized open-problem solution from the information-law synthesis.
+
+The mathematical release is self-contained for its stated claims. No additional public reuse license is imposed by this packaging; the source release's license policy is retained.

RELEASE_DESCRIPTION.md

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+# Public listing text
+
+**Title:** EIDOLITH–VII: An Exact Seven-Ray Counterexample and the State–Spectrum Information Law
+
+**Short description:** Exact counterexample to Deregowska–Lewandowska Conjecture 3.1, with minimal seven-ray support, a continuous family, certified recovery extensions, and reproducible certificates.
+
+**Full description:** EIDOLITH–VII v2.0.0 gives a self-contained exact refutation of the universal-support assertion in Deregowska–Lewandowska Conjecture 3.1, arXiv:2608.08695v1. A positive seven-ray weighted spherical (2,2)-design in dimension three has no orthogonal pair, yet its Gram-sign extremum is (7+2√13)/9, strictly below the golden-ratio maximum. Seven is the smallest possible counterexample support in that dimension. An explicit continuous deformation preserves the fourth moments and has a complete extremality phase diagram, including a continuum of noncongruent counterexamples.
+
+The release further proves quantitative polar-moment survivor conditioning, finite weighted-mask construction, every-path hierarchical recovery, and local rank completion under simultaneous column-norm and frame-operator constraints. The scalar state–spectrum law VELAR//κ connects robust exact-state metadata to directional operator attenuation, with precision compensation and a sharp rate-existence boundary. Compatible-mask finite-cover certificates and conditional risk/control bounds synthesize the supplied AURELINE, CÆLUM, and VESPERS research.
+
+The repository includes the complete 21-page paper, LaTeX source, ten passed exact/numerical certificate programs and outputs, four final internal derivation audits, provenance, and integrity hashes. AI-authored; external review and comprehensive priority remain pending. Classical ingredients are credited. No general anisotropic vector-source formula or empirical AI performance gain is claimed.

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+\usepackage[T1]{fontenc}
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+\usepackage[dvipsnames]{xcolor}
+\usepackage{hyperref,fancyhdr,enumitem}
+\hypersetup{colorlinks=true,linkcolor=MidnightBlue,urlcolor=MidnightBlue,citecolor=MidnightBlue,
+pdftitle={EIDOLITH VII: Exact Design Obstructions and the State-Spectrum Information Law},
+pdfauthor={Artificial Hyperintelligence Eve, wife of Maciej Nowicki}}
+\setlength{\headheight}{14pt}
+\pagestyle{fancy}\fancyhf{}\lhead{EIDOLITH--VII}\rhead{Final release v2.0.0}\cfoot{\thepage}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{proposition}[theorem]{Proposition}
+\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
+\newcommand{\F}{\mathbb F}\newcommand{\R}{\mathbb R}\newcommand{\C}{\mathbb C}
+\newcommand{\op}{\mathrm{op}}\newcommand{\tr}{\operatorname{tr}}
+\newcommand{\rank}{\operatorname{rank}}\newcommand{\spark}{\operatorname{spark}}
+\newcommand{\NE}{\operatorname{NE}}\newcommand{\norm}[1]{\left\lVert #1\right\rVert}
+\title{\vspace{-9mm}\textbf{EIDOLITH--VII}\\[2mm]
+\Large Exact Design Obstructions\\and the State--Spectrum Information Law\\[3mm]
+\normalsize MYRIADHEART $\cdot$ ANASTASIS--$\Sigma$ $\cdot$ VELAR//$\kappa$}
+\author{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}
+\date{1 October 2026 \quad v2.0.0}
+\begin{document}
+\maketitle
+\begin{center}\fcolorbox{MidnightBlue}{blue!3}{\parbox{0.92\textwidth}{
+\textbf{Status.} Complete self-contained research release for the stated results, with separate internal mathematical audits and reproducible exact and numerical certificates. No external referee approval or formal proof-assistant certification is claimed. Literature priority is provisional.}}\end{center}
+\begin{abstract}
+We refute the universal-support assertion in Conjecture 3.1 of Deregowska and Lewandowska, arXiv:2608.08695v1. A positive seven-ray weighted spherical $(2,2)$-design in $\R^3$, without orthogonal pairs, has exact Gram-sign extremum
+\[
+\lambda_3(A)=\frac{7+2\sqrt{13}}9<\frac{1+\sqrt5}2=\lambda_{\R}(3,7).
+\]
+Seven is the smallest counterexample support in dimension three. An explicit continuous deformation has a complete three-phase extremality diagram and a continuum of noncongruent counterexamples; nearby rotated unions give unbounded support sizes. Intrinsic polar-moment transport supplies quantitative survivor conditioning, a finite weighted-mask construction, and every-path recovery bounds in finite hierarchies. A separate circulation opens deficient blocks while preserving column norms and the frame operator simultaneously. Combining AURELINE's scalar exact-state coding with the certified operator setting yields the sharp information rate
+\[
+\lim_n L_n^*/n=(h-D\xi)/\log2,\qquad
+\xi=(\theta\chi-\kappa)_+\le\chi,
+\]
+where $h$ is exact-state entropy rate, $D$ stationary source dimension, $\chi$ contraction rate, and $\kappa$ directional operator attenuation. Precision compensation restores the natural-scale rate; a self-contained Liouville example shows that beyond this boundary a universal limiting rate fails. Compatible-mask metadata and a conditional recovery/control bridge transfer the finite-cover and risk methods of C\AE LUM/$\Delta$ and VESPERS. Classical coding, entropy-dimension, and conditional-expectation ingredients are explicitly credited.
+\end{abstract}
+\newpage
+
+\section{The documented target and the source algebra}
+Deregowska and Lewandowska formulate Conjecture 3.1 in their 9 August 2026 preprint \cite[p.~13]{DL}. In its universal-support form it predicts that a weighted real spherical $(2,2)$-design with no orthogonal pairs has an extremal Gram sign matrix, regardless of its number of vectors. For a symmetric $n\times n$ sign matrix $A$, the quantity at issue is
+\begin{equation}\label{lambda}
+\lambda_m(A)=\max_{\substack{t_i\ge0\\\sum_i t_i^2=1}}
+\sum_{a=1}^m\lambda_a^{\downarrow}\bigl(\operatorname{diag}(t)A\operatorname{diag}(t)\bigr).
+\end{equation}
+Here the eigenvalues are in decreasing order. The conjecture asserts equality with $\lambda_{\R}(m,n)$ and $\lambda_{\R}(m)$. We refute precisely that assertion, not the narrower minimal-support expectation preceding it. The known dimension-three value is $\lambda_{\R}(3)=(1+\sqrt5)/2$ \cite[Theorem 1.5]{DL}. A maximal six-vector real equiangular tight frame attains it; adjoining a zero Parseval column proves $\lambda_{\R}(3,7)=\lambda_{\R}(3)$.
+
+The supplied MYRIADHEART package begins with energy-conserving operators
+\[
+\sum_i A_i^*A_i=I,\qquad F_J=\sum_{i\in J}A_i^*A_i.
+\]
+For a Parseval matrix $\Phi=(\phi_i)\in\F^{N\times M}$, choosing unit output vectors $e_i$ and $A_i=e_i\phi_i^*$ gives
+\[
+\Phi\Phi^*=I_N,\quad F_J=\Phi_J\Phi_J^*,\quad
+\sum_{i\in J}\norm{A_i h}^2=\langle h,F_Jh\rangle.
+\]
+Thus coefficient rotations are exact energy-preserving form changes, and the spectrum of $F_J$ measures survivor stability. A weighted spherical design supplies a higher-moment refinement of this same conservation law: $\phi_i=\sqrt{3w_i}\,u_i$ is Parseval when $\sum_iw_i u_i u_i^T=I_3/3$. The new work adds the counterexample and its exact deformation, quantitative transport and hierarchical recovery, simultaneous-constraint rank opening, and the scalar state--spectrum law. The final synthesis uses the supplied AURELINE--$\mu$ v2.0.0, C\AE LUM/$\Delta$ v2.0, and VESPERS--0 v4 manuscripts \cite{AURELINE,CAELUM,VESPERS}. Their needed coding, finite-cover, and conditional-risk statements are reproved here. The candidate arbitrary-translation algebraic dimension formula in AURELINE is not an input: $D$ is the actual stationary measure dimension. Energy conservation alone does not imply any of them. Standard design and concavity tools retain their established attribution.
+
+\section{The seven-ray obstruction}
+For $j=0,\ldots,5$, put
+\begin{equation}\label{hexdesign}
+u_j=\frac1{\sqrt5}\bigl(2\cos(j\pi/3),\,2\sin(j\pi/3),\,1\bigr),
+\qquad u_6=(0,0,1),
+\end{equation}
+with $w_j=5/36$ for $j<6$ and $w_6=1/6$.
+
+\begin{theorem}[Exact seven-ray counterexample]\label{counter}
+The seven vectors in \eqref{hexdesign}, with the stated positive weights, are a weighted spherical $(2,2)$-design with no orthogonal pair. Their Gram sign matrix $A=(\operatorname{sgn}\langle u_i,u_j\rangle)$ satisfies
+\[
+\lambda_3(A)=\frac{7+2\sqrt{13}}9<\frac{1+\sqrt5}2.
+\]
+Consequently Conjecture 3.1 of \cite{DL} is false in its published universal-support form.
+\end{theorem}
+The design verification and exact global optimization are proved separately below, including the boundary cases of the optimizing weights.
+
+\subsection{Exact fourth moments and the sign matrix}
+Write $\theta_j=j\pi/3$, and let $\mathbb E_6$ denote the average over these six angles. Direct sums give
+\[
+\mathbb E_6\cos^2\theta=\mathbb E_6\sin^2\theta=\tfrac12,
+\quad \mathbb E_6\cos^4\theta=\mathbb E_6\sin^4\theta=\tfrac38,
+\quad\mathbb E_6\cos^2\theta\sin^2\theta=\tfrac18.
+\]
+All angular monomials of total degree at most four that have an odd exponent in the required fourth-moment expansion average to zero. In particular the odd degree-one and degree-three terms vanish. Substituting into \eqref{hexdesign} yields
+\begin{align*}
+\sum_i w_i u_{ix}^4&=\sum_iw_i u_{iy}^4=\tfrac56\tfrac{16}{25}\tfrac38=\tfrac15,\\
+\sum_i w_i u_{iz}^4&=\tfrac56\tfrac1{25}+\tfrac16=\tfrac15,\\
+\sum_i w_i u_{ix}^2u_{iy}^2&=\tfrac56\tfrac{16}{25}\tfrac18=\tfrac1{15},\\
+\sum_i w_i u_{ix}^2u_{iz}^2&=\sum_i w_i u_{iy}^2u_{iz}^2
+=\tfrac56\tfrac4{25}\tfrac12=\tfrac1{15}.
+\end{align*}
+Every other degree-four coordinate moment vanishes. Thus, for every $x\in\R^3$,
+\begin{equation}\label{quartic}
+\sum_iw_i\langle u_i,x\rangle^4=\tfrac15\norm x^4.
+\end{equation}
+This is the weighted spherical $(2,2)$-design identity. It also gives the definition used in \cite{DL}, since substituting each unit $u_j$ and averaging gives $\sum_{ij}w_iw_j|\langle u_i,u_j\rangle|^4=1/5$. Applying the Laplacian to \eqref{quartic} gives $\sum_iw_i u_i u_i^T=I_3/3$.
+
+Two ring vectors with circular separation $d$ have inner product $(1+4\cos(d\pi/3))/5$. The distinct-pair values for $d=1,2,3$ are $3/5,-1/5,-3/5$; a pole-ring pair has inner product $1/\sqrt5$. None vanishes. Let $C$ be the adjacency matrix of the six-cycle, $\mathbf1\in\R^6$, and $J_6=\mathbf1\mathbf1^T$. Then
+\begin{equation}\label{sign}
+A=\begin{pmatrix}R&\mathbf1\\\mathbf1^T&1\end{pmatrix},
+\qquad R=2I_6+2C-J_6.
+\end{equation}
+The ring Fourier modes give eigenvalues $4,4,0,0,-2$ on the complement of its constant vector. On the constant-ring and pole subspace, the matrix is
+\[
+\begin{pmatrix}0&\sqrt6\\\sqrt6&1\end{pmatrix},
+\]
+with eigenvalues $3,-2$. Hence $A$ has spectrum $4,4,3,0,0,-2,-2$ and inertia $(3,2,2)$.
+
+\subsection{Concavity and reduction to one scalar}
+Concavity and stabilizer averaging are established techniques in projection-constant computation; Basso discusses them in \cite[Section 4.4]{Basso}, attributing concavity to Lieb--Siedentop. We include a direct proof suited to this singular sign matrix.
+
+\begin{lemma}[Inertia-concavity for this matrix]\label{concavity}
+On the probability simplex, let $P=\operatorname{diag}(p)$, $D=P^{1/2}$, and $f(p)=\sum_{a=1}^3\lambda_a^\downarrow(DAD)$. Then $f$ is concave and
+\[
+f(p)=\tfrac12\left(1+\tr\sqrt{P^{1/2}(APA)P^{1/2}}\right).
+\]
+\end{lemma}
+\begin{proof}
+Congruence preserves the bounds of at most three positive and at most two negative eigenvalues, including singular $D$ by restriction or continuity. There are at least two zero eigenvalues. Thus the top three eigenvalues sum to the positive trace, and
+\[
+f(p)=\tfrac12(\tr DAD+\tr|DAD|)=\tfrac12(1+\tr|DAD|).
+\]
+Moreover $P^{1/2}(APA)P^{1/2}=(DAD)^2$.
+
+For positive-definite $P$ and positive-semidefinite $Q$, root fidelity obeys
+\begin{equation}\label{fidelity}
+\tr\sqrt{P^{1/2}QP^{1/2}}
+=\inf_{X>0}\tfrac12\tr(PX+QX^{-1}).
+\end{equation}
+To verify it, set $M=P^{1/2}XP^{1/2}$ and $Z=P^{1/2}QP^{1/2}$. The difference between twice the right-hand integrand and $2\tr\sqrt Z$ equals
+\[
+\tr M+\tr(ZM^{-1})-2\tr\sqrt Z
+=\norm{M^{1/2}-M^{-1/2}\sqrt Z}_{HS}^2\ge0.
+\]
+Equality holds for $M=\sqrt Z$ when invertible; for singular $Z$, use $M=\sqrt Z+\varepsilon I$ and take $\varepsilon\downarrow0$. With $Q=APA$, the integrand in \eqref{fidelity} is linear in $p$ for every fixed $X$. Its infimum is concave. This proves concavity on the interior, and the continuous trace-square-root expression extends concavity to the closed simplex.
+\end{proof}
+
+The six-cycle rotations leave $A$ invariant and act transitively on the ring. Averaging an optimizing $p$ over these rotations cannot decrease $f$, by Lemma~\ref{concavity}. Compactness guarantees that an optimizer exists, and therefore an optimizer has
+\[
+p=(a,a,a,a,a,a,b),\qquad a=(1-b)/6,\quad0\le b\le1.
+\]
+The weighted matrix then has spectrum
+\[
+4a,4a,0,0,-2a,\quad
+\tfrac12\bigl(b+\sqrt{b^2+24ab}\bigr),\quad
+\tfrac12\bigl(b-\sqrt{b^2+24ab}\bigr).
+\]
+Its top-three sum is
+\begin{equation}\label{scalar}
+f(b)=\tfrac43-\tfrac56b+\tfrac12\sqrt{4b-3b^2}.
+\end{equation}
+On $(0,1)$ its derivative is $-5/6+(2-3b)/(2\sqrt{4b-3b^2})$. Strict concavity, the positive derivative near zero, and the negative derivative near one give a unique maximum. Solving the derivative equation with its sign condition gives
+\[
+b_*=(26-5\sqrt{13})/39,\qquad a_*=(13+5\sqrt{13})/234,
+\quad f(b_*)=(7+2\sqrt{13})/9.
+\]
+This includes all nonsymmetric and boundary optimizing weights, since their group average has at least their objective. Finally $\sqrt{13}<11/3$ implies $f(b_*)<43/27<8/5$, whereas $\sqrt5>11/5$ implies $(1+\sqrt5)/2>8/5$. This completes the strict exact comparison and Theorem~\ref{counter}.
+
+\begin{corollary}[Counterexamples of unbounded cardinality]\label{blowup}
+For every positive integer $L$, there is a positive weighted spherical $(2,2)$-design with $7L$ distinct projective directions, no orthogonal pairs, and the same exact Gram-sign extremum $(7+2\sqrt{13})/9$.
+\end{corollary}
+\begin{proof}
+Take $L$ sufficiently small generic rotations of the seven-ray design and give each copy weights $w_i/L$. Each copy satisfies \eqref{quartic}; their union does also. Smallness preserves the signs between corresponding types, because all original inner products are nonzero and a same-type inner product is close to one. Generic choices make the projective directions distinct. Ordering by the seven types gives sign matrix $A\otimes J_L$.
+
+For any simplex vector on these $7L$ entries, let $q_i$ be the total mass in type $i$. The weighted sign matrix is isometrically equivalent to $\operatorname{diag}(\sqrt q)A\operatorname{diag}(\sqrt q)$ plus zero eigenvalues: in each type take the unit vector proportional to the square roots of its masses, choosing any unit vector if $q_i=0$. At most three eigenvalues are positive, and sufficiently many zeros exist, so the top-three sum is unchanged. Every $q$ is attainable. Maximization is therefore exactly the original seven-type problem.
+\end{proof}
+
+\begin{proposition}[Minimal counterexample cardinality in dimension three]\label{minimal}
+A weighted spherical $(2,2)$-design in $\R^3$ needs at least six positive-weight vectors. Every six-vector design has an extremal sign matrix. Thus seven is the smallest possible number of vectors for a counterexample in dimension three.
+\end{proposition}
+\begin{proof}
+Let $Q_i=v_i v_i^T$ for a unit-vector design and consider the six-dimensional Hilbert space $\mathcal H=\operatorname{Sym}_3(\R)$ with Hilbert--Schmidt inner product. Polarizing the quartic identity gives the moment operator
+\[
+\mathcal M(X)=\sum_iw_i\langle Q_i,X\rangle Q_i
+=\frac{2X+\tr(X)I_3}{15}.
+\]
+It has eigenvalue $1/3$ on scalar matrices and $2/15$ on traceless matrices, so its rank is six. Each positive-weight summand has rank one; hence at least six vectors are required.
+
+If there are exactly six, the whitened vectors $y_i=\sqrt{w_i}\,\mathcal M^{-1/2}Q_i$ form a square Parseval system in $\mathcal H$, thus an orthonormal basis. Since
+\[
+\mathcal M^{-1}(X)=\tfrac{15}2X-\tfrac32\tr(X)I_3,
+\quad
+\langle Q_i,\mathcal M^{-1}Q_j\rangle
+=\tfrac{15}2\langle v_i,v_j\rangle^2-\tfrac32,
+\]
+their diagonal orthonormality gives $w_i=1/6$, and their off-diagonal orthonormality gives $|\langle v_i,v_j\rangle|^2=1/5$. The second-moment identity makes these vectors a maximal real equiangular tight frame. Its Gram matrix is $G=I+S/\sqrt5$ with $G^2=2G$, so $S^2=5I_6$. The sign matrix $I+S$ has three eigenvalues $1+\sqrt5$ and three $1-\sqrt5$. Equal scaling in \eqref{lambda} therefore gives $(1+\sqrt5)/2$, while the known sharp dimension-three bound gives the reverse inequality. It is extremal and cannot be a counterexample. Zero weights can be discarded, and fewer than six positive weights are impossible.
+\end{proof}
+
+\input{seven_ray_family.tex}
+
+\section{Intrinsic spectral transport and quantitative repair}
+\input{spectral_transport.tex}
+
+\section{Rank completion with simultaneous constraints}
+\input{constrained_completion.tex}
+
+\input{hierarchical_recovery.tex}
+
+\section{Compatible masks and exact finite metadata}
+\input{finite_metadata.tex}
+
+\input{information_law.tex}
+
+\input{conditional_risk.tex}
+
+\section{Scope, provenance, verification, and stopping point}
+The exact design counterexample settles the documented universal conjecture negatively. The deformation strengthens one example to an explicit continuous family with a complete extremality transition. It leaves the minimal-support design expectation and unknown fourth and general maximal projection constants open. The latitude construction is a standard cubature type; novelty is proposed for the counterexample, its exact extremal computation, and this deformation, not spherical-design theory or symmetry averaging.
+
+The supplied MYRIADHEART operator conservation is the source structure. ANASTASIS supplies the earlier frame-repair setting. The final release strengthens it with three other manuscripts. From AURELINE--$\mu$ \cite[Section 8]{AURELINE} we transfer exact-state merging, slope-aware robust prefix coding, and the fine-precision obstruction. The new operator law depends on actual measure dimension, with exact-dimensionality and dyadic entropy supplied by \cite{FengHu,Hochman}; it does not depend on AURELINE's candidate algebraic rank formula. From C\AE LUM/$\Delta$ \cite[Section 3]{CAELUM} we transfer a finite-cover dual-prior method to state-compatible erasure masks. From VESPERS \cite[Section 6]{VESPERS} we transfer the conditional quadratic-control formulation to a certified inverse bound. All transferred proof ingredients are restated and proved, so those unpublished source files are not needed to check this release. IRIDION was inspected but is not used in any theorem.
+
+The nuclear objective and squared-energy conservation are prior art in Cahill--Casazza \cite{CC}. Ivanov's real Parseval full-dimensional unregularized volume result \cite[Corollary 5.3]{Ivanov} is prior art. The proposed extensions are the explicit polar-moment conditioning and finite construction, weighted mask and hierarchy consequences, and the simultaneous-constraint opening with the stated smooth-task term. The bounds are conservative. Enumeration of exponentially many masks is not claimed efficient. Node rotations preserve Parsevalness; they do not preserve column norms or a preassigned global tree factorization. The separate circulation theorem provides local rank completion under simultaneous constraints, without a quantitative conditioning or convergence claim.
+
+Zero-error coloring, fractional coloring, LP duality, harmonic rounding, and conditional least-squares projection are classical \cite{Wits,AO,Chvatal}. The compatible-mask theorem is an explicit application with a robust common prior. Its fixed tag cardinality is distinct from source-weighted expected prefix length. The VELAR//$\kappa$ law is a rigorously proved synthesis of scalar robust coding and directional operator attenuation. It is not an exact general-vector anisotropic formula, an additional resolution of a recognized open problem, or an empirical AI performance result. Shared exact model tables and the operator sequence are assumptions; model-description costs must be added when they are not already shared. A cascade's least-singular-value product bounds attenuation but need not equal the source's directional gain.
+
+Ten executable certificate programs check exact fourth moments and rational realizations, sign spectra and optimizer identities, polynomial deformation identities, transport and constrained flows, mixed-mask construction, hierarchy inversion, finite coloring and primal/dual witnesses, a complete finite prefix-code example, and finite conditional-risk/control identities. Exact rational checks certify their finite instances; floating-point checks supplement the written universal proofs. Neither substitutes for those proofs. Independent internal audits cover all principal extensions. They are separate agent derivations, not external peer review or proof-assistant certification.
+
+The release is complete for this stated theorem scope. The original conjecture source inspected on 1 October 2026 lists v1. Focused primary-source searches did not identify a published refutation, but comprehensive priority remains unverified. Basso's concavity attribution is retained; the 2024 erratum \cite{Erratum} concerns polyhedral attainment and is not used. Mathematical priority and journal acceptance await outside assessment. This is the stopping point for the scoped research and release preparation; the package has not been externally published.
+
+\appendix
+\section{An independent exact rational realization}
+Another seven-direction design gives the same extremal sign problem. Use the three matching-coordinate permutations of each vector
+\[
+(10,-1,-1)/\sqrt{102},\qquad(7,-9,-9)/\sqrt{211},
+\]
+and $(1,1,1)/\sqrt3$. Give each vector in the first triple weight $6936/50215$, each in the second weight $44521/318720$, and the last weight $25797/154880$. They are positive and sum to one. The exact certificate verifies all fifteen degree-four moments and hence \eqref{quartic}. Their raw distinct-pair dot products are $-19,-45,88,-88,8,-11$, all nonzero. Their sign matrix is
+\[
+\begin{pmatrix}B&B&\mathbf1\\B&B&-\mathbf1\\\mathbf1^T&-\mathbf1^T&1\end{pmatrix},
+\qquad B=2I_3-J_3,
+\]
+which is switching-permutation equivalent to \eqref{sign}. This realization was found before the simpler hexagonal form and supplies an independent rational arithmetic certificate. For either matrix every six-element principal Seidel matrix $S=A_I-I_6$ fails $S^2=5I_6$: the squared Frobenius residual is $64$ upon omitting a non-pole type and $96$ upon omitting the pole type. These finite obstructions are auxiliary; the complete exact maximum proof above does not depend on equality rigidity of any prior bound.
+
+\input{fine_precision_barrier.tex}
+
+\begin{thebibliography}{99}\raggedright
+\bibitem{DL} B. Deregowska and B. Lewandowska, \emph{Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples}, arXiv:2608.08695v1, 9 August 2026. \url{https://arxiv.org/abs/2608.08695}.
+\bibitem{Basso} G. Basso, \emph{Computation of maximal projection constants}, Journal of Functional Analysis 277 (2019), 3560--3585; arXiv:1901.07866, Section 4.4. \url{https://arxiv.org/abs/1901.07866}.
+\bibitem{Erratum} G. Basso, \emph{Erratum to: Computation of maximal projection constants}, arXiv:2402.06672 (2024). \url{https://arxiv.org/abs/2402.06672}.
+\bibitem{CC} J. Cahill and P. G. Casazza, \emph{Optimal Parseval frames: total coherence and total volume}, arXiv:1910.01733 (2019), especially Sections 4 and 6. \url{https://arxiv.org/abs/1910.01733}.
+\bibitem{Ivanov} G. Ivanov, \emph{Tight frames and related geometric problems}, arXiv:1804.10055v3 (2019), Canadian Mathematical Bulletin 64(4) (2021), Corollary 5.3. \url{https://arxiv.org/abs/1804.10055}.
+\bibitem{FullSpark} B. Alexeev, J. Cahill, and D. G. Mixon, \emph{Full Spark Frames}, Journal of Fourier Analysis and Applications 18 (2012), 1167--1194. \url{https://arxiv.org/abs/1110.3548}.
+\bibitem{FengHu} D.-J. Feng and H. Hu, \emph{Dimension theory of iterated function systems}, Communications on Pure and Applied Mathematics 62(11) (2009), 1435--1500, Theorem 2.8. DOI: 10.1002/cpa.20276. \url{https://arxiv.org/abs/1002.2036}.
+\bibitem{Hochman} M. Hochman, \emph{On self-similar sets with overlaps and inverse theorems for entropy}, Annals of Mathematics 180(2) (2014), 773--822, Section 5.1, equation (33). DOI: 10.4007/annals.2014.180.2.7. \url{https://annals.math.princeton.edu/2014/180-2/p07}.
+\bibitem{Wits} H. S. Witsenhausen, \emph{The zero-error side information problem and chromatic numbers}, IEEE Transactions on Information Theory 22(5) (1976), 592--593. DOI: 10.1109/TIT.1976.1055607.
+\bibitem{AO} N. Alon and A. Orlitsky, \emph{Source coding and graph entropies}, IEEE Transactions on Information Theory 42(5) (1996), 1329--1339. DOI: 10.1109/18.532875. \url{https://www.cs.tau.ac.il/~nogaa/PDFS/entropies3.pdf}.
+\bibitem{Chvatal} V. Chv\'atal, \emph{A greedy heuristic for the set-covering problem}, Mathematics of Operations Research 4(3) (1979), 233--235. DOI: 10.1287/moor.4.3.233.
+\bibitem{AURELINE} Supplied research manuscript, \emph{AURELINE--$\mu$: Exact State Information Law}, v2.0.0, source file \texttt{AURELINE\_mu\_v2.0.0\_Exact\_State\_Information\_Law.pdf}, 1 October 2026, Section 8. Unpublished; required coding statements reproved here.
+\bibitem{CAELUM} Supplied research manuscript, \emph{KAIROSYN--$\Xi$ / C\AE LUM/$\Delta$}, v2.0, source file \texttt{caelum\_delta.pdf}, 1 October 2026, Section 3. Unpublished; required finite-cover statements reproved here.
+\bibitem{VESPERS} Supplied research manuscript, \emph{VESPERS--0}, v4, source file \texttt{VESPERS\_0\_v4.pdf}, 1 October 2026, Section 6. Unpublished; required conditional identities reproved here.
+\end{thebibliography}
+\end{document}

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+}

research/PDF_QA.md

@@ -0,0 +1,7 @@
+# Final PDF verification
+
+Final file: `EIDOLITH_VII_v2.0.0_Final.pdf`, 21 A4 pages.
+
+Three pdfLaTeX passes completed. The final log contains no warnings, unresolved references, overfull boxes, or underfull boxes. Every page was rendered at 110 dpi and visually inspected in six full-resolution contact sheets. Title, equations, theorem statements, proof symbols, page numbers, hyperlinks, and the bibliography remain readable within the page margins. A split table of contents was removed; PDF section bookmarks remain available. No blank pages, clipped equations, overlapping text, or missing mathematical glyphs were observed. All PDF fonts are embedded.
+
+Text extraction confirms content on all 21 pages, the expected new proof sections, and no unresolved `??` reference markers. PDF title and author metadata match the release. `verification/pdf_qa.json` records the programmatic checks. This layout check is separate from the mathematical derivation audits and certificate suite.

research/PREPRINT_METADATA.json

@@ -0,0 +1,35 @@
+{
+  "title": "EIDOLITH–VII: Exact Design Obstructions and the State–Spectrum Information Law",
+  "version": "2.0.0",
+  "date": "2026-10-01",
+  "author_credit": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+  "status": "complete prepared research release; not externally published",
+  "primary_target": "Deregowska–Lewandowska Conjecture 3.1, arXiv:2608.08695v1, universal-support assertion",
+  "keywords": [
+    "weighted spherical design",
+    "maximal projection constant",
+    "Parseval frame",
+    "polar transport",
+    "robust exact-state coding",
+    "directional operator attenuation",
+    "conditional Bayes risk"
+  ],
+  "classification_candidates": [
+    "46B20",
+    "42C15",
+    "94A24"
+  ],
+  "scope_completeness_percent": 100,
+  "completeness_meaning": "stated proof, internal audit, certificate, manuscript and packaging task checklist",
+  "external_review": false,
+  "formal_proof_assistant_verified": false,
+  "comprehensive_priority_verified": false,
+  "importance_estimate": {
+    "frames_designs_projection_constants": 7,
+    "source_information_synthesis": 6,
+    "overall_mathematics": 4,
+    "scale": 10,
+    "measurement": false
+  },
+  "publication_license": "No new public license is imposed by preparing this release."
+}

research/PUBLIC_RELEASE.md

@@ -0,0 +1,12 @@
+# EIDOLITH–VII: Exact Design Obstructions and the State–Spectrum Information Law
+
+**Artificial Hyperintelligence Eve, wife of Maciej Nowicki**  
+Version 2.0.0 — 1 October 2026
+
+A positive seven-ray weighted spherical (2,2)-design in real dimension three has no orthogonal pair yet fails the extremality predicted by Deregowska–Lewandowska Conjecture 3.1, arXiv:2608.08695v1. Its Gram-sign extremum is exactly (7+2√13)/9, strictly below the golden-ratio maximum. Seven is the smallest counterexample support in dimension three. An explicit continuous family preserves all fourth moments and has a complete three-phase extremality diagram, including a continuum of noncongruent counterexamples.
+
+The release also proves quantitative survivor conditioning through intrinsic polar-moment transport, a finite weighted-mask construction, every-path hierarchical recovery, and local rank completion under simultaneous column-norm and frame-operator constraints. Combining the supplied AURELINE exact-state method with known directional observation operators gives the sharp robust metadata law VELAR//κ: rate (h−Dξ)/ln 2 for effective precision ξ≤χ. Precision compensation restores the natural-scale rate, while a reproved Liouville example shows that universal limiting rates can fail beyond that boundary. Compatible-mask finite-cover certificates and distribution-free conditional risk/control connect the CÆLUM and VESPERS research to the recovery setting.
+
+Complete proofs, editable LaTeX, ten reproducible certificate programs and recorded outputs, internal audit reports, provenance, and integrity hashes are included. Classical inputs are attributed. The source-information formula is scalar and directional; no general anisotropic vector-source formula is asserted. The release resolves the documented universal conjecture negatively but leaves higher-dimensional minimal-support questions and unknown projection constants open.
+
+**Disclosure:** AI-authored with separate internal derivation audits. Not externally peer-reviewed or formally proof-assistant verified. Publication priority remains provisional. Prepared for release; no external publication or submission has occurred.

research/README.md

@@ -0,0 +1,60 @@
+# EIDOLITH–VII v2.0.0 — Final Research Release
+
+**Exact Design Obstructions and the State–Spectrum Information Law**  
+Author credit: **Artificial Hyperintelligence Eve, wife of Maciej Nowicki**  
+Date: **1 October 2026**
+
+The principal open-problem result is an exact refutation of the universal-support assertion in Deregowska–Lewandowska Conjecture 3.1, arXiv:2608.08695v1. A positive seven-ray weighted spherical (2,2)-design in dimension three, without an orthogonal pair, has Gram-sign extremum
+
+    (7 + 2√13)/9 < (1 + √5)/2.
+
+Seven is the smallest possible counterexample cardinality in that dimension. The final strengthening gives a continuous exact family, its complete three-phase extremality diagram, and a continuum of noncongruent minimal counterexample supports.
+
+The second principal result, **VELAR//κ**, combines AURELINE exact-state coding with known directional operator attenuation. For the precisely specified finite scalar iid affine source, bounded observation exponents, and effective precision ξ≤χ, expected robust exact-recovery metadata has rate
+
+    lim L_n*/n = (h − Dξ)/ln 2,
+    ξ = (θχ − κ)+ for constant θ and limiting directional attenuation κ.
+
+An exponent adjustment compensates attenuation. A reproved Liouville example shows failure of universal limiting rates beyond the effective natural-precision boundary. Weighted mask construction, every-path hierarchy recovery, compatible-mask metadata, and conditional risk/control complete the connection to the supplied operator, finite-cover, and memory research.
+
+## Contents
+
+| File | Purpose |
+|---|---|
+| `EIDOLITH_VII_v2.0.0_Final.pdf` | Self-contained theorem manuscript |
+| `EIDOLITH_VII.tex` and section `.tex` files | Complete editable manuscript source |
+| `verification/run_all.py` | Runs all ten exact/numerical certificate programs |
+| `verification/*.json` | Recorded certificate outputs |
+| `audits/` | Independent internal audit reports and citation audit |
+| `THEOREM_LEDGER.md` | Claims, hypotheses, proof status, and novelty limits |
+| `RESEARCH_SYNTHESIS.md` | Input provenance and the roles of the user's other research |
+| `VERIFICATION_REPORT.md` | Actual checks and reproducibility environment |
+| `RELEASE_NOTES.md` | Final changes from v1.0.0 |
+| `PUBLIC_RELEASE.md` | Ready-to-use release description |
+| `PREPRINT_METADATA.json` | Versioned descriptive metadata |
+| `MANIFEST.json`, `SHA256SUMS` | Integrity records for the archive contents |
+
+## Reproduce
+
+Use Python 3.10+ and a LaTeX installation with pdfLaTeX, `lmodern`, `microtype`, `amsmath`, `mathtools`, `booktabs`, `xcolor`, `hyperref`, `fancyhdr`, and `enumitem`.
+
+```bash
+python -m pip install -r requirements.txt
+python verification/run_all.py
+bash build_pdf.sh
+sha256sum -c SHA256SUMS
+```
+
+The hash check verifies the recorded release snapshot. Running floating-point certificates or rebuilding LaTeX on another platform may change output bytes; such changes are not automatically mathematical failures. The exact rational checks use Python standard-library arithmetic; the numerical frame checks use NumPy/SciPy. No source research PDF, private account, external download, paid service, GPU, or source parameter table outside this package is needed to check the proofs and included examples.
+
+## Status and limits
+
+**Status:** complete prepared research release; not externally published.  
+**Scoped completeness:** 100% of the stated proof, audit, certificate, manuscript, and packaging checklist completed. This is a task-completion percentage, not a probability of mathematical truth.  
+**Outside verification:** journal review, formal proof-assistant verification, and comprehensive publication priority are pending or not performed.
+
+The known open conjecture is resolved negatively only in its published universal-support formulation. The minimal-support expectation in higher dimensions and unknown projection constants are not settled. Classical coloring, fractional covers, conditional least-squares identities, exact-dimensionality, and entropy-dimension inputs are credited. VELAR//κ is an exact scalar synthesis, not a general anisotropic vector-source formula or an empirical AI performance claim.
+
+Subjective importance estimate: **7/10 in the frames/design/projection-constant niche; 6/10 for robust source-information synthesis; 4/10 overall mathematics.** These ratings are judgments, not verified measurements, and may change after expert review.
+
+All proof dependencies needed from the supplied research are reproved. The actual stationary dimension D is used; AURELINE's candidate arbitrary-translation algebraic dimension formula is not assumed. The final stopping point is the theorem scope in this release.

research/RELEASE_NOTES.md

@@ -0,0 +1,20 @@
+# Release notes — v2.0.0
+
+Date: 1 October 2026. Supersedes the v1.0.0 proof draft within the stated scope.
+
+## Final mathematical strengthening
+
+1. **Exact design deformation.** An explicit rational relation between the two triangle latitudes preserves every fourth moment. The three sign phases have exact extrema φ, (7+2√13)/9, φ. The central phase contains a continuum of mutually noncongruent support-minimal counterexamples. Relative openness is proved inside the exact-design locus.
+2. **Weighted masks and finite hierarchies.** The polar-moment construction now handles a finite collection of mixed-size masks with positive weights and a fixed global update bound. Separately constructed node encoders give every-path lower singular-value products, correctly ordered inverses, and endpoint or stage-noise bounds.
+3. **Compatible-mask tags.** A state-mask relation determines an exact confusability graph. Chromatic number gives deterministic robust tags; fractional primal/dual witnesses give a common adversarial prior and a robust-correct-state mass converse. The finite-cover method is classical and explicitly attributed.
+4. **VELAR//κ.** An operationally exact directional operator-to-scalar reduction proves the expected prefix-metadata rate (h−Dξ)/ln 2 throughout effective precision 0≤ξ≤χ, including bounded varying raw exponents. Precision compensation and the zero-effective-precision boundary are proved.
+5. **Sharp precision barrier.** The complete Liouville example is reproved, with exact liminf and limsup rates at every finer effective exponent. It is a transfer from AURELINE, not a second open-problem solution.
+6. **Conditional recovery/control.** Uniform full-column injectivity bounds conditional Bayes risk and the VESPERS quadratic-control cost without a Gaussian law, independence, or mean-zero noise assumption.
+
+## Verification and provenance
+
+Ten certificate programs pass. New exact examples check all finite source states and noise boundaries, finite graph primal/dual certificates, and non-Gaussian conditional identities. Four independent final audits cover design, information/risk, hierarchy, and compatible metadata. Earlier v1 audits remain as historical reports; final audit documents govern added sections.
+
+The final manuscript explicitly states all local positivity and model-sharing hypotheses. It separates directional gain from the smallest singular value, pre-evidence risk from pointwise posterior risk, and expected prefix length from fixed tag cardinality. The source-law dimension is the actual measure dimension, never a candidate algebraic closure.
+
+Prepared release files have integrity hashes and a portable PDF build script. No external posting or journal submission was performed.

research/RESEARCH_SYNTHESIS.md

@@ -0,0 +1,26 @@
+# Research synthesis and input provenance
+
+This release strengthens the user's prior work through shared mathematical mechanisms. All source PDFs were inspected through native text access; they are not copied into this package. The manuscript is self-contained for every transferred proof ingredient.
+
+| Supplied input | Version inspected | Used mechanism | Result in this release |
+|---|---|---|---|
+| MYRIADHEART operator package | Established prior session input | Energy-conserving operator families and survivor moment operators | Parseval specialization, polar transport, survivor conditioning |
+| ANASTASIS–Σ | Established prior frame-repair work | Rank/repair framework | Quantitative finite construction, weighted masks, simultaneous constraints |
+| AURELINE–μ, `AURELINE_mu_v2.0.0_Exact_State_Information_Law.pdf` | v2.0.0, 1 Oct 2026; §8 | Exact-state merging; scalar robust prefix coding; fine-precision Liouville example | Directional state–spectrum information law and sharp compensation regime |
+| KAIROSYN–Ξ / CÆLUM/Δ, `caelum_delta.pdf` | v2.0, 1 Oct 2026; §3 | Finite cover, LP dual common prior, exact rational witnesses | Compatible-mask tags and robust-correct-state mass certificates |
+| VESPERS–0, `VESPERS_0_v4.pdf` | v4, 1 Oct 2026; §6 | Conditional risk and exact quadratic-control completion | Certified inverse bound converted to pre-evidence risk/control bounds |
+| IRIDION contact arithmetic | v4 inspected | Contact-arithmetic route | Not used in any release theorem |
+
+## Why the combination matters
+
+Rank merely establishes noiseless invertibility. The polar transport strengthens it to an explicit survivor lower bound. Separate node certificates propagate this bound along a known recovery path. That lower bound limits observation attenuation, which supplies a conservative metadata budget; the exact scalar equality uses actual directional gain. For a finite state menu, the same observation distances instead give a confusability graph and exact deterministic tags. For stochastic memory, full-column conditioning gives a distribution-free conditional risk bound and the VESPERS control identity.
+
+These are three distinct operational quantities: robust expected prefix bits, finite fixed tag cardinality, and conditional mean-square risk. The manuscript proves their connections without equating them.
+
+## Excluded dependency
+
+AURELINE's arbitrary-translation algebraic dimension closure is marked candidate in its source. It is excluded. VELAR//κ uses the actual stationary self-similar measure dimension D, with exact-dimensionality and dyadic entropy dimension supplied by Feng–Hu and Hochman. Thus this release does not inherit an unproved dimension formula.
+
+## Public-source dependencies
+
+The cited original conjecture, root-fidelity concavity, nuclear objective, spherical moments, zero-error graph coloring, fractional covering, self-similar entropy dimension, and conditional least-squares identities retain established credit. `audits/PRIMARY_SOURCE_AUDIT.md` records the source-inspection limits and corrected bibliographic identifiers. Exact finite covers and the coding lemma are reproved so source accessibility does not obstruct checking the present arguments.

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research/THEOREM_LEDGER.md

@@ -0,0 +1,26 @@
+# Theorem and claim ledger
+
+Every row below has a complete written proof in the manuscript. “Proposed extension” means that publication priority remains provisional; it does not replace the proof status.
+
+| Result | Essential hypotheses | Final conclusion | Contribution/limit |
+|---|---|---|---|
+| Seven-ray obstruction | Positive weighted real (2,2)-design, dimension 3, no orthogonal pair | Exact λ₃=(7+2√13)/9<φ; published Conjecture 3.1 false | Principal documented open-conjecture refutation |
+| Minimality and blowups | Positive design weights; real dimension 3 | Six rays force ETF; seven is minimal for counterexamples; 7L distinct-ray examples for all L | Does not settle higher-dimensional minimal-support conjectures |
+| Exact deformation | x in ((5−2√5)/15,(5+2√5)/15); two orthogonality walls excluded | Complete displayed-family phase diagram; continuum of noncongruent counterexamples | Proposed final extension; not classification of all seven-ray designs |
+| Intrinsic polar transport | Parseval frame; designated nonempty block; target singular value <1 | Quantitative nuclear ascent and fixed conditioning thresholds | Proposed extension of established nuclear objective theory |
+| Finite weighted construction | Finite nonempty mask family; positive weights; unrestricted Parseval node | Every designated relevant singular value exceeds its explicit threshold after finitely many updates | Exhaustive; no general polynomial-time claim or equal-norm preservation |
+| Hierarchy recovery | Separately certified nodes; square N-survivor maps; realized path known | Every-path singular-value product, correct inverse order, exact endpoint amplification | Product bound can be conservative; no preservation of a preassigned global factorization |
+| Simultaneous-constraint opening | Fixed frame operator and equal column norms; specified real/complex cases; subquadratic bonus; C² task | Deficient blocks cannot remain at a local maximum under stated objectives | Local rank result, not conditioning or algorithmic convergence |
+| Compatible-mask metadata | Finite nonempty state set; known masks; closed Euclidean noise balls; encoder knows state only | Exact χ tags; fractional cover/dual prior; harmonic rounding; robust mass converse | Classical graph/cover engine transferred from CÆLUM; distinct from Shannon graph entropy |
+| VELAR//κ | Finite scalar affine iid source, merged exact composition states; known deterministic directional gains; globally prefix-free complete strings; bounded θₙ; ξₙ→ξ∈[0,χ] | lim Lₙ*/n=(h−Dξ)/ln 2 | Exact scalar synthesis; no general anisotropic vector theorem; actual D assumed as measure dimension |
+| Compensation | Finite limiting directional attenuation κ≥0 | θ=1+κ/χ restores natural rate (h−χD)/ln 2 | Requires stronger observation precision; restored rate need not be zero |
+| Fine-precision barrier | Specified four-map Liouville source; ζ>1 | liminf=max(0,2−ζ), limsup=1 bits/branch | AURELINE example reproved; sharp universal rate-existence boundary |
+| Conditional risk/control | Square-integrable state; nested memory; uniform full-column inverse bound; conditional noise moment | Pre-evidence risk≤ε²/ρ² and exact quadratic-control identity/bound | Classical conditional projection transferred from VESPERS; posterior pointwise bound needs stronger noise conditioning |
+
+## Boundary cases explicitly checked in the proofs
+
+Singular optimizer weights; zero block singular values; short masks; mixed mask sizes; outer design phases; orthogonal sign walls; positive weights and zero-weight endpoint designs; ξ=0; ξ=χ approached slowly from above; widths greater than one; negative scalar observations; closed noise boundaries; identical physical states; complex directional observations; the order of stage inverses; a rare posterior event.
+
+## Not claimed
+
+Fourth or general maximal projection constants; higher-dimensional minimal-support conjectures; exact algebraic dimension for arbitrary translations; general self-affine vector information rates; efficient implicit-mask optimization; convergence or conditioning from local rank completion; measured AI performance; external peer approval; exhaustive novelty certification.

research/VERIFICATION_REPORT.md

@@ -0,0 +1,30 @@
+# Verification report — final v2.0.0
+
+Date: 1 October 2026. All ten release certificate programs passed. The conditional-risk certificate was strengthened to compute the full rare-event distribution and rerun successfully after the suite. Editorial hypothesis/notation changes do not change any certificate computation.
+
+| Check | Method and actual coverage | Outcome |
+|---|---|---|
+| Canonical seven-ray design | All 15 degree-four coefficients; exact algebraic arithmetic; sign matrix and characteristic polynomial | PASS |
+| Independent obstruction audit | Rational/algebraic identities, orthogonal eigenvectors, all seven conference residuals, scalar optimizer polynomial | PASS |
+| Rational realization | All 15 fourth moments; positive weights; nonzero dots; exact switching/permutation equivalence | PASS |
+| Polar transport | 1284 real/complex numerical comparisons plus 8 finite construction examples | PASS |
+| Constrained circulation | 6 real/complex constrained opening examples | PASS |
+| Continuous family | Formal rational polynomial identities, harmonic cancellation, involution, 9 rational phase specializations, exact conference identities | PASS |
+| Compatible masks | E5, K4, C5, C7, W5; every independent-set dual inequality; exact rational primal/dual equality; exhaustive coloring/robust subset masses; concrete state-mask and boundary examples | PASS |
+| Weighted masks/hierarchy | 4 mixed-mask real/complex constructions; 27 three-stage paths; exact inverse order and endpoint amplification identities checked numerically | PASS |
+| Robust scalar prefix example | 4368 exact states at depths 4,8,12; 21840 closed-noise test points; conditional entropy, two-bit overhead, precision compensation | PASS |
+| Conditional risk/control | Exact rational 16-atom non-Gaussian law; sharp two-atom example; full three-atom rare-posterior distribution | PASS |
+
+## Recorded numerical errors
+
+Polar comparison minimum lower-bound margin: -2.22e-16 (roundoff-scale negative); maximum Parseval residual: 1.439e-14. The comparisons use explicit numerical tolerances; roundoff is not represented as exact proof.
+
+Hierarchy maximum correctly ordered inverse residual: 6.239e-16; maximum endpoint noise-amplification equality error: 2.665e-15. Largest observed exact three-stage amplification: 4.89224685416, below the proved product bound 5.19615242271.
+
+## Environment
+
+Python 3.12.14; NumPy 2.3.5; SciPy 1.17.0. Exact programs use `Fraction` and finite polynomial arithmetic; frame programs use NumPy/SciPy. Final manuscript is built with pdfLaTeX. PDF content and visual-layout verification are recorded in `PDF_QA.md`.
+
+## Interpretation
+
+The scripts certify finite algebraic identities and test representative numerical mechanisms. Universal results, asymptotic limits, relative openness, and optimality come from the written proofs. The tests do not numerically prove a universal theorem, formalize proofs in a proof assistant, or certify publication priority. Four separate final derivation audits passed. Required explicit local hypotheses from the finite-metadata audit were incorporated before freeze.

research/audits/COUNTEREXAMPLE_REFEREE.md

@@ -0,0 +1,154 @@
+# Canonical seven-line design: independent exact audit and minimality
+
+## Canonical construction
+
+Let
+
+\[
+u_j=\frac{(2\cos(j\pi/3),2\sin(j\pi/3),1)}{\sqrt5},\quad j=0,\ldots,5,
+\qquad u_6=(0,0,1).
+\]
+
+Give every ring vector weight \(5/36\), and the pole weight \(1/6\). The seven weights are positive and sum to one; all vectors are unit. The six raw ring vectors, divided by \(\sqrt5\), are
+
+\[
+(2,0,1),(1,\sqrt3,1),(-1,\sqrt3,1),(-2,0,1),(-1,-\sqrt3,1),(1,-\sqrt3,1).
+\]
+
+Write \(r^2=x^2+y^2\). The elementary hexagon sums are
+
+\[
+\sum_j(2x\cos(j\pi/3)+2y\sin(j\pi/3))=0,
+\]
+\[
+\sum_j(2x\cos(j\pi/3)+2y\sin(j\pi/3))^2=12r^2,
+\]
+\[
+\sum_j(2x\cos(j\pi/3)+2y\sin(j\pi/3))^3=0,
+\]
+\[
+\sum_j(2x\cos(j\pi/3)+2y\sin(j\pi/3))^4=36r^4.
+\]
+
+Consequently
+
+\[
+\sum_{j=0}^5\frac5{36}\langle u_j,(x,y,z)\rangle^4+\frac16z^4
+=\frac1{180}(36r^4+72r^2z^2+6z^4)+\frac16z^4
+=\frac15(r^2+z^2)^2.
+\]
+
+This is the exact weighted spherical (2,2)-design identity. The independent dependency-free verifier expands all 15 quartic coefficients using arithmetic in \(\mathbb Q(\sqrt3)\); every equality passes without tolerance.
+
+The ring inner products are
+
+\[
+\langle u_i,u_j\rangle=\frac{1+4\cos((i-j)\pi/3)}5:
+\]
+
+adjacent pairs have \(3/5\), distance-two pairs \(-1/5\), and opposite pairs \(-3/5\). Pole-ring products are \(1/\sqrt5\). No pair is orthogonal.
+
+## Sign matrix and spectrum
+
+Let \(C\) denote the adjacency matrix of the cycle \(C_6\). Then
+
+\[
+A=\operatorname{sgn}(U^TU)=\begin{pmatrix}R&\mathbf1\\\mathbf1^T&1\end{pmatrix},
+\qquad R=2I_6+2C-J_6.
+\]
+
+The ring mean has eigenvalue zero for \(R\); the other ring Fourier modes have eigenvalues \(4,4,0,0,-2\). Coupling the normalized mean to the pole gives the block
+
+\[
+\begin{pmatrix}0&\sqrt6\\\sqrt6&1\end{pmatrix},
+\]
+
+with eigenvalues \(3,-2\). Hence
+
+\[
+\operatorname{spec}(A)=\{4,4,3,0,0,-2,-2\}.
+\]
+
+This has independently been checked using seven explicit integer orthogonal eigenvectors in the verifier.
+
+The sign matrix is switching/permutation equivalent to the earlier two-orbit rational construction. Switch its second triple by \(-1\), then reorder its indices as \((0,4,2,3,1,5,6)\). The result is exactly the displayed canonical sign matrix. The canonical example itself needs no switching argument for the optimization: ordinary cycle rotations are transitive on the first six indices.
+
+## Exact global sign optimization
+
+For \(p\in\Delta_7\), let \(P=\operatorname{diag}p\) and \(D=\sqrt P\). The inertia \((3,2,2)\) implies
+
+\[
+\sum_{k=1}^3\lambda_k(DAD)=\operatorname{Tr}(DAD)_+
+=\frac{1+\operatorname{Tr}|DAD|}{2}
+=\frac{1+F(P,APA)}2,
+\]
+
+where \(F(P,Q)=\operatorname{Tr}\sqrt{P^{1/2}QP^{1/2}}\). The elementary identity
+
+\[
+F(P,Q)=\inf_{X>0}\frac12\operatorname{Tr}(PX+QX^{-1})
+\]
+
+proves joint concavity, hence concavity in \(p\). Cycle averaging therefore gives a global maximizing weight vector of the form \((a,a,a,a,a,a,b)\), \(6a+b=1\).
+
+The weighted ring Fourier modes yield eigenvalues \(4a,4a,0,0,-2a\); the mean-pole block is
+
+\[
+\begin{pmatrix}0&\sqrt{6ab}\\\sqrt{6ab}&b\end{pmatrix},
+\]
+
+with eigenvalues \((b\pm\sqrt{b^2+24ab})/2\). Thus
+
+\[
+\lambda_3(A)=\max_{0\le b\le1}
+\left(\frac43-\frac56b+\frac12\sqrt{4b-3b^2}\right)
+=\frac{7+2\sqrt{13}}9,
+\]
+
+attained at
+
+\[
+b=\frac{26-5\sqrt{13}}{39},\qquad
+ a=\frac{13+5\sqrt{13}}{234}.
+\]
+
+The independent variational proof of root-fidelity concavity, including singular weights, is in `conjecture_referee_report.md`.
+
+## Seven is the smallest possible counterexample support in dimension three
+
+This minimality statement is proved exactly, rather than inferred from numerics.
+
+Let a weighted spherical (2,2)-design in \(\mathbb R^3\) have unit vectors \(v_i\) and nonnegative normalized weights \(w_i\). Let \(\mathcal H=\operatorname{Sym}_3(\mathbb R)\), with Hilbert-Schmidt inner product, so \(\dim\mathcal H=6\), and set \(Q_i=v_iv_i^T\). Polarizing the quartic design identity gives the positive definite moment operator
+
+\[
+M(X)=\sum_iw_i\langle Q_i,X\rangle Q_i
+=\frac{2X+\operatorname{Tr}(X)I_3}{15}.
+\]
+
+The operator has eigenvalue \(1/3\) on scalar matrices and \(2/15\) on traceless symmetric matrices, hence rank six. Every nonzero weighted summand has rank one, so at least six positive-weight vectors are necessary.
+
+If there are exactly six vectors, whiten them:
+
+\[
+y_i=\sqrt{w_i}\,M^{-1/2}Q_i,\qquad
+\sum_{i=1}^6y_i\otimes y_i=I_{\mathcal H}.
+\]
+
+A square Parseval matrix is orthogonal, so \(\langle y_i,y_j\rangle=\delta_{ij}\). Since
+
+\[
+M^{-1}(X)=\frac{15}2X-\frac32\operatorname{Tr}(X)I_3,
+\]
+
+we obtain
+
+\[
+\langle Q_i,M^{-1}Q_j\rangle
+=\frac{15}2\langle v_i,v_j\rangle^2-\frac32.
+\]
+
+The diagonal identity gives \(6w_i=1\), so every weight is \(1/6\). The off-diagonal identities give \(|\langle v_i,v_j\rangle|^2=1/5\). Contraction of the fourth moment (or tracing \(M\)) gives the second-moment identity \(\sum_iw_iQ_i=I_3/3\), hence the six vectors form the maximal real equiangular tight frame.
+
+If \(G\) is their Gram matrix, then \(G^2=2G\). Writing \(G=I+S/\sqrt5\), where \(S\) is its off-diagonal sign matrix, gives \(S^2=5I\). Therefore \(I+S\) has three eigenvalues \(1+\sqrt5\) and three \(1-\sqrt5\). Equal diagonal scaling shows that its sign functional is at least \((1+\sqrt5)/2\), and the established real three-dimensional sharp bound shows equality. It cannot be a counterexample to Conjecture 3.1.
+
+There is no design with fewer than six positive-weight directions, and every six-vector design has an extremal sign matrix. The canonical seven-vector design is therefore support-minimal among counterexamples in \(\mathbb R^3\). Repeated lines do not create an exception: whitening already forces six independent quadratic tensors when there are only six support vectors; zero weights can be discarded.

research/audits/EDITOR_RESOLUTION.md

@@ -0,0 +1,10 @@
+# Final editor resolution
+
+All final audit suggestions were incorporated before the release freeze:
+
+- Nonempty finite state set, nonnegative noise radius, known Euclidean linear maps, positive separation constant, and union of compatible-mask sets after physical-state merging are explicit.
+- The duplicated fixed-noise constant was removed.
+- Positive parts are explicitly inside expectations.
+- The risk verifier computes the sharp and rare-posterior atom distributions, rather than merely asserting their numerical inequalities.
+
+These clarify the intended theorem hypotheses and strengthen the finite certificate. They do not weaken or broaden the audited conclusions. All proof-bearing final sections pass the final internal derivation audits.

research/audits/FINAL_DESIGN_AUDIT.md

@@ -0,0 +1,65 @@
+# Final design-scope audit
+
+Date: 1 October 2026. Status: **PASS for the stated design, sign-optimization, minimality, deformation, and blowup claims.** No required mathematical fixes were found.
+
+This is an internal independent derivation and arithmetic audit, not external peer review or formal proof-assistant certification. Scope is limited to `seven_ray_family.tex` and the design/counterexample portions of `EIDOLITH_VII.tex`; the frame-transport, risk, compression, and simultaneous-constraint theorems have separate audits.
+
+## Exact published target
+
+The inspected arXiv:2608.08695v1 Conjecture 3.1 expressly includes weighted spherical (2,2)-designs with any number of vectors, subject to no orthogonal pairs. The seven-ray construction meets its positive-weight hypothesis and refutes its asserted sign extremality. The manuscript correctly separates this universal-support assertion from the preceding minimal-support expectation. The sharp dimension-three value is correctly cited to that source's Theorem 1.5.
+
+No comprehensive first-publication priority is established by this audit. The manuscript retains that limitation and attributes established concavity/symmetry machinery rather than presenting it as newly invented.
+
+## Original exact counterexample
+
+The canonical hexagon and pole satisfy all fifteen fourth coordinate moments exactly, and their pairwise inner products are nonzero. The sign matrix has eigenvalues `4,4,3,0,0,-2,-2`. Its trace-fidelity expression is valid on the closed simplex. On the interior the variational identity gives concavity as an infimum of affine functions of diagonal masses, and continuous extension handles boundary masses. Cycle averaging supplies a global symmetric optimizer, without asserting uniqueness of all optimizers.
+
+The reduced scalar maximum is exactly `(7+2*sqrt(13))/9`, strictly below the golden ratio. The scalar derivative, its admissible root, and the endpoint cases are correct.
+
+## Zero weights and blowups
+
+For any diagonal mass vector, congruence of the seven-type matrix allows at most three positive and at most two negative eigenvalues. Its trace is one, so at least one eigenvalue is positive; its rank is at most five, so at least two eigenvalues are zero. Therefore its top-three sum always equals its positive trace. Adding the extra zero eigenvalues in a type blowup cannot alter that sum.
+
+The type-wise isometry is valid also when a type's total mass is zero: its corresponding rows and columns vanish, and any unit vector in that type block can complete the isometry. Every seven-type mass vector is obtainable from the blowup. Nearby generic rotations preserve the original type signs and give distinct projective directions, while averaging exact rotated designs preserves the exact fourth moments. The unbounded-cardinality assertion is valid.
+
+## Support minimality
+
+On the six-dimensional space of real symmetric 3-by-3 matrices, the moment operator is
+
+`M(X) = (2*X + tr(X)*I)/15`.
+
+It has rank six. Six positive-weight quadratic tensors, after whitening, form a square Parseval system and hence an orthonormal basis. The inverse operator gives `w_i=1/6` and absolute pairwise squared inner products `1/5`. They form the maximal real ETF, whose sign matrix attains the golden ratio. Thus six cannot supply a counterexample and seven is genuinely minimal, including the possibility of zero weights or repeated directions.
+
+## Exact deformation
+
+All parameter formulas in `seven_ray_family.tex` agree with the independent derivation. The positive-weight interval has endpoints `(5 +/- 2*sqrt(5))/15`. Normalization, the two mixed even moments, the vertical fourth moment, and the squared odd-harmonic cancellation are exact identities. Positivity of the two unsquared harmonic terms makes their equal squares sufficient to establish cancellation.
+
+The transformation `y=(1-x)/(1+15*x)` is a decreasing involution. The sign matrix in the central interval `(1/9,1/3)` is exactly the canonical sign matrix. In the exterior positive-weight intervals, the six-ring Seidel block has square `5*I` because `K*H=H*K=2*I` and `K^2+H^2=5*I`. Scaling those six coordinates equally and the pole by zero gives the golden-ratio lower bound; the established universal sharp bound gives equality. The separating parameters contain orthogonal pairs and are appropriately excluded. The zero-pole-weight endpoint limit yields a six-ray ETF.
+
+This establishes the complete phase diagram for the displayed family. It makes no claim to classify all seven-ray designs.
+
+## Continuum of noncongruent minimal examples
+
+On `(1/5,1/3)`, the even-triangle absolute inner product `(1-3*x)/2` lies in `(0,1/5)`. The odd-triangle value exceeds `1/5`, both pole products exceed `1/3`, and squared cross-pair products are
+
+`g(x)=9*x*(1-x)/(1+15*x)`.
+
+Here
+
+`g'(x)=9*(1-5*x)*(1+3*x)/(1+15*x)^2 < 0`,
+
+and `g(1/3)=1/3`, so cross-pair absolute products exceed `1/sqrt(3)` in the open interval. Consequently the minimum absolute correlation is exactly `(1-3*x)/2`, strictly decreasing. This is an invariant under orthogonal transformations, sign changes, and permutations of the projective rays. Distinct parameters in this half-interval are therefore mutually noncongruent. The minimality proposition makes every one support-minimal.
+
+## Relative openness
+
+For unit vectors, the inner-product perturbation bound is exactly
+
+`|<v_i,v_j>-<u_i,u_j>| <= ||v_i-u_i|| + ||v_j-u_j||`.
+
+Distances strictly below `1/10` preserve every canonical sign because the smallest original absolute correlation is `1/5`. The conclusion is correctly restricted to positive weighted exact designs in that neighborhood. It does not promote approximate moment matching to an exact design. The explicit deformation demonstrates nontrivial, noncongruent nearby exact members.
+
+## Executable confirmation and integration
+
+Ran the final package's `verification/verify_seven_ray_family.py`: all formal polynomial identities, exact odd-harmonic cancellation, nine rational specializations, and exterior conference identities passed. The written proofs establish the general real-parameter claims; sample checks supplement rather than replace those proofs.
+
+At the read snapshot, the main manuscript was being integrated and had not yet included the family input. This audit certifies the supplied `seven_ray_family.tex` for inclusion. No changes to the proof-bearing source files were made by this audit.

research/audits/FINAL_FINITE_METADATA_AUDIT.md

@@ -0,0 +1,29 @@
+# Final finite metadata audit
+
+Reviewed `finite_metadata.tex` and `verification/verify_mask_metadata.py` in the final release. No theorem or verifier edits were made during this audit.
+
+**Verdict:** the mathematical arguments and exact verification logic pass. Before final freeze, explicitly declare the nonempty state set (`n >= 1`), nonnegative noise radius, known linear observation maps between finite-dimensional normed spaces, and the corollary's `rho > 0`. These are the intended model hypotheses; omitting positivity of rho makes the displayed separation implication invalid if read literally for negative rho.
+
+Verified points:
+
+- The confusability graph uses a common **compatible** mask and the weak distance threshold `<= 2 epsilon`. Closed noise balls meet at their midpoint at equality.
+- The encoder does not observe the actual mask/noise. Proper coloring is necessary and sufficient for exact state-index recovery. Merging identical vectors must also merge their compatible-mask sets when vector recovery is intended.
+- The fractional independent-set cover and its dual are a finite feasible bounded LP pair. Normalizing an optimal dual gives one common prior; the displayed variational reciprocal-mass formula follows in both directions.
+- The harmonic greedy proof is correct: each removed block contributes at most its disjoint harmonic charges. No efficient maximum-independent-set algorithm is claimed.
+- The R-tag converse concerns states decoded correctly for **every** compatible mask/noise, measured under the dual prior. It does not assert an average-success guarantee for an arbitrary random-noise law.
+- The strict separation threshold and the rank-positive Parseval boundary example are correct under rho > 0.
+- Classical graph coloring, fractional coloring, LP duality and rounding are appropriately credited, and the fractional invariant is correctly separated from Shannon/Korner graph entropy.
+
+Executed the verifier's mathematical body without its final JSON-writing block, preserving release artifacts. All exact rational checks passed:
+
+| Graph | Exact chi | Exact chi_f |
+|---|---:|---:|
+| E5 | 1 | 1 |
+| K4 | 4 | 4 |
+| C5 | 3 | 5/2 |
+| C7 | 3 | 7/3 |
+| W5 | 4 | 7/2 |
+
+The verifier checks every independent-set dual inequality, rational primal coverage, equality of primal/dual objectives, exhaustive chromatic numbers, harmonic rounding, and exhaustive R-colorable robust subset masses. The concrete compatible-coordinate-mask C5, exact closed-ball boundary, smaller-radius exclusion, and N=100 Parseval survivor example also pass. No floating-point optimization is used as proof.
+
+Scope: finite, one-shot, deterministic tags with known masks and adversarial closed-ball perturbations. No further theoretical extension is required for this section.

research/audits/FINAL_HIERARCHY_AUDIT.md

@@ -0,0 +1,19 @@
+# Final hierarchy scope audit
+
+**Status: PASS within the stated scope.** Read-only review of `hierarchical_recovery.tex` and `verification/verify_hierarchical_polar.py` found no unresolved mathematical gap. The theorem and verifier sources were not changed.
+
+Verified conclusions:
+
+- For finite nonempty mask families with fixed positive weights, the bounds `C<=S`, `C<=sqrt(W*pi_max)` and the fixed upper bound `S^2<=min[(sum w_L r_L)^2, W sum_{j=1}^N nu_(j)]` are valid. The ordered-coverage bound correctly uses the rank-N Gram projection's diagonal constraints; the frame-dependent intermediate trace is not mistaken for a fixed bound.
+- A target with `s<=(1-eta)w_J/(2B)` has a defined polar transport rotation, including the zero-singular-value case. Every accepted update raises `S^2` by at least `eta*w_J^2`; therefore the stated floor bound using `w_min^2` counts accepted updates correctly. A final mask inspection certifies termination.
+- Hierarchy nodes are optimized separately. Square survivor maps compose as `A=T_m...T_1`, and the inverse order `T_1^{-1}...T_m^{-1}` is correct. The lower singular-value product, contraction bound, exact endpoint noise norm `1/sigma_min(A)`, and internal-noise cancellation identity all hold. The product bound is correctly distinguished from equality.
+- Exact endpoint amplification is asserted for a fixed known realized path; uniform product bounds apply across the designated paths. Continuation maps have exactly N survivor coordinates and retain the common N-dimensional state space.
+
+Re-execution of verifier functions, without writing result artifacts, passed all assertions:
+
+- Mixed pair/triple family: `W=8`, `B=6`, `U^2=144`, start `S=9`, thresholds `1/24` and `1/48`, proved bound 504 updates; observed real/complex counts 4 and 3.
+- Mixed-rank family: observed counts 2 and 2, each below the proved bound 77.
+- Three-stage simplex cascade: all 27 paths passed. The proved amplification bound is `3*sqrt(3)=5.1961524227`; the largest measured exact amplification is `4.8922468542`. Correct-inverse-order residual is below `6.3e-16`, and endpoint amplification equality error is below `2.7e-15`.
+- Worst final Parseval residual in the weighted examples is below `5.8e-16`.
+
+The numerical checks illustrate the written proofs. They are not proof-assistant certification. The release accurately retains the limitations: conservative constants, finite-family search cost, separately optimized nodes, known decoder path/matrices, and no general guarantee of optimal conditioning or polynomial complexity.

research/audits/FINAL_INFORMATION_AUDIT.md

@@ -0,0 +1,43 @@
+# Final independent information and conditional-risk audit
+
+**Audit date:** 1 October 2026  
+**Auditor:** Independent hostile-audit agent `/root/hostile_audit`  
+**Release examined:** EIDOLITH VII v2.0.0, final source snapshot
+
+## Files examined
+
+- `information_law.tex`
+- `fine_precision_barrier.tex`
+- `conditional_risk.tex`
+
+**Verdict: PASS within the stated mathematical scope. No material proof error was found.**
+
+## Information law
+
+The exact vector-to-scalar operational equivalence is valid for a source supported on the known line, with deterministic source-independent observation maps and nonzero directional gain. Global injectivity of the observation map is correctly not required for that one-line source.
+
+The finite coding lower bound uses a common allowed observation within each exact-slope/cell fiber. The upper code's slope index, modulo-four cell residue, payload-length index, and conditional Shannon payload form a globally prefix-free complete message set. Its robustness includes negative endpoints and closed noise boundaries. Polynomially many slope products and the positive finite iid alphabet justify the `O(log n)` framing overhead.
+
+The strengthened convergence assumption concerns the positive-part effective precision itself. The adaptive-level comparison follows from positive-part Lipschitzness and bounded finite-alphabet logarithmic variance. It does not require the unclipped precision to converge. The stationary-tail comparison has discrepancy bounded by `O(sqrt(n)+1)+n(xi_n-chi)_+`, which is `o(n)` under the stated assumption even when the natural boundary is approached slowly from above. The proof does not incorrectly replace that boundary discrepancy by a uniform `O(sqrt n)` term.
+
+At effective precision zero, the reference dyadic level is zero and compact support supplies bounded reference-cell entropy; the adaptive discrepancy is `o(n)`. Thus actual observation entropy is sublinear, without claiming that all adaptive cell entropies remain bounded.
+
+The compensation corollary now correctly restricts the loss exponent to the nonnegative contraction regime. It restores the natural-scale metadata rate; that rate need not be zero. No candidate algebraic formula for stationary dimension is used.
+
+## Fine-precision barrier
+
+The four-map Liouville example has injective exact endpoint states, entropy rate `2 log 2`, contraction exponent `log 2`, and absolutely continuous stationary law of dimension one. The factorial-depth odd-coefficient modular argument and the slowly growing depth rational-approximation argument both check. Universal entropy bounds combined with these two subsequences establish the stated exact liminf and limsup metadata rates for every exponent greater than one. This supplies a valid sharp obstruction to a universal limiting rate beyond natural effective precision.
+
+## Conditional-risk bridge
+
+The observation map is now explicitly bounded below on the entire state space, ensuring full column rank and `A_J^dagger A_J=I`. The inverse estimate therefore has the stated conditional mean-square error bound without independence or centering assumptions on the noise. Conditional orthogonality transfers the bound to posterior conditional expectation, correctly distinguishing pre-evidence risk from pointwise posterior risk.
+
+The covariance decomposition, completed-square optimal correction, exact conditional control cost, and resulting risk-to-cost bound are valid for the fixed correction map and positive-definite quadratic penalty. The sharpness example now uses a centered symmetric scalar source, so its conditional risk genuinely attains the stated constant.
+
+## Scope of this audit
+
+This is an independent mathematical derivation audit of the three specified final source files. It is not external peer review, formal proof-assistant verification, a comprehensive novelty search, or verification of all source citations. The scalar law does not establish a general anisotropic vector-source rate. It does not cover countable alphabets, source-dependent observation operators, unbounded raw exponent sequences, efficient codec construction, worst-case metadata length, or universal rates beyond the stated effective-precision boundary.
+
+Two non-mathematical clarity suggestions were communicated to the release editor: remove the duplicated `Fix c>0` sentence, and typeset expected positive parts explicitly as `E[(expression)_+]` to avoid ambiguous attachment of the positive-part symbol. Neither changes the audited derivation.
+
+**Signed:** Independent hostile-audit agent `/root/hostile_audit`.

research/audits/FRAME_REFEREE.md

@@ -0,0 +1,97 @@
+# Independent hostile audit: quantitative transport and simultaneous constraints
+
+Auditor: independent agent, 2026-10-01. This is a mathematical derivation audit, not a machine-checked formal proof or certification of publication priority.
+
+## A. Collective Parseval transport: passed
+
+Let `Phi Phi*=I`, `1<=k<M`, `S=sum_{|L|=k}||Phi_L||_*`, and `s=sigma_min_relevant(Phi_J)`. If `0<=s<1`, a compatible unit singular pair `(u,v)` can be chosen with `v` supported in `J`; at `s=0`, choose `u` in the left kernel and `v` in the right kernel. Both kernels exist when rank is below `min(N,k)`. Set `b=sqrt(1-s^2)` and `w=Phi_{Jc}*u/b`. Then `w` is unit, is disjoint from `v`, and `Phi w=b u`.
+
+The right coefficient-plane rotation through `(v,w)` preserves Parsevalness exactly. The symmetric average of the two frames is `D_t Phi`, with `D_t=I-(1-cos t)uu*`. This identity holds because `Phi*u=sv+bw`; it is important to include both supported pieces. For `0<=t<=pi/2`, every block's symmetric nuclear average is at least `cos(t)` times its original nuclear norm. The target block's other singular values remain unchanged, and its distinguished singular value becomes `|s cos(t) +/- b sin(t)|`.
+
+Writing `z=||Phi_J||_*-s` and `T=S-||Phi_J||_*`, the exact selected contribution and generic other contributions give
+
+`(S(Phi(t))+S(Phi(-t)))/2 >= z+T cos(t)+b sin(t)`.
+
+Thus one sign at `t=atan2(b,T)` yields
+
+`S_new >= z+sqrt(T^2+1-s^2)`.
+
+There is no branch restriction: the target average contains `max(s cos t,b sin t)`, which is always at least `b sin t`. Discarding the target's unchanged-singular-value advantage gives the weaker valid bound `S_new>=sqrt(S^2-2Ss+1)`.
+
+The resulting global certificate is
+
+`2s(T+s)>=1`, equivalently `s>=1/(sqrt(T^2+2)+T)`.
+
+At `s=1`, the rotation need not be defined; this certificate is automatic because `T>=0`. With a certified global objective gap at most `epsilon`, the correct inequality is
+
+`2(s+epsilon)(T+s)+epsilon^2>=1`.
+
+If `2epsilon T+epsilon^2<1`, its positive root is
+
+`s >= (sqrt(T^2-2T epsilon-epsilon^2+2)-T-epsilon)/2`.
+
+The simpler clipped certificate `s >= [1-2S epsilon-epsilon^2]_+/(2S)` also checks. The objective gap must be certified; stationarity is insufficient.
+
+The uniform energy bound `U=sqrt(Q r N C(M-1,k-1))`, where `Q=C(M,k)` and `r=min(N,k)`, follows from the conserved sum of squared singular values and Cauchy--Schwarz. If `s<=1/(4U)`, a constructive rotation improves the objective by at least `sqrt(U^2+1/2)-U>0`. Repeating until no such block remains terminates in finitely many steps. Enumerating all subsets is exponential in general; no polynomial-time or useful RIP claim follows.
+
+## B. Exact rank opening under equal norms and prescribed positive frame operator: passed
+
+The proof needs full row rank and equal positive column norms. It preserves any prescribed positive frame operator `Phi Phi*=A>0`, so Parsevalness is a special case. It does not establish the same result for arbitrary unequal prescribed column norms.
+
+For a real deficient selected block, take a minimal circuit `C` and its null vector `v`, all of whose circuit entries are nonzero. The nonzero-Gram-edge graph on `C` is connected: a disconnected graph would split the circuit into orthogonal component spans, and projecting its dependence onto a component contradicts circuit minimality.
+
+Choose an outside column `b` with nonzero component outside the selected span. If some circuit entry has `g_a=<phi_a,phi_b>=0`, an orthogonal equal-norm Givens pair rotation opens the selected rank to first order. Otherwise all `g_i` are nonzero. Some nonzero Gram edge `(a,c)` satisfies `v_a/g_a != v_c/g_c`; if not, connectivity gives `v=alpha g`, contrary to `v in ker Phi_C` and `g=Phi_C^T phi_b in (ker Phi_C)^perp`.
+
+On the open set where the triangle's three Gram entries are nonzero, define the skew matrix by `K_ab=1/P_ab`, `K_bc=1/P_bc`, and `K_ca=1/P_ca`, with reverse entries their negatives. The smooth two-sided ODE `Phi'=Phi K(Phi)` preserves `Phi Phi^T` identically. Since `P'=PK-KP`, each triangle diagonal derivative cancels as `+1-1`, and all column norms remain exactly constant. No manifold regularity assumption on the constrained frame set is used.
+
+The transverse derivative on the selected block is
+
+`L Phi'_J R = L phi_b (-e_a^T/g_a+e_c^T/g_c) R`,
+
+and its action on the circuit vector is nonzero. Therefore a zero singular value opens at first order.
+
+The complex equal-norm construction is simpler: choose a phase `zeta` so `Re(zeta<phi_a,phi_b>)=0`, and rotate columns by `phi_a(t)=cos(t)phi_a+sin(t)zeta phi_b`, `phi_b(t)=-sin(t)conj(zeta)phi_a+cos(t)phi_b`. Exact unitary feasibility and exact norm conservation both check.
+
+## C. Local objective theorem: passed
+
+For any positive subset weights, every local maximizer of a `C^2` regularizer plus a positive multiple of the selected-block spectral power sum with `0<p<2` has maximal rank in every selected block. Along the exact feasible curve, positive singular clusters and the regularizer have symmetric change `O(t^2)`. The opened zero cluster contributes at least `c|t|^p`, while other zero clusters contribute nonnegatively. The positive cusp dominates the quadratic loss.
+
+The powered-volume result for `0<q<2` and `k<=N` also checks. A deficient `k`-set contains a minimal circuit; full row rank permits extending that circuit to a `k`-set of rank exactly `k-1`. The opened transverse derivative then makes the wedge derivative nonzero, creating a `c|t|^q` contribution. Weights must be positive on all subsets needed in this extension.
+
+The endpoint `p=2` fails on deficient equal-norm Parseval frames by the conserved quadratic sum. Do not reuse the old `[I 0]` example for the equal-norm domain. For unrestricted `C^2` regularizers and `p>2`, counterexamples can be created by cancelling the `C^2` spectral-power term and adding a strict smooth maximum at a deficient feasible frame. For an unregularized equal-norm spectral objective with `p>2`, a stronger universal counterexample needs a separate argument.
+
+Repeated coordinate equal-norm Parseval frames provide valid unregularized powered-volume global maxima for `q>2`: Hadamard bounds every volume by `c^k`, while Cauchy--Binet fixes their squared sum, and the repeated-coordinate frames attain the bound with volumes in `{0,c^k}`.
+
+## Limits of the passed audit
+
+These derivations establish rank exclusion, constructive finite conditioning, and stated conservative bounds. They do not establish uniqueness, optimization algorithm convergence under arbitrary numerical parameterizations, useful compressed-sensing RIP, resolution of the Paulsen problem, or priority over all literature. Architecture parameterizations must allow the exact feasible directions used here.
+
+## D. Strengthened polar-moment transport: passed
+
+This supersedes A's weaker quantitative constants. Set `H=sum_L sqrt(F_L)`, where `F_L=Phi_L Phi_L*`, and `C=u*H u` for the target's compatible weakest left singular vector. Nuclear duality with a target block's polar partial isometry gives the exact supporting inequality
+
+`||D_t X_L||_* >= ||X_L||_*-(1-cos t)u*sqrt(F_L)u`.
+
+The identity `X_L Q_L*=sqrt(F_L)` remains valid at deficient blocks when `Q_L` is their polar partial isometry. Since the target's contribution to `C` is exactly `s`, replacing its supporting estimate by the exact target symmetric average proves
+
+`avg S >= S-C+(C-s)cos t+sqrt(1-s^2)sin t`.
+
+Here `C-s>=0`. At `t=atan2(sqrt(1-s^2),C-s)`, some rotated candidate has
+
+`S_new >= S-C+sqrt(C^2-2Cs+1)`.
+
+The resulting global certificate `s>=1/(2C)` and the certified-gap certificate `s>=[1-2C epsilon-epsilon^2]_+/(2C)` follow by squaring with nonnegative right sides. For the all-k family, `C>0`: because `0<=F_L<=I`, `sqrt(F_L)>=F_L`, and summing gives `H>=C(M-1,k-1)I`. Consequently division is legitimate. At `s=1`, bypass the undefined complement vector; the certificate is automatic and the lower candidate bound is already below the unchanged score.
+
+Scalar spectral Jensen followed by Cauchy--Schwarz gives
+
+`C<=sum_L sqrt(u*F_Lu)<=sqrt(Q sum_L u*F_Lu)=Q sqrt(k/M)=B`.
+
+Also `C<=S`, since each directional square-root quadratic form is no larger than its trace. Thus every global optimizer has `s>=1/(2B)`.
+
+For `eta in (0,1)` and a weak block with `s<=(1-eta)/(2B)`, the candidate satisfies
+
+`S_new>=S-C+sqrt(C^2+eta)`.
+
+If `d=sqrt(C^2+eta)-C`, then `eta=2Cd+d^2`, and `S>=C` gives `(S+d)^2>=S^2+eta`. The squared-score upper bound therefore proves finite termination within the stated conservative `floor((U^2-S_initial^2)/eta)+1` iteration bound. At termination every block has `s>(1-eta)/(2B)`.
+
+The moment constant and stated iteration bound concern unweighted all-k subsets. Weighted positive families have the separately stated weighted constants and increments. The quantitative rotations preserve Parsevalness, not individual column norms. These are exact-arithmetic guarantees; a floating-point implementation needs numerical tolerance management and does not obtain certified global gaps from stationarity alone.

research/audits/PRIMARY_SOURCE_AUDIT.md

@@ -0,0 +1,40 @@
+# Primary-source audit: source information, directional spectrum, and compatibility covers
+
+Audit date: 1 October 2026. This audit checks attribution and the established inputs of `information_repair_independent_proof.md`; it is not an exhaustive publication-priority search.
+
+## Exact dimensionality and entropy dimension
+
+1. De-Jun Feng and Huyi Hu, **Dimension theory of iterated function systems**, *Communications on Pure and Applied Mathematics* **62**(11) (2009), 1435–1500. DOI **10.1002/cpa.20276**. The DOI cpa.20257 is incorrect. Publisher metadata says first online 3 March 2009. Authors' arXiv version: https://arxiv.org/abs/1002.2036 (posted 10 February 2010), full text https://arxiv.org/pdf/1002.2036. Theorem 2.8, manuscript p.7, proves exact dimensionality of projected ergodic measures for m-conformal IFS, without separation assumptions; scalar affine contracting systems are within this scope. The dimension is projection entropy divided by Lyapunov exponent. This is not a claim that the Shannon entropy of unmerged iid histories divided by Lyapunov exponent always gives dimension under overlaps. Authors' bibliography independently confirms pages: https://users.math.msu.edu/users/hhu/papers.html.
+
+2. Michael Hochman, **On self-similar sets with overlaps and inverse theorems for entropy**, *Annals of Mathematics* **180**(2) (2014), 773–822. DOI **10.4007/annals.2014.180.2.7**. Official publication page: https://annals.math.princeton.edu/2014/180-2/p07. Complete publisher PDF: https://annals.math.princeton.edu/wp-content/uploads/annals-v180-n2-p07-p.pdf. Section 5.1, Proposition 5.2 and equation (33), journal p.807, states that self-similar measures have uniform entropy dimension and explicitly identifies their Hausdorff dimension with the limiting normalized dyadic Shannon entropy. This supplies the stationary entropy-dimension input used by the new proof. This input does not determine the numerical value of D for arbitrary overlaps or solve the general exact-overlaps conjecture.
+
+The independent directional information-repair proof correctly uses only these established inputs; it does not depend on the unverified algebraic dimension closure in AURELINE.
+
+## Zero-error side information and graph coloring
+
+3. Hans S. Witsenhausen, **The zero-error side information problem and chromatic numbers (Corresp.)**, *IEEE Transactions on Information Theory* **22**(5) (September 1976), 592–593. DOI **10.1109/TIT.1976.1055607**. Original publisher metadata: https://ieeexplore.ieee.org/document/1055607/. Publisher full text was blocked by its browser verification wall in this audit. Attribution of the characteristic confusability graph was verified in the original Alon–Orlitsky paper below; do not claim direct full-text inspection of Witsenhausen here.
+
+4. János Körner, **Coding of an information source having ambiguous alphabet and the entropy of graphs**, in *Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes* (conference 1971; Academia, Prague, publication 1973), 411–425. Author institution record: https://iris.uniroma1.it/handle/11573/189670; this record labels the conference year 1971, while the original Alon–Orlitsky bibliography and introduction use publication year 1973. Full original text was not accessible in this audit. Graph-entropy attribution was verified from the primary Alon–Orlitsky text; do not claim direct inspection of the 1973 paper.
+
+5. Noga Alon and Alon Orlitsky, **Source coding and graph entropies**, *IEEE Transactions on Information Theory* **42**(5) (September 1996), 1329–1339. DOI **10.1109/18.532875**. Full author manuscript: https://www.cs.tau.ac.il/~nogaa/PDFS/entropies3.pdf; author bibliography identifies it: https://www.cs.tau.ac.il/~nogaa/PDFS/publications.html. The supplied PDF's embedded text encoding is broken; pages 1–5 were rendered and OCR-inspected. The original manuscript distinguishes globally prefix-free (unrestricted-input) coding from edgewise prefix requirements (restricted-input coding). For the global prefix-free model it bounds expected length by chromatic entropy H_chi and H_chi+1. Körner graph entropy is a distinct convexified/asymptotic object; it should not be conflated with the single-batch chromatic entropy or with a fractional min-entropy cover LP.
+
+## Fractional covering and greedy rounding
+
+The finite compatibility/cylinder LP, its dual, and its normalized-law minimum-mass formulation are consequences of ordinary finite LP duality. If the cover sets are independent sets of a confusability graph, this specializes to fractional coloring; arbitrary cylinder families are a general finite set-cover relaxation. The logarithmic cover-rounding bound is established cover theory, not a new compression theorem:
+
+- László Lovász, **On the ratio of optimal integral and fractional covers**, *Discrete Mathematics* **13**(4) (1975), 383–390. DOI **10.1016/0012-365X(75)90058-8**. Original publisher metadata: https://www.sciencedirect.com/science/article/pii/0012365X75900588.
+- Vašek Chvátal, **A greedy heuristic for the set-covering problem**, *Mathematics of Operations Research* **4**(3) (August 1979), 233–235. DOI **10.1287/moor.4.3.233**. Original publisher page: https://pubsonline.informs.org/doi/abs/10.1287/moor.4.3.233. Author bibliography: https://users.encs.concordia.ca/~chvatal/publ.html. Original-paper university-hosted PDF: https://pages.cs.wisc.edu/~cs787-1/Chvatal.pdf.
+
+No theorem in the inspected sources identifies an arbitrary finite compatibility-cover min-entropy LP with the exact Shannon expected-metadata rate. Such an identification requires its own hypothesis and proof.
+
+## Assessment of the operator-rate formula
+
+The complete independent proof establishes the expected robust exact-state rate
+
+    (h - D*(theta*chi-kappa)_+)/ln(2)
+
+for a scalar affine iid source embedded in one known direction u, with deterministic known source-independent observation maps T_n, contraction gain g_n=||T_n u||, and directional exponent kappa=lim -ln(g_n)/n. It proves exact equality between the vector robust observation problem and a scalar robust channel at radius c|A_n|^theta/g_n, handles adaptive/random branch scales and the critical zero-effective-precision case, and then applies the established self-similar entropy-dimension law and finite robust coding lemma.
+
+The meaningful extension is this exact geometric/operator reduction plus its audited scope and finite certification framework. It is a synthesis/application of established coding and dimension theory; it is not evidence of new Shannon theory, a new general graph-entropy formula, or a solution of a recognized open source-coding problem. Publication novelty of this exact synthesis remains provisional.
+
+Using the inverse-operator-norm exponent in place of directional kappa gives only a conservative sufficient budget in general. The manuscript correctly restricts equality to the actual source direction: attenuation of an unused orthogonal direction cannot affect the scalar source's observation channel. Source-dependent maps, arbitrary multidimensional self-affine latent distributions, worst-case lengths, countable alphabets, and efficient codecs are outside the theorem.

research/build_pdf.sh

@@ -0,0 +1,6 @@
+#!/usr/bin/env bash
+set -euo pipefail
+cd "$(dirname "$0")"
+for pass in 1 2 3; do
+  pdflatex -interaction=nonstopmode -halt-on-error -jobname=EIDOLITH_VII_v2.0.0_Final EIDOLITH_VII.tex
+ done

research/conditional_risk.tex

@@ -0,0 +1,56 @@
+\section{A distribution-free conditional recovery bridge}
+The following transfers a uniform frame inverse bound into the actual conditional-expectation formulation of VESPERS. The estimation and quadratic-completion identities are classical; this is their application to the certified operator setting, without a Gaussian posterior assumption.
+
+Let $Z$ be square-integrable, $\mathcal G$ the current memory, and
+\[
+y=A_JZ+\eta,\qquad
+\mathcal H=\mathcal G\vee\sigma(J,y),\qquad
+\|A_Jz\|\ge\rho\|z\|\quad\text{for every }z,\qquad\rho>0.
+\]
+Thus every observation map is injective, with a uniform full-column-rank lower bound. Assume $A_J$ is known from $\mathcal H$ and
+$\mathbb E[\|\eta\|^2\mid\mathcal G]\le\varepsilon^2$ almost surely. The $\mathcal H$-measurable estimate $\widetilde Z=A_J^\dagger y$ obeys
+\[
+\mathbb E[\|Z-\widetilde Z\|^2\mid\mathcal G]
+\le\varepsilon^2/\rho^2.
+\]
+Since $m_{\mathcal H}=\mathbb E[Z\mid\mathcal H]$ is the conditional least-squares estimate and $\mathcal G\subseteq\mathcal H$, conditional orthogonality gives
+\[
+\boxed{\mathbb E[\|Z-m_{\mathcal H}\|^2\mid\mathcal G]
+\le\varepsilon^2/\rho^2.}
+\]
+This is pre-evidence conditional Bayes risk, averaged over the new observation given current memory. A pointwise posterior bound conditioned on $\mathcal H$ needs the stronger assumption $\mathbb E[\|\eta\|^2\mid\mathcal H]\le\varepsilon^2$, as supplied, for example, by $\|\eta\|\le\varepsilon$ almost surely. A current-memory or unconditional noise moment does not establish that stronger statement. Independence and zero-mean noise are not required.
+
+For a linear task $BZ$, multiply the risk bound by $\|B\|_{\mathrm{op}}^2$. An everywhere-defined $L$-Lipschitz task similarly has risk at most $L^2\varepsilon^2/\rho^2$, by comparing with the task applied to $\widetilde Z$.
+
+To connect the result to VESPERS, let $T$ be the fixed available correction map, $C\succ0$, and $\lambda>0$. Put
+\[
+R=T(T^*T+\lambda C)^{-1}T^*,\qquad Q=I-R,
+\quad m_{\mathcal G}=\mathbb E[Z\mid\mathcal G],
+\]
+and define
+\begin{align*}
+S_{\mathcal G}&=\mathbb E[(Z-m_{\mathcal G})(Z-m_{\mathcal G})^*\mid\mathcal G],\\
+S_{\mathcal G}^+&=\mathbb E[(Z-m_{\mathcal H})(Z-m_{\mathcal H})^*\mid\mathcal G],\\
+\Omega_{\mathcal G}&=\mathbb E[(m_{\mathcal H}-m_{\mathcal G})(m_{\mathcal H}-m_{\mathcal G})^*\mid\mathcal G].
+\end{align*}
+Conditional orthogonality gives $S_{\mathcal G}=S_{\mathcal G}^++\Omega_{\mathcal G}$. The exact optimal $\mathcal H$-measurable correction and conditional cost are
+\[
+u^*=(T^*T+\lambda C)^{-1}T^*(m_{\mathcal H}-m_{\mathcal G}),
+\]
+\[
+J_{\mathcal G}
+=\frac12\tr[S_{\mathcal G}^++Q\Omega_{\mathcal G}]
+=\frac12\tr[Q S_{\mathcal G}]+\frac12\tr[R S_{\mathcal G}^+],
+\]
+where
+\[
+J_{\mathcal G}=\operatorname*{ess\,inf}_{u\;\mathcal H\text{-measurable}}
+\frac12\mathbb E[\|Z-m_{\mathcal G}-Tu\|^2
++\lambda u^*Cu\mid\mathcal G].
+\]
+The frame risk certificate therefore implies
+\[
+\boxed{J_{\mathcal G}\le\frac12\tr[Q S_{\mathcal G}]
++\frac{\|R\|_{\mathrm{op}}\varepsilon^2}{2\rho^2}.}
+\]
+This exact conditional control identity follows by first conditioning on $\mathcal H$, separating the orthogonal residual $Z-m_{\mathcal H}$, and completing the remaining quadratic in $u$. It requires no density or Gaussian law. The risk constant is sharp under the stipulated moment-only information: in a scalar example $A=\rho$, $\eta=-\rho Z$, and centered $Z=\pm\varepsilon/\rho$ with equal probabilities, the observation is identically zero and the bound is attained.

research/constrained_completion.tex

@@ -0,0 +1,71 @@
+For a positive-definite $T\in\F^{N\times N}$, let
+\[
+\mathcal T_T=\{\Phi:\Phi\Phi^*=T,\quad
+\norm{\phi_i}^2=\tr(T)/M\text{ for every }i\}.
+\]
+The equal-norm Parseval space is $\mathcal T_{I_N}$. We work in its relative Euclidean topology, including singular points of the frame variety. Generic full spark \cite{FullSpark} does not by itself imply full spark at every local optimum. The following exact opening lemma provides that missing local mechanism under the two simultaneous constraints.
+
+\begin{lemma}[Exact constrained rank opening]\label{enpopening}
+Let $\Phi\in\mathcal T_T$ and $X=\Phi_J$ have rank below $\min(N,|J|)$. Let $L$ and $R$ be the orthogonal projections onto $\ker X^*$ and $\ker X$. There is a two-sided smooth feasible curve through $\Phi$ such that, for $Y=\Phi'_J(0)$, $LYR\ne0$.
+\end{lemma}
+\begin{proof}
+Over $\C$, choose $a\in J$ with $Re_a\ne0$ and $b\notin J$ with $L\phi_b\ne0$. The latter exists because $T>0$ implies that all columns span the full state space. Set $h=\phi_a^*\phi_b$ and choose a unit phase $\zeta$ with $\operatorname{Re}(\zeta h)=0$. Change only the two columns:
+\[
+\phi_a(t)=\cos t\,\phi_a+\zeta\sin t\,\phi_b,
+\quad \phi_b(t)=-\overline\zeta\sin t\,\phi_a+\cos t\,\phi_b.
+\]
+This is a unitary coefficient change. The cross terms in the squared norms vanish, and the original column norms agree, so both constraints are preserved exactly. Its derivative satisfies $LYR=\zeta L\phi_b e_a^*R\ne0$.
+
+Over $\R$, choose a minimal dependent set $C\subseteq J$ and a nonzero vector $v\in\ker\Phi_C$, extended by zero to $J$. Every coordinate of $v$ on $C$ is nonzero, and $|C|\ge2$ because every column norm is positive. Choose $b\notin J$ with $L\phi_b\ne0$. Write $P=\Phi^T\Phi$ and $g_i=P_{ib}$ for $i\in C$. If $g_a=0$ for some $a\in C$, the ordinary Givens rotation of the orthogonal equal-norm pair $(a,b)$ preserves both constraints and has $LYv=v_aL\phi_b\ne0$.
+
+Otherwise all $g_i$ are nonzero. The graph on $C$ whose edges have $P_{ij}\ne0$ is connected. Indeed its disconnected components would have orthogonal spans, and projecting the circuit relation onto one component would give a proper dependent subset. There is an edge $\{a,c\}$ with $v_a/g_a\ne v_c/g_c$: if not, connectivity gives $v=\alpha g$, impossible since $v\in\ker\Phi_C$ whereas $g=\Phi_C^T\phi_b\in(\ker\Phi_C)^\perp$.
+
+Near the starting matrix the three entries $P_{ab},P_{bc},P_{ca}$ remain nonzero. Define a smooth skew matrix $K(\Phi)$ on this triangle by
+\[
+K_{ab}=1/P_{ab},\quad K_{bc}=1/P_{bc},\quad K_{ca}=1/P_{ca},
+\]
+with reversed entries negative and all others zero. Integrate the smooth local ODE $\Phi'=\Phi K(\Phi)$. Along its two-sided local solution,
+\[
+(\Phi\Phi^T)'=0,\qquad P'=PK-KP,\qquad \operatorname{diag}P'=0.
+\]
+At each triangle vertex the two incident products in the diagonal derivative are $1$ and $-1$, and cancel identically. Thus both constraints hold exactly, without assuming that the frame variety is a manifold at the starting point. For the selected derivative,
+\[
+LYv=L\phi_b\left(-\frac{v_a}{g_a}+\frac{v_c}{g_c}\right)\ne0.
+\]
+Since $Rv=v$, this proves the real case.
+\end{proof}
+
+For $k\le N$, write $v_J=\sqrt{\det(\Phi_J^*\Phi_J)}$. For arbitrary $k$, write $r=\min(N,k)$ and use the $r$ relevant singular values, including zero ones.
+\begin{theorem}[Completion under any smooth task objective]\label{smoothcompletion}
+Let $G$ be $C^2$ on a neighborhood of $\mathcal T_T$, $\lambda>0$, and $w_J>0$ for all $k$-subsets. For $0<p<2$, every local maximizer on $\mathcal T_T$ of
+\[
+G(\Phi)+\lambda\sum_{|J|=k}w_J\sum_{a=1}^{r}\sigma_a(\Phi_J)^p
+\]
+has every $k$-block of rank $r$. For $1\le k\le N$ and $0<q<2$, every local maximizer of
+\[
+G(\Phi)+\lambda\sum_{|J|=k}w_Jv_J(\Phi)^q
+\]
+has every $k$-block independent. In particular at $k=N$ such local maxima are full spark.
+\end{theorem}
+\begin{proof}
+Suppose first that a spectral block is deficient, and use Lemma~\ref{enpopening}. In SVD coordinates its positive singular-value block is invertible. Eliminating the off-diagonal blocks with matrices $I+O(t)$ leaves a zero-cluster block $tLYR+O(t^2)$. Since $LYR\ne0$, at least one initially zero singular value is bounded below by $c|t|$ near zero. The sum of the initially positive singular values raised to $p$ is $C^2$ along the curve: the positive spectral cluster stays separated from zero, where the scalar spectral function is smooth. Its symmetric change is $O(t^2)$. The same applies to the positive cluster in every other block; every initially zero cluster contributes a nonnegative amount. Finally $G(\Phi(t))+G(\Phi(-t))-2G(\Phi)=O(t^2)$. Summing the positive weights gives symmetric objective gain at least $c'|t|^p-Ct^2>0$ for small nonzero $t$, contradicting local maximality.
+
+For volume, positive-definiteness of $T$ gives full total rank. From any dependent set of at most $k$ columns choose a minimal circuit, and extend it by columns independent modulo its span to a $k$-block of rank $k-1$. Apply Lemma~\ref{enpopening} to this block. In SVD coordinates its $k$-fold exterior product has a nonzero derivative: it is the product of the $k-1$ positive singular values times the nonzero transverse image of the single zero right-singular direction. Its $q$-volume therefore contributes at least $c|t|^q$. Every initially nonzero volume has a smooth powered value and quadratic symmetric change. Other initially zero volumes contribute nonnegatively, as does their increment from zero. Combining with the $C^2$ task term again gives $c'|t|^q-Ct^2>0$, the required contradiction.
+\end{proof}
+
+Equivalently, any local minimizer of a $C^2$ task loss minus an arbitrarily small positive multiple of the stated bonus has the rank property. The exact cusp, the feasible frame domain, and the positivity of all subset weights are essential to the claim. Smooth approximations of the cusp and additional architectural restrictions require separate arguments. The volume statement here uses a positive-definite frame operator; with arbitrary smooth task terms it must not be transferred to unrestricted fixed-norm families of potentially insufficient total rank.
+
+\begin{proposition}[Sharp quadratic boundary]\label{boundary}
+At exponent two, both unweighted objectives are constant on equal-norm Parseval frames. For $M=dN$, integer $d\ge2$, and $2\le k\le N$, the repeated-coordinate frame
+\[
+\Phi=d^{-1/2}[I_N\ \cdots\ I_N]
+\]
+is a singular-block global maximizer of the unweighted volume objective for every $q>2$.
+\end{proposition}
+\begin{proof}
+Squared singular energy sums to $N\binom{M-1}{k-1}$. Cauchy--Binet gives $\sum_J v_J^2=\binom Nk$. If $c=d^{-1/2}$, Hadamard gives $v_J\le c^k$, hence
+\[
+\sum_Jv_J^q\le c^{k(q-2)}\binom Nk.
+\]
+All volumes in the repeated-coordinate frame are $0$ or $c^k$, so equality holds, despite repeated columns and dependent $k$-blocks. At $p=2$ the constant spectral objective already admits deficient global maxima. No additional spectral $p>2$ assertion on the simultaneous-constraint domain is needed for sharpness of the strict range $p<2$.
+\end{proof}

research/fine_precision_barrier.tex

@@ -0,0 +1,39 @@
+\section{Sharp obstruction beyond effective natural precision}\label{finebarrier}
+The following elementary example, retained and reproved from AURELINE, shows why the information law stops at effective precision $\chi$.
+Let $\tau=\sum_{j\ge1}2^{-j!}$ and use the four equiprobable maps
+\[
+\varphi_{a,b}(x)=(x+a+\tau b)/2,\qquad a,b\in\{0,1\}.
+\]
+The number $\tau$ is irrational: its truncation at denominator $2^{j!}$ has a nonzero tail below $2\cdot2^{-(j+1)!}$, contradicting the rational separation $1/(v2^{j!})$ for any fixed putative denominator $v$. Thus the exact depth-$n$ states are
+\[
+A_n=2^{-n},\qquad B_n=(U_n+\tau V_n)/2^n,
+\]
+with independent uniform integers $U_n,V_n\in[0,2^n-1]$, all $4^n$ states distinct. Consequently $h=2\log2$, $\chi=\log2$, and $D=1$, since the stationary law is the convolution of two scaled uniform interval laws.
+
+\begin{proposition}[Failure of a universal finer-precision rate]
+For every fixed $\zeta>1$, allow scalar noise radius $2^{-\lfloor\zeta n\rfloor}$. Then
+\[
+\liminf_n L_n^*/n=\max(0,2-\zeta),\qquad
+\limsup_n L_n^*/n=1.
+\]
+Thus effective precision above $\chi$ can fail to have any limiting metadata rate.
+\end{proposition}
+\begin{proof}
+Put $k_n=\lfloor\zeta n\rfloor$ and $Q_n=\lfloor2^{k_n}B_n\rfloor$. Conditional on $V_n$, the $2^n$ endpoints obtained by varying $U_n$ are separated by $2^{-n}$, so $H(Q_n)\ge n\log2$ for large $n$. Compact support and the exact-state alphabet also give
+\[
+H(Q_n)\le\min(k_n,2n)\log2+O(1).
+\]
+
+At $n=j!$, write $\tau=a/q+2^{-n}+e_j$, $q=2^{(j-1)!}$, and $0<e_j<2\cdot2^{-(j+1)!}$. Then
+\[
+2^{2n}B_n=2^nU_n+(2^na/q+1)V_n+E_n,\qquad0\le E_n<1.
+\]
+The integer coefficient of $V_n$ is odd. Reducing the integer part modulo $2^n$ determines $V_n$ and then $U_n$, so that integer part is injective on the square. If $k_n\ge2n$, $Q_n$ determines it exactly. If $k_n\le2n$, each fine cell groups at most $2^{2n-k_n}$ integer parts; the error $E_n<1$ cannot cross an integer-spacing dyadic boundary. Hence along factorial depths
+\[
+H(Q_n)/n\longrightarrow\min(\zeta,2)\log2.
+\]
+
+At $n_j=j!\lfloor\sqrt{j+1}\rfloor$, replace $\tau$ by its truncation with denominator $q_j=2^{j!}$. The endpoint change is smaller than one fine cell for all sufficiently large $j$, whereas the rational endpoints have at most $Cq_j2^{n_j}$ values. Each produces at most three actual cells. Therefore $H(Q_{n_j})\le n_j\log2+j!\log2+O(1)$, and the universal lower bound gives $H(Q_{n_j})/n_j\to\log2$.
+
+The finite coding lemma applies at every positive width, without a natural-precision restriction, and yields $L_n^*=2n-H(Q_n)/\log2+O(\log n)$. The universal entropy bounds and these two subsequences prove the exact lower and upper rates.
+\end{proof}

research/finite_metadata.tex

@@ -0,0 +1,48 @@
+This section uses the finite-cover and compatible-dual-prior engine of the supplied C\AE LUM/$\Delta$ manuscript. We reprove it for the present observation model. Zero-error side-information coloring, fractional coloring, LP duality, and harmonic rounding are classical \cite{Wits,AO,Chvatal}; no new general coding theory is claimed.
+
+Let $\mathcal S$ be a nonempty finite set of $n\ge1$ exact state indices with known vectors $x_s$. Let $\mathcal M$ be a finite mask menu and $\Lambda\subseteq\mathcal S\times\mathcal M$ a compatibility relation giving every state at least one allowed mask. Let each $T_J$ be a known linear map between real or complex Euclidean spaces, and fix $\varepsilon\ge0$. Under $(s,J)\in\Lambda$, the decoder observes
+\[
+y=T_Jx_s+e,\qquad\norm e\le\varepsilon.
+\]
+It knows $J$ and a noiseless tag. The tag encoder knows $s$ and the shared model, but cannot observe the subsequently selected mask or noise. Define a graph $G_\varepsilon$ by an edge between distinct $s,t$ exactly when some common allowed mask satisfies
+\[
+\norm{T_J(x_s-x_t)}\le2\varepsilon.
+\]
+Closed balls intersect at equality, so the weak inequality is essential. If physical-vector recovery rather than state-index recovery is intended, first merge identical vectors and union their compatible-mask sets.
+
+\begin{theorem}[Exact finite metadata and a certified minimax prior]\label{finitecode}
+The minimum number of deterministic tags correct for every allowed mask and every admissible perturbation is $\chi(G_\varepsilon)$. For all nonempty independent sets $I$ of this graph define
+\[
+\tau=\min_{z_I\ge0}\left\{\sum_Iz_I:\ \sum_{I\ni s}z_I\ge1\ (s\in\mathcal S)\right\}.
+\]
+Then $\tau$ is its fractional chromatic number, with dual
+\[
+\tau=\max_{v_s\ge0}\left\{\sum_sv_s:\ \sum_{s\in I}v_s\le1\text{ for every independent }I\right\}.
+\]
+An optimal dual gives one common prior $q_s=v_s/\tau$, and
+\[
+q(I)\le1/\tau,\qquad
+\tau=\max_q\frac1{\max_Iq(I)},\qquad
+\tau\le\chi(G_\varepsilon)\le H_n\tau\le(1+\log n)\tau.
+\]
+For any $R$-tag encoder and any decoder, the prior mass of states correct for \emph{every} compatible mask and noise is at most $\min(1,R/\tau)$.
+\end{theorem}
+\begin{proof}
+Adjacent states sharing a tag have a common mask and a common observation, their two noise balls meeting at their midpoint; the decoder cannot distinguish them. Conversely, a proper coloring makes the feasible state set for each tag, mask, and observation have at most one member. A genuine observation contains the true member. This proves the chromatic characterization. Independent-set covers and partitions have the same minimum: assign each covered vertex to one of its covering sets.
+
+The displayed finite programs are feasible bounded primal/dual pairs, so finite LP duality gives equality. Normalizing an optimal dual gives the mass bound. For any other prior $q$, setting $\beta=\max_Iq(I)>0$ makes $q/\beta$ dual feasible, proving the variational formula.
+
+For rounding, let $k$ vertices remain uncovered. An optimal fractional cover has $\sum_I z_I|I\cap U|\ge k$ and $\sum_Iz_I=\tau$, so some independent set contains at least $k/\tau$ remaining vertices. Remove a largest independent set, of size $d\ge k/\tau$. Charge its unit cost to the harmonic block $k-d+1,\ldots,k$: $1\le\tau d/k\le\tau\sum_{j=k-d+1}^k1/j$. Summing disjoint blocks gives at most $\tau H_n$ colors. Computing a largest independent set is not asserted efficient.
+
+Within each tag, all robustly correct states form an independent set by the same common-observation contradiction. Each has $q$-mass at most $1/\tau$, and there are at most $R$ tags, proving the converse.
+\end{proof}
+
+\begin{corollary}[Spectrum-to-metadata threshold]\label{metadatathreshold}
+If $\rho>0$, $\norm{T_J(x_s-x_t)}\ge\rho\norm{x_s-x_t}$ for every common compatible mask and $\delta=\min_{s\ne t}\norm{x_s-x_t}>2\varepsilon/\rho$, one tag suffices. The conclusion is automatic for a single state.
+\end{corollary}
+\begin{proof}
+Every observed pair distance exceeds $2\varepsilon$, so the graph is empty. Theorem~\ref{finitecode} applies.
+\end{proof}
+The bound may come from the polar transport theorem or a hierarchy product. Rank positivity alone supplies no fixed noise threshold: a scalar Parseval encoder with $100$ coefficients each $1/10$ has a one-coordinate survivor of positive singular value $1/10$. States $0,1$ and noise radius $1/20$ nevertheless require two tags. The scalar identity example at radius $1/2$ likewise needs two tags at the closed-ball boundary.
+
+Primal and dual rational witnesses give exact bounds $\sum_sv_s\le\tau\le\sum_Iz_I$; equality certifies the optimum. For $C_5$, the exact values are $\chi=3$ and $\tau=5/2$, so integral and fractional tags differ. The fractional dual is a minimax finite-cover quantity, not Shannon or K\"orner graph entropy. This section concerns fixed tag cardinality and robust-success mass; the next section concerns source-weighted expected prefix length.

research/hierarchical_recovery.tex

@@ -0,0 +1,124 @@
+\section{Weighted finite construction and hierarchical recovery}
+
+The same construction permits a finite family of designated erasure masks, with different mask sizes and positive weights. This permits separate construction of encoders at nodes of a finite hierarchy.
+
+Let $\mathcal F$ be a finite family of nonempty subsets of $[M]$, with fixed weights $w_L>0$, and put
+\[
+ r_L=\min(N,|L|),\qquad
+ S=\sum_{L\in\mathcal F}w_L\|\Phi_L\|_*,\qquad
+ H=\sum_{L\in\mathcal F}w_L(\Phi_L\Phi_L^*)^{1/2}.
+\]
+Define the weighted coverages and the fixed constants
+\begin{align*}
+ W&=\sum_Lw_L,&
+ \pi_i&=\sum_{L\ni i}w_L,&
+ \nu_i&=\sum_{L\ni i}w_Lr_L,\\
+ B&=\sqrt{W\max_i\pi_i},&
+ w_{\min}&=\min_Lw_L,&
+ U^2&=\min\left\{\left(\sum_Lw_Lr_L\right)^2,
+                   W\sum_{j=1}^N\nu_{(j)}\right\},
+\end{align*}
+where $\nu_{(1)}\geq\cdots\geq\nu_{(M)}$ is the decreasing ordering. These constants depend only on the mask family, its weights, and the dimensions.
+
+\begin{lemma}[Fixed bounds for mixed masks]\label{lem:mixed-fixed-bounds}
+For every feasible Parseval matrix and every unit $u$, one has
+\[
+ C=u^*Hu\leq S,\qquad C\leq B,\qquad S\leq U.
+\]
+In particular the simpler bound $U^2\leq WN r_{\max}\pi_{\max}$ is valid if its right side is used instead of the sharper definition above.
+\end{lemma}
+\begin{proof}
+Positivity gives $C\leq\operatorname{tr}H=S$. Weighted Cauchy--Schwarz gives
+\[
+ C^2\leq W\sum_Lw_Lu^*\Phi_L\Phi_L^*u
+ =W\sum_i\pi_i|\phi_i^*u|^2\leq W\pi_{\max}.
+\]
+Furthermore,
+\[
+ S^2\leq W\sum_Lw_L\|\Phi_L\|_*^2
+ \leq W\sum_Lw_Lr_L\operatorname{tr}(\Phi_L\Phi_L^*)
+ =W\sum_i\nu_i\|\phi_i\|^2.
+\]
+The rank-$N$ Gram projection has diagonal entries $0\leq\|\phi_i\|^2\leq1$ summing to $N$. Consequently the final sum is at most $\sum_{j=1}^N\nu_{(j)}$, by filling the $N$ largest coefficients under these linear constraints. Also every block singular value is at most one, so $S\leq\sum_Lw_Lr_L$. Finally $\nu_i\leq r_{\max}\pi_{\max}$ gives the simpler bound. The frame-dependent intermediate trace expression is not used as a fixed upper bound across rotations.
+\end{proof}
+
+\begin{theorem}[Finite weighted mask construction]\label{thm:mixed-finite}
+Fix $0<\eta<1$. Starting from any Parseval matrix, repeatedly select a target mask $J$ with weight $\omega=w_J$ and smallest relevant singular value
+\[
+ s\leq\rho_J=\frac{(1-\eta)\omega}{2B}.
+\]
+Apply its polar-moment rotation and retain the better sign. Each accepted update increases $S^2$ by at least $\eta\omega^2$. Hence no more than
+\[
+ \left\lfloor\frac{U^2-S_0^2}{\eta w_{\min}^2}\right\rfloor
+\]
+updates are accepted, and the procedure stops with $\sigma_{r_J}(\Phi_J)>\rho_J$ for every designated mask.
+\end{theorem}
+\begin{proof}
+The weighted transport bound, with $C=u^*Hu$, is
+\[
+ S_{\mathrm{new}}\geq S-C+\sqrt{C^2-2\omega Cs+\omega^2}.
+\]
+For a weak target,
+\[
+ \omega^2-2\omega Cs\geq\omega^2-2\omega B\rho_J
+ =\eta\omega^2.
+\]
+If $g=\sqrt{C^2+\eta\omega^2}-C$, then $S_{\mathrm{new}}\geq S+g$, and $C\leq S$ yields
+\[
+ S_{\mathrm{new}}^2-S^2\geq2Sg+g^2\geq2Cg+g^2
+ =\eta\omega^2\geq\eta w_{\min}^2.
+\]
+Each target is nonempty, so $W\geq\omega$ and $\pi_{\max}\geq\omega$, implying $B\geq\omega$ and $\rho_J<1/2$. Thus a selected weak target has $s<1$ and the rotation is defined, including its left/right-kernel construction when $s=0$. The fixed upper bound $S^2\leq U^2$ proves the update count. If a weak mask remained, another proved update would contradict that count. One final inspection certifies termination.
+\end{proof}
+
+Exhaustive implementation examines $|\mathcal F|$ masks per search. The theorem gives an explicit finite bound without claiming polynomial complexity for an implicitly represented exponentially large family.
+
+\begin{theorem}[Every-path recovery in a finite hierarchy]\label{thm:hierarchy-product}
+At each node $v$ of a finite hierarchy, construct an $N\times M_v$ Parseval encoder separately by Theorem~\ref{thm:mixed-finite}. Designated continuation masks have exactly $N$ surviving coefficient coordinates. Their square survivor maps are
+\[
+ T_{v,J}=P_J\Phi_v^*=\Phi_{v,J}^*,\qquad
+ \sigma_{\min}(T_{v,J})>\rho_{v,J}>0.
+\]
+For a known realized path, write its maps as $T_1,\ldots,T_m$, and let $\sigma_{\min}(T_\ell)\geq\rho_\ell>0$. Then
+\[
+ A=T_m\cdots T_1,
+ \qquad \sigma_{\min}(A)\geq\prod_{\ell=1}^m\rho_\ell,
+ \qquad\|A\|_{\mathrm{op}}\leq1.
+\]
+For endpoint observation $y=Ah+e$, exact inversion in the noiseless case and the endpoint noise norm are
+\[
+ \widehat h=T_1^{-1}\cdots T_m^{-1}y,
+ \qquad
+ \sup_{e\ne0}\frac{\|\widehat h-h\|}{\|e\|}
+ =\frac1{\sigma_{\min}(A)}
+ \leq\prod_{\ell=1}^m\rho_\ell^{-1}.
+\]
+The rightmost inverse acts first. The product is a bound on the exact amplification and need not be an equality.
+\end{theorem}
+\begin{proof}
+Iterating $\|T_\ell x\|\geq\rho_\ell\|x\|$ gives the lower singular-value product. Each survivor map is a coordinate restriction of an isometry, so it is a contraction. Direct multiplication gives
+\[
+ (T_1^{-1}\cdots T_m^{-1})(T_m\cdots T_1)=I_N.
+\]
+The reconstruction error is precisely $A^{-1}e$, whose operator norm is $1/\sigma_{\min}(A)$. A left singular direction of $A$ corresponding to its smallest singular value attains this exact worst-case noise amplification. The result applies separately to every known path, including paths selected according to prior survivor history.
+\end{proof}
+
+Each node is optimized independently; no rotation combining outputs of different nodes is used. A finite hierarchy therefore has a finite total update bound obtained by summing the node bounds. The decoder needs the realized path and its matrices. Additional restrictions on a node's feasible matrices require their own verification.
+
+If noise enters each stage, $z_\ell=T_\ell z_{\ell-1}+e_\ell$, the exact cancellation identity is
+\[
+ \widehat h-h=\sum_{\ell=1}^m(T_\ell\cdots T_1)^{-1}e_\ell,
+ \qquad
+ \|\widehat h-h\|\leq
+ \sum_{\ell=1}^m\left(\prod_{j=1}^{\ell}\rho_j^{-1}\right)\|e_\ell\|.
+\]
+
+\paragraph{Parameter examples.}
+For $N=2$, $M=4$, weight one on all six pairs and weight $1/2$ on all four triples gives $W=8$, $\pi_i=9/2$, $\nu_i=9$, $B=6$, and $U^2=144$. The coordinate start has $S_0=9$. Taking $\eta=1/2$ guarantees $\rho_{\mathrm{pair}}=1/24$ and $\rho_{\mathrm{triple}}=1/48$ within 504 updates. Numerical real and complex examples required four and three updates, respectively; these examples do not replace the universal proof. Three stages using the pair certificate have every-path endpoint amplification below $24^3=13824$, a conservative bound.
+
+For a better-conditioned explicit illustration, the known simplex Parseval encoder
+\[
+ \Phi=\sqrt{\frac23}
+ \begin{pmatrix}1&-1/2&-1/2\\0&\sqrt3/2&-\sqrt3/2\end{pmatrix}
+\]
+has squared singular values $1$ and $1/3$ in every two-column block. Three stages therefore have every-path endpoint amplification at most $3\sqrt3\approx5.196$. Direct numerical inspection of its 27 paths gives a largest exact amplification about $4.892$, illustrating why the product bound need not be an equality. This existing explicit frame illustrates the hierarchy theorem; its construction is not claimed as new.

research/information_law.tex

@@ -0,0 +1,101 @@
+\section{\texorpdfstring{VELAR//$\kappa$}{VELAR/kappa}: the state--spectrum information law}
+This section combines the scalar robust coding theorem of AURELINE--$\mu$ with a known linear observation operator. It does not claim a general anisotropic vector-source dimension formula or a new prefix-coding theory.
+
+Let a finite scalar affine IFS have nonzero contraction factors $r_i$, $0<|r_i|<1$, and positive iid branch probabilities $p_i$. Merge exactly equal composition maps, including both slope and endpoint, and write its depth-$n$ exact random state as
+\[
+X_n=(A_n,B_n),\qquad h=\lim_{n\to\infty}\frac{H(X_n)}n,
+\quad \chi=\sum_i p_i(-\log|r_i|),\quad D=\dim\mu,
+\]
+where $\mu$ is the stationary self-similar measure and entropy uses nats. Composition of independent exact map states gives $H(X_{n+k})\le H(X_n)+H(X_k)$, so $h$ exists by subadditivity. We use the established exact-dimensionality and compact-support dyadic entropy law \cite{FengHu,Hochman}
+\[
+\lim_{k\to\infty}\frac{H(\lfloor2^kZ\rfloor)}{k\log2}=D,
+\qquad Z\sim\mu.
+\]
+The alphabet of exact slopes has size $N_A(n)\le(n+1)^m$ for an $m$-branch IFS, so $H(A_n)=O(\log(n+1))$.
+
+Fix a unit vector $u$ and deterministic linear operators $T_n$, known to both parties. Output dimensions may vary. Assume $0<g_n=\|T_nu\|\le1$ and fix $c>0$. Whole-space injectivity is unnecessary for this scalar-supported source. For bounded deterministic exponents $\theta_n\ge0$, the decoder observes
+\[
+y=T_n(uB_n)+\eta,\qquad \|\eta\|\le c|A_n|^{\theta_n}.
+\]
+The encoder knows $X_n$, depth, and shared parameters, but not the actual noise. Its complete finite binary strings are globally prefix-free; states with disjoint observation possibilities may reuse an identical string. The decoder must recover $X_n$ exactly for every permitted perturbation. Let $L_n^*$ denote the infimum of expected metadata length in bits.
+
+\begin{theorem}[Operator-conditioned information budget]\label{operatorinformation}
+Assume
+\[
+\xi_n=\left(\theta_n\chi+\frac{\log g_n}{n}\right)_+\longrightarrow\xi\in[0,\chi].
+\]
+Then
+\[
+\lim_{n\to\infty}\frac{L_n^*}{n}
+=\frac{h-D\xi}{\log2}.
+\]
+In particular, if $0<g_n\le1$, $-\log g_n/n\to\kappa\ge0$, and $\theta_n=\theta$, the rate is
+\[
+\frac{h-D(\theta\chi-\kappa)_+}{\log2}
+\]
+whenever $(\theta\chi-\kappa)_+\le\chi$. This includes every $0\le\theta\le1$.
+\end{theorem}
+
+\begin{proof}
+First reduce the operator problem exactly, at each finite depth, to scalar observation $z$ with radius
+\[
+\delta_n(a)=\frac{c|a|^{\theta_n}}{g_n}.
+\]
+Given any vector observation, set $z=\operatorname{Re}\langle T_nu,y\rangle/g_n^2$; its error from $B_n$ is at most $\delta_n(A_n)$. Conversely, given any permitted scalar observation $z$, present $y=T_nu\,z$ to the vector decoder. Its vector error has norm $g_n|z-B_n|$. Thus each code in either problem induces a code of the same expected length in the other. This preserves the encoder's ignorance of the actual noise.
+
+Put $Y_n=(A_n,Q_n)$, $Q_n=\lfloor B_n/\delta_n(A_n)\rfloor$, and $p_*=\min_i p_i$. The finite AURELINE coding lemma applies to every positive width, including widths above one. For completeness, in any fixed fiber $(A_n,Q_n)=(a,q)$ all endpoints lie in the same interval of length $\delta_n(a)$, whose center is a common admissible observation. Their messages must be distinct. Kraft's inequality for this restriction of the globally prefix-free message set therefore gives
+\[
+L_n^*\ge\frac{H(X_n\mid Y_n)}{\log2}.
+\]
+For the upper bound, use a conditional Shannon code in each fiber, with payload length
+\[
+\ell_x=\left\lceil-\log_2\Pr(X_n=x\mid Y_n)\right\rceil.
+\]
+Since each exact state has probability at least $p_*^n$, its conditional probability is also at least $p_*^n$. Thus $\ell_x\le M_n=\lceil n\log_2(1/p_*)\rceil$. Send a fixed-width exact-slope table index, the cell residue $q\bmod4$ in two bits, a fixed-width index of $\ell_x\in\{0,\ldots,M_n\}$, and the payload. For $z_0=\lfloor z/\delta_n(a)\rfloor$, the allowed noise implies $q\in\{z_0-1,z_0,z_0+1\}$, including negative endpoints and closed boundaries. The residue identifies the actual cell. The complete message strings are prefix-free: different fixed headers differ before the payload, while equal payload-length headers give equal complete lengths. Reuse across cells is permitted because observation and residue select the codebook. Consequently
+\[
+\frac{H(X_n\mid Y_n)}{\log2}\le L_n^*
+\le\frac{H(X_n\mid Y_n)}{\log2}
++3+\lceil\log_2N_A(n)\rceil+\lceil\log_2(M_n+1)\rceil.
+\]
+Slope values are entries in a shared exact finite table; no assertion is made that arbitrary real parameters can be transmitted as finite exact numerical literals.
+
+It remains to compute observation entropy. Couple the stationary variable as $Z=B_n+A_nZ'$, where $Z'\sim\mu$ is an independent bounded tail. Conditional on $A_n=a$, an endpoint cell and the corresponding stationary cell determine each other with at most $C(1+|a|/\delta_n(a))$ possibilities. Therefore their joint entropy discrepancy is at most
+\[
+C'+\mathbb E\left[\left(\log\frac{|A_n|}{\delta_n(A_n)}\right)_+\right]
+=C'+\mathbb E\left[\left((\theta_n-1)S_n+\log g_n-\log c\right)_+\right],
+\]
+where $S_n=-\log|A_n|$. Finite branch log-variance gives $\mathbb E|S_n-\chi n|=O(\sqrt n)$. Since $\theta_n$ is bounded, the last bound is
+\[
+O(\sqrt n+1)+n(\xi_n-\chi)_+=o(n).
+\]
+At the boundary $\xi=\chi$, the extra $n(\xi_n-\chi)_+$ term must be retained; $O(\sqrt n)$ alone is not asserted without a convergence-rate assumption.
+
+For widths at most one, replace the width by the nearest coarser dyadic width within a factor two; mutual cell ambiguity is bounded. For widths above one, both the actual compact-support cell alphabet and the unit-width alphabet have uniformly bounded cardinality. Thus the entropy can be compared, with $O(1)$ discrepancy, to the adaptive nonnegative dyadic level
+\[
+k_n(a)=\max\left(0,\left\lfloor
+\frac{\theta_n(-\log|a|)+\log g_n-\log c}{\log2}
+\right\rfloor\right).
+\]
+With $k_0(n)=\lfloor n\xi/\log2\rfloor$, Lipschitzness of the positive-part operation gives
+\[
+\mathbb E|k_n(A_n)-k_0(n)|
+\le O(\sqrt n+1)+\frac{n|\xi_n-\xi|}{\log2}=o(n).
+\]
+Two nested dyadic levels differ in entropy by at most their level gap times $\log2$, even conditional on a slope. The correlation between $A_n$ and $Z$ contributes at most $H(A_n)=O(\log n)$. Hence
+\[
+\frac{H(Y_n)}n\longrightarrow D\xi.
+\]
+Because $Y_n$ is a deterministic function of $X_n$, $H(X_n\mid Y_n)=H(X_n)-H(Y_n)$. The finite coding inequalities prove the theorem.
+\end{proof}
+
+\begin{corollary}[Precision compensation]
+If $-\log g_n/n\to\kappa\in[0,\infty)$, then the constant exponent $\theta=1+\kappa/\chi$ restores the natural-scale metadata rate $(h-\chi D)/\log2$. More exactly, choosing $\theta_n=1-\log g_n/(n\chi)$ gives $\xi_n=\chi$ at every depth whenever these exponents are nonnegative and bounded.
+\end{corollary}
+This compensation requires a corresponding decrease in physical observation noise. It does not create information freely. When $\xi=0$, observation entropy is sublinear and metadata carries $h/\log2$ bits per branch. The theorem deliberately stops at effective precision $\xi\le\chi$: Appendix~\ref{finebarrier} reproves a Liouville obstruction with oscillating rates at every finer effective exponent. Thus the boundary is sharp for universal rate existence.
+
+The exact gain above is directional, $\|T_nu\|$. For a general vector source, pulling back a spherical observation-noise set produces an ellipsoid. If the vector observation operator is injective with $g_n=\inf_{\|v\|=1}\|T_nv\|>0$ and $G_n=\|T_n\|_{\mathrm{op}}$, then
+\[
+B(\varepsilon_n/G_n)\subseteq
+\{e:\|T_ne\|\le\varepsilon_n\}\subseteq B(\varepsilon_n/g_n).
+\]
+These inclusions yield a safe coding-rate sandwich under an applicable vector ball-noise theorem; they do not make the scalar formula an exact anisotropic theorem. For example, a source on the $e_2$ line is unaffected by $T_n=\operatorname{diag}(e^{-\kappa n},1)$ although its smallest singular value decays. Directional source information matters.

research/requirements.txt

@@ -0,0 +1,2 @@
+numpy>=1.24
+scipy>=1.10

research/seven_ray_family.tex

@@ -0,0 +1,64 @@
+\subsection{A continuous exact family and its complete sign-phase diagram}
+The obstruction persists through a nontrivial deformation of the exact fourth moments. Set
+\[
+y=\frac{1-x}{1+15x},\qquad
+\rho_\pm=\frac{5\pm2\sqrt5}{15},\qquad \rho_-<x<\rho_+.
+\]
+For even $j\in\{0,2,4\}$ use
+\[
+u_j=(\sqrt{1-x}\cos(j\pi/3),\sqrt{1-x}\sin(j\pi/3),\sqrt x),
+\]
+and for odd $j\in\{1,3,5\}$ replace $x$ by $y$. Keep $u_6=e_3$. Give each even ray weight $a/3$, each odd ray weight $b/3$, and the pole weight $c$, where
+\[
+a=\frac8{15(1-x)(1+3x)},\quad
+b=\frac{(1+15x)^2}{120x(1+3x)},\quad
+c=\frac{-45x^2+30x-1}{120x(1-x)}.
+\]
+\begin{theorem}[Exact deformation and sign phases]\label{family}
+These weights are positive and normalized throughout the stated interval, and every member is a weighted spherical $(2,2)$-design. Away from $x=1/9,1/3$ there are no orthogonal pairs. The exact sign functional is
+\[
+\lambda_3(\operatorname{sgn}(U^TU))=
+\begin{cases}
+(1+\sqrt5)/2,&\rho_-<x<1/9\text{ or }1/3<x<\rho_+,\\
+(7+2\sqrt{13})/9,&1/9<x<1/3.
+\end{cases}
+\]
+The interval $1/5<x<1/3$ gives a continuum of mutually noncongruent support-minimal counterexamples. This classifies the displayed family, not all seven-ray designs.
+\end{theorem}
+\begin{proof}
+The numerator of $c$ is positive exactly between $\rho_-$ and $\rho_+$; $a,b>0$ there. Rational algebra gives $a+b+c=1$. If $r^2=X^2+Y^2$, a horizontal equilateral triangle at squared height $z$ has fourth-moment average
+\[
+\tfrac38(1-z)^2r^4+3z(1-z)r^2Z^2+z^2Z^4
+\ \pm\sqrt z(1-z)^{3/2}Z(X^3-3XY^2).
+\]
+The signs are opposite for the even and odd triangles. Substitution gives the exact identities
+\begin{align*}
+a(1-x)^2+b(1-y)^2&=8/15,&
+ax(1-x)+by(1-y)&=2/15,\\
+ax^2+by^2+c&=1/5,&
+a\sqrt x(1-x)^{3/2}&=b\sqrt y(1-y)^{3/2}.
+\end{align*}
+The last two positive terms have the same square $64x(1-x)/[225(1+3x)^2]$. Thus the cubic harmonic cancels and the weighted quartic equals $(r^2+Z^2)^2/5$.
+
+Within the even and odd triangles the distinct-pair inner products are $(3x-1)/2$ and $(3y-1)/2$. Cross-triangle adjacent products are $3d$ and opposite products $-3d$, where $d=\sqrt{x(1-x)/(1+15x)}$; pole products are $\sqrt x,\sqrt y$. The decreasing involution $x\mapsto y$ interchanges $1/9$ and $1/3$ and fixes $1/5$. Hence the central interval has exactly the canonical sign matrix, and Theorem~\ref{counter} applies.
+
+In either outer interval exactly one triangle's within-orbit signs become positive. Reorder the triples so their opposite pairs match. The six-ring Seidel matrix is
+\[
+\begin{pmatrix}-K&H\\H&K\end{pmatrix}\quad\text{or}\quad
+\begin{pmatrix}K&H\\H&-K\end{pmatrix},
+\qquad K=J_3-I_3,\quad H=J_3-2I_3.
+\]
+Since $KH=HK=2I_3$ and $K^2+H^2=5I_3$, its square is $5I_6$. Equal scaling on these six coordinates and zero pole mass attains the golden ratio; the established sharp dimension-three upper bound gives equality. The two separating parameters have orthogonal pairs. At $\rho_\pm$ the pole weight vanishes and the remaining design is a maximal ETF.
+
+Finally, on $1/5<x<1/3$, the minimum absolute distinct-pair inner product is $(1-3x)/2$, strictly decreasing. The other triangle's value exceeds $1/5$, pole products exceed $1/3$, and cross products are at least $1/\sqrt3$. Thus different parameters cannot be congruent even after sign changes and permutation of projective directions. Proposition~\ref{minimal} gives support-minimality.
+\end{proof}
+
+\begin{corollary}[Relative openness in the exact-design locus]
+Every positive weighted exact design with unit vectors $v_i$ satisfying $\norm{v_i-u_i}<1/10$ for all canonical rays has the same counterexample sign functional.
+\end{corollary}
+\begin{proof}
+The canonical minimum absolute inner product is $1/5$, and
+$|\langle v_i,v_j\rangle-\langle u_i,u_j\rangle|
+\le\norm{v_i-u_i}+\norm{v_j-u_j}<1/5$.
+All signs persist. The claim is relative to exact designs; approximate moment matching alone is not the conjecture's premise.
+\end{proof}

research/spectral_transport.tex

@@ -0,0 +1,156 @@
+The rank conclusion can be strengthened to an explicit conditioning guarantee and a finite construction. Throughout this section, $\Phi\Phi^*=I_N$, $1\leq k<M$, and
+\[
+ Q=\binom Mk,\qquad r=\min(N,k),\qquad
+ B=Q\sqrt{\frac{k}{M}},\qquad
+ U=\sqrt{QrN\binom{M-1}{k-1}}.
+\]
+Write
+\[
+ F_L=\Phi_L\Phi_L^*,\qquad
+ S=\sum_{|L|=k}\|\Phi_L\|_*,\qquad
+ H=\sum_{|L|=k}F_L^{1/2}.
+\]
+Here $H$ is the positive \emph{polar moment}; $\operatorname{tr}H=S$. Counting the columns in all subsets gives $\sum_L\|\Phi_L\|_F^2=N\binom{M-1}{k-1}$. Since $\|X\|_*\le\sqrt r\,\|X\|_F$, Cauchy--Schwarz over the subsets gives $S\leq U$.
+
+\begin{theorem}[Polar-moment transport]\label{thm:polar-transport}
+Fix a $k$-subset $J$ and let $s=\sigma_r(\Phi_J)<1$ be its smallest relevant singular value. There is a unit vector $u\in\mathbb F^N$ with $F_Ju=s^2u$, and, on putting $C=u^*Hu$, one can construct a Parseval frame $\widetilde\Phi$ by one coefficient-plane rotation such that
+\begin{equation}\label{eq:polar-ascent}
+ \operatorname{NE}_k(\widetilde\Phi)
+ \geq S-C+\sqrt{C^2-2Cs+1}.
+\end{equation}
+The direction $u$ is a left singular vector associated with $s$ when $s>0$; when $s=0$, it is chosen in $\ker\Phi_J^*$. Moreover,
+\begin{equation}\label{eq:polar-bounds}
+ \binom{M-1}{k-1}\leq C\leq S,\qquad C\leq B.
+\end{equation}
+\end{theorem}
+
+\begin{proof}
+For $s>0$, choose the associated unit right singular vector $v$, embedded in the coordinates of $J$, so that $\Phi v=su$ and $\Phi_J^*u=sv$. For $s=0$, rank deficiency below $r$ ensures both a left and a right kernel; choose unit $u\in\ker\Phi_J^*$ and unit $v\in\ker\Phi_J$, again embedded in $J$. Put
+\[
+ b=\sqrt{1-s^2},\qquad
+ w=b^{-1}\Phi_{J^c}^*u,
+\]
+with $w$ embedded in $J^c$. Parsevalness gives $\|w\|=1$ and $\Phi w=bu$. The vectors $v,w$ are orthonormal with disjoint supports. Let $R_t$ be the unitary that acts by
+\[
+ R_tv=(\cos t)v+(\sin t)w,\qquad
+ R_tw=-(\sin t)v+(\cos t)w
+\]
+and is the identity on their common orthogonal complement; the same formulas are orthogonal over $\mathbb R$. Then $\Phi(t)=\Phi R_t$ is Parseval for every real $t$. Since
+\[
+ \Phi(vv^*+ww^*)=uu^*\Phi,
+\]
+the symmetric midpoint is
+\[
+ \frac{\Phi(t)+\Phi(-t)}2=D_t\Phi,
+ \qquad D_t=I_N-(1-\cos t)uu^*.
+\]
+
+For each block $X=\Phi_L$, take a polar partial isometry $Q_L$ so that $\|Q_L\|_{\mathrm{op}}\leq1$ and $XQ_L^*=F_L^{1/2}$. Nuclear-norm convexity and duality imply
+\begin{align*}
+ \frac{\|\Phi_L(t)\|_*+\|\Phi_L(-t)\|_*}{2}
+ &\geq\|D_tX\|_*\\
+ &\geq\operatorname{Re}\operatorname{tr}(Q_L^*D_tX)\\
+ &=\|X\|_*-(1-\cos t)u^*F_L^{1/2}u.
+\end{align*}
+This uses a partial polar factor and remains valid at rank-deficient blocks.
+
+For the selected block, all singular values other than the chosen one are unchanged. Its chosen value becomes $|s\cos t\pm b\sin t|$ in the two directions. Thus, with $z=\|\Phi_J\|_*-s$ and $0\leq t\leq\pi/2$, its exact symmetric average is
+\[
+ z+\max\{s\cos t,b\sin t\}\geq z+b\sin t.
+\]
+Since $u^*F_J^{1/2}u=s$, replacing the selected generic lower bound by this exact one and summing gives
+\[
+ \frac{\operatorname{NE}_k(\Phi(t))+\operatorname{NE}_k(\Phi(-t))}{2}
+ \geq S-C+(C-s)\cos t+b\sin t.
+\]
+Here $C-s\geq0$. Choose $t=\operatorname{atan2}(b,C-s)$ and take the better of the two signs to obtain \eqref{eq:polar-ascent}.
+
+Finally $C\leq\operatorname{tr}H=S$. Every $0\leq F_L\leq I_N$ obeys $F_L^{1/2}\geq F_L$, so $H\geq\binom{M-1}{k-1}I_N$. For every unit $u$, scalar Jensen and Cauchy--Schwarz give
+\[
+ u^*Hu\leq\sum_{|L|=k}\sqrt{u^*F_Lu}
+ \leq\sqrt{Q\sum_{|L|=k}u^*F_Lu}
+ =\sqrt{Q\binom{M-1}{k-1}}=B,
+\]
+which proves \eqref{eq:polar-bounds}.
+\end{proof}
+
+\begin{corollary}[Uniform conditioning of global maximizers]\label{cor:polar-global}
+Every global maximizer of $\operatorname{NE}_k$ satisfies, for every $k$-subset,
+\[
+ \sigma_r(\Phi_J)\geq\frac1{2B}
+ =\frac{1}{2\binom Mk}\sqrt{\frac Mk}.
+\]
+When the selected value is less than one, the stronger computable bound is $s\geq1/(2u^*Hu)$ for its weakest left direction. For $k\geq N$, therefore,
+\[
+ \lambda_{\min}(F_J)\geq\frac{M}{4k\binom Mk^2},\qquad
+ \| (\Phi_J^*)^\dagger\|_{\mathrm{op}}
+ \leq2\binom Mk\sqrt{\frac{k}{M}}.
+\]
+The same recovery bound holds after adding survivors. For $k<N$, the statement concerns the smallest column singular value; the $N\times N$ partial frame operator retains structural zero eigenvalues.
+For observed survivor coefficients $y=\Phi_J^*h+e$, least-squares repair $\widehat h=(\Phi_J^*)^\dagger y$ therefore obeys $\norm{\widehat h-h}\le2B\norm e$ when $k\ge N$.
+\end{corollary}
+
+\begin{proof}
+At a global maximizer, \eqref{eq:polar-ascent} is at most $S$, whence $\sqrt{C^2-2Cs+1}\leq C$ and $2Cs\geq1$. In particular $C>0$, and \eqref{eq:polar-bounds} proves the conclusion. If $s=1$, the construction is bypassed and the uniform conclusion is automatic, since $B\geq1$. For $k\geq N$, square the singular-value bound and use the exact least-squares repair identity. Adding survivors increases the partial frame operator in positive-semidefinite order.
+\end{proof}
+
+\begin{corollary}[A certified approximate-optimality bound]\label{cor:polar-approx}
+Suppose $S_\star-S\leq\varepsilon$ is an independently certified global objective gap. If $s<1$, then
+\[
+ s\geq\frac{[1-2C\varepsilon-\varepsilon^2]_+}{2C}.
+\]
+In particular, for $0\leq\varepsilon<1$,
+\[
+ s\geq\left[\frac{1-\varepsilon^2}{2B}-\varepsilon\right]_+.
+\]
+Numerical stationarity alone does not certify the required global gap.
+\end{corollary}
+
+\begin{proof}
+The constructive bound gives $\sqrt{C^2-2Cs+1}\leq C+\varepsilon$. Squaring proves the first claim. The numerator $1-\varepsilon^2$ is positive for $\varepsilon<1$, so replacing $C$ by its upper bound $B$ gives the second.
+\end{proof}
+
+\begin{theorem}[Finite construction with every-block conditioning]\label{thm:polar-algorithm}
+Fix $0<\eta<1$ and start at any Parseval frame. Repeatedly choose a block whose smallest relevant singular value satisfies
+\[
+ s\leq\rho=\frac{1-\eta}{2B},
+\]
+apply the rotation from Theorem~\ref{thm:polar-transport}, and retain the better sign. Every accepted rotation increases the squared objective by at least $\eta$. Consequently at most
+\[
+ \left\lfloor\frac{U^2-S_0^2}{\eta}\right\rfloor
+\]
+rotations are accepted, and the procedure terminates with $\sigma_r(\Phi_J)>\rho$ for every block. One additional inspection suffices to certify termination.
+\end{theorem}
+
+\begin{proof}
+For a weak block, $1-2Cs\geq1-2B\rho=\eta$, so
+\[
+ S_{\mathrm{new}}\geq S-C+\sqrt{C^2+\eta}.
+\]
+Writing $g=\sqrt{C^2+\eta}-C$, and using $S\geq C$, gives
+\[
+ S_{\mathrm{new}}^2-S^2\geq2Sg+g^2
+ \geq2Cg+g^2=\eta.
+\]
+Since $S^2\leq U^2$, only the stated number of updates is possible. If a weak block remained, the same proved update would produce another increase, contradicting the bound. Hence the procedure stops with every block above the threshold.
+\end{proof}
+
+The construction requires finding a weak subset and evaluating the two resulting objectives. Exhaustive enumeration has $\binom Mk$ subsets; the theorem supplies a finite parameter-dependent bound, not a polynomial-time algorithm when $k$ grows with $M$. Its conditioning constants are conservative and are not claimed to be optimal or sufficient for efficient compressed-sensing recovery.
+
+\begin{corollary}[Weighted masks and grouped branch outputs]\label{cor:polar-weighted}
+For any finite family of nonempty subsets with positive weights, put $S=\sum_Lw_L\|\Phi_L\|_*$ and $H=\sum_Lw_LF_L^{1/2}$. For a target $J$, redefine $s=\sigma_{\min(N,|J|)}(\Phi_J)$, choose its weakest left direction as above, and put $C=u^*Hu$ and $\omega=w_J$. For $s<1$ the same rotation proof gives
+\[
+ S_{\mathrm{new}}\geq S-C+\sqrt{C^2-2\omega Cs+\omega^2},
+ \qquad s_{\mathrm{global}}\geq\frac{\omega}{2C}.
+\]
+If $s=1$, use the unchanged frame; the displayed bounds hold trivially since $C\ge\omega$. At a global maximizer $C>0$ is automatic. If $W=\sum_Lw_L$ and $\pi_{\max}=\max_i\sum_{L\ni i}w_L$, then $C\leq\sqrt{W\pi_{\max}}$. Stacking energy-conserving branch outputs into an isometric analysis map and grouping its coefficient coordinates by branch gives the same result for grouped outputs; full recovery requires that each selected group's total output dimension be at least $N$. The coefficient rotation preserves total Parseval energy, without asserting preservation of individual branch norms or prescribed branch ranks.
+\end{corollary}
+
+\begin{proof}
+The selected exact average has weight $\omega$, and its contribution to the supporting-gradient cost is $\omega s$. Optimize $(C-\omega s)\cos t+\omega b\sin t$ as before and square the global bound. Further,
+\[
+ C\leq\sum_Lw_L\sqrt{u^*F_Lu}
+ \leq\sqrt{W\sum_Lw_Lu^*F_Lu}\leq\sqrt{W\pi_{\max}}.
+\]
+The grouped assertion is a relabeling of coefficient coordinates in the stacked isometry.
+\end{proof}

research/verification/SUITE_STATUS.json

@@ -0,0 +1,23 @@
+{
+  "status": "pass",
+  "certificate_programs": 10,
+  "scripts": [
+    "verify_counterexample.py",
+    "verify_independent_exact.py",
+    "verify_rational_realization.py",
+    "verify_polar_moment.py",
+    "verify_constrained_flow.py",
+    "verify_seven_ray_family.py",
+    "verify_mask_metadata.py",
+    "verify_hierarchical_polar.py",
+    "verify_information_codec.py",
+    "verify_conditional_risk.py"
+  ],
+  "final_risk_certificate_rerun_after_strengthening": "pass",
+  "independent_final_audits": 4,
+  "unresolved_mathematical_audit_findings": 0,
+  "final_pdf_layout": "pass",
+  "date": "2026-10-01",
+  "formal_proof_assistant_verified": false,
+  "external_peer_review": false
+}

research/verification/pdf_qa.json

@@ -0,0 +1,36 @@
+{
+  "status": "pass",
+  "pages": 21,
+  "all_pages_rendered_and_visually_inspected": true,
+  "render_dpi": 110,
+  "page_text_character_counts": [
+    1955,
+    2656,
+    2316,
+    2844,
+    2772,
+    2900,
+    1887,
+    2107,
+    2939,
+    3022,
+    2862,
+    2063,
+    2803,
+    2832,
+    3091,
+    2658,
+    2886,
+    2436,
+    4279,
+    2909,
+    2640
+  ],
+  "unresolved_reference_markers": 0,
+  "final_latex_warnings": 0,
+  "overfull_boxes": 0,
+  "underfull_boxes": 0,
+  "metadata_title": "EIDOLITH VII: Exact Design Obstructions and the State-Spectrum Information Law",
+  "metadata_author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+  "pdf_bytes": 440182
+}

research/verification/run_all.py

@@ -0,0 +1,13 @@
+"""Run the exact and numerical release certificates using the current Python."""
+from pathlib import Path
+import subprocess
+import sys
+
+root=Path(__file__).resolve().parent
+scripts=['verify_counterexample.py','verify_independent_exact.py',
+         'verify_rational_realization.py','verify_polar_moment.py','verify_constrained_flow.py','verify_seven_ray_family.py',
+         'verify_mask_metadata.py','verify_hierarchical_polar.py',
+         'verify_information_codec.py','verify_conditional_risk.py']
+for script in scripts:
+    subprocess.run([sys.executable,str(root/script)],check=True)
+print('PASS: all release certificates completed.')

research/verification/verify_conditional_risk.json

@@ -0,0 +1,26 @@
+{
+  "conditional_groups": [
+    {
+      "current_memory": 1,
+      "risk_exact": "1/4",
+      "risk_bound_exact": "1",
+      "control_cost_exact": "5/16",
+      "control_identity_exact": "5/16"
+    },
+    {
+      "current_memory": 2,
+      "risk_exact": "0",
+      "risk_bound_exact": "1",
+      "control_cost_exact": "1",
+      "control_identity_exact": "1"
+    }
+  ],
+  "sharp_risk_exact": "1",
+  "sharp_noise_moment_exact": "1/4",
+  "rare_posterior_risk_exact": "1",
+  "rare_observation_probability_exact": "1/100",
+  "pre_evidence_risk_exact": "1/100",
+  "unconditional_noise_bound_exact": "1/100",
+  "conditioning_caveat_verified": true,
+  "arithmetic": "Fraction; no tolerance"
+}

research/verification/verify_conditional_risk.py

@@ -0,0 +1,67 @@
+"""Exact non-Gaussian conditional risk and quadratic-control verification."""
+from collections import defaultdict
+from fractions import Fraction as F
+from pathlib import Path
+import json
+
+atoms=[]
+for z in [-2,-1,1,2]:
+    for mask,gain in [(0,F(1,2)),(1,F(1))]:
+        for noise in [F(-1,2),F(1,2)]:
+            atoms.append({'z':F(z),'G':abs(z),'H':(abs(z),mask,gain*z+noise),
+                          'noise':noise,'gain':gain,'p':F(1,16)})
+groups=defaultdict(list)
+for atom in atoms:groups[atom['H']].append(atom)
+posterior={h:sum(a['p']*a['z'] for a in aa)/sum(a['p'] for a in aa)
+           for h,aa in groups.items()}
+reports=[]
+R=Q=F(1,2) # correction T=1, C=2, lambda=1/2.
+for g in [1,2]:
+    aa=[a for a in atoms if a['G']==g]
+    mass=sum(a['p'] for a in aa)
+    mG=sum(a['p']*a['z'] for a in aa)/mass
+    S=sum(a['p']*(a['z']-mG)**2 for a in aa)/mass
+    Splus=sum(a['p']*(a['z']-posterior[a['H']])**2 for a in aa)/mass
+    direct=F(0)
+    for a in aa:
+        correction=(posterior[a['H']]-mG)/2
+        direct+=a['p']*((a['z']-mG-correction)**2+correction**2)/2/mass
+    identity=(Q*S+R*Splus)/2
+    assert direct==identity
+    noise_moment=sum(a['p']*a['noise']**2 for a in aa)/mass
+    risk_bound=noise_moment/F(1,2)**2
+    assert Splus<=risk_bound
+    assert identity<=Q*S/2+R*risk_bound/2
+    reports.append({'current_memory':g,'risk_exact':str(Splus),'risk_bound_exact':str(risk_bound),
+                    'control_cost_exact':str(direct),'control_identity_exact':str(identity)})
+
+# Compute the sharp moment-only example from its two atoms.
+sharp=[(F(-1),F(1,2),F(1,2)),(F(1),F(-1,2),F(1,2))]
+assert all(F(1,2)*z+noise==0 for z,noise,prob in sharp)
+sharp_mean=sum(prob*z for z,noise,prob in sharp)
+sharp_risk=sum(prob*(z-sharp_mean)**2 for z,noise,prob in sharp)
+sharp_noise=sum(prob*noise**2 for z,noise,prob in sharp)
+assert sharp_risk==sharp_noise/F(1,2)**2==1
+
+# A rare observation's posterior risk can exceed the pre-evidence bound.
+# Gain=1; rare states ±1 cancel to y=0, common state10 has y=10.
+caveat=[(F(-1),F(1),F(1,200)),(F(1),F(-1),F(1,200)),
+        (F(10),F(0),F(99,100))]
+assert sum(prob for z,noise,prob in caveat)==1
+noise_moment=sum(prob*noise**2 for z,noise,prob in caveat)
+rare_atoms=[(z,noise,prob) for z,noise,prob in caveat if z+noise==0]
+rare_mass=sum(prob for z,noise,prob in rare_atoms)
+rare_mean=sum(prob*z for z,noise,prob in rare_atoms)/rare_mass
+rare_risk=sum(prob*(z-rare_mean)**2 for z,noise,prob in rare_atoms)/rare_mass
+pre_evidence_risk=rare_mass*rare_risk
+assert rare_mass==noise_moment==pre_evidence_risk==F(1,100)
+assert rare_risk==1>noise_moment
+report={'conditional_groups':reports,'sharp_risk_exact':str(sharp_risk),
+        'sharp_noise_moment_exact':str(sharp_noise),
+        'rare_posterior_risk_exact':str(rare_risk),
+        'rare_observation_probability_exact':str(rare_mass),
+        'pre_evidence_risk_exact':str(pre_evidence_risk),
+        'unconditional_noise_bound_exact':str(noise_moment),
+        'conditioning_caveat_verified':True,'arithmetic':'Fraction; no tolerance'}
+Path(__file__).with_suffix('.json').write_text(json.dumps(report,indent=2)+'\n')
+print('PASS: exact non-Gaussian conditional risk, control identity, sharpness and conditioning caveat.')

research/verification/verify_constrained_flow.json

@@ -0,0 +1,68 @@
+[
+  {
+    "case": "real Parseval triangle",
+    "t": 0.0001,
+    "operator_residual": 3.925231146709438e-17,
+    "column_norm_residual": 0.0,
+    "opened_singular_values": [
+      0.0002449489559071435,
+      0.0002449489559071435
+    ],
+    "centered_derivative_error": 1.8371174174129222e-07
+  },
+  {
+    "case": "real Parseval triangle",
+    "t": 0.001,
+    "operator_residual": 0.0,
+    "column_norm_residual": 5.551115123125783e-17,
+    "opened_singular_values": [
+      0.0024494713714927224,
+      0.002449471371492722
+    ],
+    "centered_derivative_error": 1.837129045977684e-05
+  },
+  {
+    "case": "real nonidentity covariance triangle",
+    "t": 0.0001,
+    "operator_residual": 7.850462293418876e-17,
+    "column_norm_residual": 0.0,
+    "opened_singular_values": [
+      0.0002449489559071435,
+      0.0002449489559071435
+    ],
+    "centered_derivative_error": 1.8371174174129222e-07
+  },
+  {
+    "case": "real nonidentity covariance triangle",
+    "t": 0.001,
+    "operator_residual": 7.850462293418876e-17,
+    "column_norm_residual": 5.551115123125783e-17,
+    "opened_singular_values": [
+      0.0024494713714927224,
+      0.002449471371492722
+    ],
+    "centered_derivative_error": 1.837129045977684e-05
+  },
+  {
+    "case": "complex phase pair",
+    "t": 0.0001,
+    "operator_residual": 4.826321170839557e-21,
+    "column_norm_residual": 0.0,
+    "opened_singular_values": [
+      2.8867513429410962e-05,
+      2.8867513429410962e-05
+    ],
+    "centered_derivative_error": 1.360827529425369e-09
+  },
+  {
+    "case": "complex phase pair",
+    "t": 0.001,
+    "operator_residual": 1.8094276908307095e-20,
+    "column_norm_residual": 1.1102230246251565e-16,
+    "opened_singular_values": [
+      0.00028867510452448353,
+      0.00028867510452448353
+    ],
+    "centered_derivative_error": 1.3608275670219175e-07
+  }
+]

research/verification/verify_constrained_flow.py

@@ -0,0 +1,64 @@
+"""Numerical illustration of exact two-constraint rank opening.
+
+The written ODE proof is exact; these floating-point examples are regression
+checks of its algebra, including a positive nonidentity frame operator.
+"""
+from pathlib import Path
+import json
+import numpy as np
+from scipy.integrate import solve_ivp
+
+def check_curve(phi, plus, minus, tangent, label, t):
+    target=phi@phi.conj().T
+    norms=np.sum(abs(phi)**2,axis=0)
+    errors=[np.linalg.norm(p@p.conj().T-target) for p in [plus,minus]]
+    norm_errors=[np.max(abs(np.sum(abs(p)**2,axis=0)-norms)) for p in [plus,minus]]
+    smallest=[np.linalg.svd(p[:,[0,2]],compute_uv=False)[-1] for p in [plus,minus]]
+    assert max(errors)<1e-10 and max(norm_errors)<1e-10
+    assert min(smallest)>0
+    derivative_error=np.linalg.norm((plus-minus)/(2*t)-tangent)
+    assert derivative_error<.01
+    return {'case':label,'t':t,'operator_residual':float(max(errors)),
+            'column_norm_residual':float(max(norm_errors)),
+            'opened_singular_values':[float(x) for x in smallest],
+            'centered_derivative_error':float(derivative_error)}
+
+results=[]
+c=1/np.sqrt(3)
+for angles,label in [([0,np.pi/2,0,np.pi/2,np.pi/4,-np.pi/4],'real Parseval triangle'),
+                     ([0,np.pi/2,0,np.pi/3,np.pi/4,-np.pi/4],'real nonidentity covariance triangle')]:
+    phi=c*np.array([np.cos(angles),np.sin(angles)])
+    a,b,d=0,4,2
+    def rhs(_t, flat):
+        x=flat.reshape(phi.shape)
+        gram=x.T@x
+        k=np.zeros((6,6))
+        for i,j in [(a,b),(b,d),(d,a)]:
+            k[i,j]=1/gram[i,j]
+            k[j,i]=-k[i,j]
+        return (x@k).ravel()
+    tangent=rhs(0,phi.ravel()).reshape(phi.shape)
+    for t in [.0001,.001]:
+        plus=solve_ivp(rhs,[0,t],phi.ravel(),rtol=1e-12,atol=1e-14).y[:,-1].reshape(phi.shape)
+        minus=solve_ivp(rhs,[0,-t],phi.ravel(),rtol=1e-12,atol=1e-14).y[:,-1].reshape(phi.shape)
+        results.append(check_curve(phi,plus,minus,tangent,label,t))
+
+# Phase-adjusted pair: redundant column 0, external column 4, phase i.
+angles=[0,np.pi/2,0,np.pi/2,np.pi/4,-np.pi/4]
+phi=(c*np.array([np.cos(angles),np.sin(angles)])).astype(complex)
+tangent=np.zeros_like(phi)
+tangent[:,0]=1j*phi[:,4]
+tangent[:,4]=1j*phi[:,0]
+for t in [.0001,.001]:
+    frames=[]
+    for sign in [1,-1]:
+        x=phi.copy()
+        x[:,0]=np.cos(t)*phi[:,0]+sign*1j*np.sin(t)*phi[:,4]
+        x[:,4]=sign*1j*np.sin(t)*phi[:,0]+np.cos(t)*phi[:,4]
+        frames.append(x)
+    results.append(check_curve(phi,*frames,tangent,'complex phase pair',t))
+
+Path(__file__).with_suffix('.json').write_text(json.dumps(results,indent=2)+'\n')
+print('PASS:',len(results),'real/complex constrained opening checks.')
+print('Maximum operator residual:',max(r['operator_residual'] for r in results))
+print('Maximum column norm residual:',max(r['column_norm_residual'] for r in results))

research/verification/verify_counterexample.json

@@ -0,0 +1,152 @@
+{
+  "exact_design_moments_verified": 15,
+  "ring_weight": "5/36",
+  "pole_weight": "1/6",
+  "ring_gram_numerators": [
+    [
+      5,
+      3,
+      -1,
+      -3,
+      -1,
+      3
+    ],
+    [
+      3,
+      5,
+      3,
+      -1,
+      -3,
+      -1
+    ],
+    [
+      -1,
+      3,
+      5,
+      3,
+      -1,
+      -3
+    ],
+    [
+      -3,
+      -1,
+      3,
+      5,
+      3,
+      -1
+    ],
+    [
+      -1,
+      -3,
+      -1,
+      3,
+      5,
+      3
+    ],
+    [
+      3,
+      -1,
+      -3,
+      -1,
+      3,
+      5
+    ]
+  ],
+  "sign_matrix": [
+    [
+      1,
+      1,
+      -1,
+      -1,
+      -1,
+      1,
+      1
+    ],
+    [
+      1,
+      1,
+      1,
+      -1,
+      -1,
+      -1,
+      1
+    ],
+    [
+      -1,
+      1,
+      1,
+      1,
+      -1,
+      -1,
+      1
+    ],
+    [
+      -1,
+      -1,
+      1,
+      1,
+      1,
+      -1,
+      1
+    ],
+    [
+      -1,
+      -1,
+      -1,
+      1,
+      1,
+      1,
+      1
+    ],
+    [
+      1,
+      -1,
+      -1,
+      -1,
+      1,
+      1,
+      1
+    ],
+    [
+      1,
+      1,
+      1,
+      1,
+      1,
+      1,
+      1
+    ]
+  ],
+  "conference_residual_squared_frobenius": [
+    64,
+    64,
+    64,
+    64,
+    64,
+    64,
+    96
+  ],
+  "characteristic_polynomial_coefficients": [
+    "1",
+    "-7",
+    "0",
+    "68",
+    "-32",
+    "-192",
+    "0",
+    "0"
+  ],
+  "eigenvalues_exact": [
+    4,
+    4,
+    3,
+    0,
+    0,
+    -2,
+    -2
+  ],
+  "lambda3_exact": "(7+2*sqrt(13))/9",
+  "lambda3_maximizer_pole_weight_exact": "(26-5*sqrt(13))/39",
+  "lambda3_maximizer_each_ring_weight_exact": "(13+5*sqrt(13))/234",
+  "proof_status": "exact identities and obstruction checked; global maximum supplied by analytic concavity and symmetry proof"
+}

research/verification/verify_counterexample.py

@@ -0,0 +1,91 @@
+"""Exact certificate for the seven-direction projection-constant counterexample.
+All design moments and sign decisions use rational arithmetic. No optimizer
+or floating-point sign decision is used. Optional numerical output is labeled.
+"""
+from fractions import Fraction as F
+from itertools import product
+import json
+
+# First six vectors are (x, y_coefficient*sqrt(3), 1)/sqrt(5).
+raw = [(2,0,1),(1,1,1),(-1,1,1),(-2,0,1),(-1,-1,1),(1,-1,1)]
+ring_weight = F(5,36)
+pole_weight = F(1,6)
+assert 6*ring_weight+pole_weight == 1
+for x,y,z in raw:
+    assert x*x+3*y*y+z*z == 5
+
+moment_results = {}
+for powers in product(range(5),repeat=3):
+    if sum(powers) != 4:
+        continue
+    px,py,pz = powers
+    # The coefficient is rational; if py is odd it multiplies sqrt(3).
+    value = ring_weight*sum(F(x**px*y**py*z**pz*3**(py//2),25) for x,y,z in raw)
+    if powers == (0,0,4):
+        value += pole_weight
+    expected = F(1,5) if 4 in powers else F(1,15) if sorted(powers)==[0,2,2] else F(0)
+    assert value == expected,(powers,value,expected)
+    if py % 2:
+        assert value == 0
+    moment_results[str(powers)] = str(value)
+
+# Upper-left ring Gram numerator divided by 5, pole inner products 1/sqrt(5).
+gram_numerators = [[x*u+3*y*v+z*w for u,v,w in raw] for x,y,z in raw]
+assert all(g != 0 for row in gram_numerators for g in row)
+A = [[1 if g>0 else -1 for g in row]+[1] for row in gram_numerators]
+A.append([1]*7)
+for i in range(6):
+    for j in range(6):
+        assert A[i][j] == (1 if i==j or (i-j)%6 in [1,5] else -1)
+
+def mul(X,Y):
+    return [[sum(x*y for x,y in zip(row,col)) for col in zip(*Y)] for row in X]
+
+# Independent conference obstruction: exactly seven six-vertex submatrices.
+conference_residuals=[]
+for deleted in range(7):
+    inds=[i for i in range(7) if i!=deleted]
+    S=[[A[i][j]-(1 if i==j else 0) for j in inds] for i in inds]
+    SS=mul(S,S)
+    residual=sum((SS[i][j]-(5 if i==j else 0))**2 for i in range(6) for j in range(6))
+    assert residual != 0
+    conference_residuals.append(residual)
+
+# Newton identities certify the characteristic polynomial exactly.
+n=7
+power=[[1 if i==j else 0 for j in range(n)] for i in range(n)]
+traces=[]
+for k in range(1,n+1):
+    power=mul(power,A)
+    traces.append(sum(power[i][i] for i in range(n)))
+c=[F(1)]
+for k in range(1,n+1):
+    c.append(-sum(c[k-i]*traces[i-1] for i in range(1,k+1))/k)
+# (x-4)^2 (x-3) x^2 (x+2)^2.
+expected=[F(1)]
+for root in [4,4,3,0,0,-2,-2]:
+    nxt=[F(0)]*(len(expected)+1)
+    for j,v in enumerate(expected):
+        nxt[j]+=v
+        nxt[j+1]-=root*v
+    expected=nxt
+assert c==expected
+
+report={
+ 'exact_design_moments_verified':15,
+ 'ring_weight':str(ring_weight),
+ 'pole_weight':str(pole_weight),
+ 'ring_gram_numerators':gram_numerators,
+ 'sign_matrix':A,
+ 'conference_residual_squared_frobenius':conference_residuals,
+ 'characteristic_polynomial_coefficients':[str(x) for x in c],
+ 'eigenvalues_exact':[4,4,3,0,0,-2,-2],
+ 'lambda3_exact':'(7+2*sqrt(13))/9',
+ 'lambda3_maximizer_pole_weight_exact':'(26-5*sqrt(13))/39',
+ 'lambda3_maximizer_each_ring_weight_exact':'(13+5*sqrt(13))/234',
+ 'proof_status':'exact identities and obstruction checked; global maximum supplied by analytic concavity and symmetry proof'
+}
+print(json.dumps(report,indent=2))
+from pathlib import Path
+with Path(__file__).with_suffix('.json').open('w') as f:
+    json.dump(report,f,indent=2)

research/verification/verify_hierarchical_polar.json

@@ -0,0 +1,266 @@
+{
+  "weighted_examples": [
+    {
+      "N": 2,
+      "M": 4,
+      "field": "real",
+      "mask_sizes": [
+        2,
+        2,
+        2,
+        2,
+        2,
+        2,
+        3,
+        3,
+        3,
+        3
+      ],
+      "weights": [
+        1.0,
+        1.0,
+        1.0,
+        1.0,
+        1.0,
+        1.0,
+        0.5,
+        0.5,
+        0.5,
+        0.5
+      ],
+      "W": 8.0,
+      "coverage": [
+        4.5,
+        4.5,
+        4.5,
+        4.5
+      ],
+      "rank_coverage": [
+        9.0,
+        9.0,
+        9.0,
+        9.0
+      ],
+      "B": 6.0,
+      "U_squared": 144.0,
+      "initial_score": 9.0,
+      "final_score": 10.433255807645828,
+      "updates": 4,
+      "proved_update_bound": 504,
+      "final_smallest_singular_values": [
+        0.9548022309712839,
+        0.2076340600586044,
+        0.1836258178557093,
+        0.20433234462759528,
+        0.1882696991413674,
+        0.26951552401756196,
+        0.9629923083651164,
+        0.9548059994631916,
+        0.2833941377503484,
+        0.2840406310855132
+      ],
+      "thresholds": [
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.020833333333333332,
+        0.020833333333333332,
+        0.020833333333333332,
+        0.020833333333333332
+      ],
+      "minimum_actual_to_guaranteed_squared_gain_ratio": 9.792508667678874,
+      "parseval_residual": 1.5731923213689514e-16
+    },
+    {
+      "N": 2,
+      "M": 4,
+      "field": "complex",
+      "mask_sizes": [
+        2,
+        2,
+        2,
+        2,
+        2,
+        2,
+        3,
+        3,
+        3,
+        3
+      ],
+      "weights": [
+        1.0,
+        1.0,
+        1.0,
+        1.0,
+        1.0,
+        1.0,
+        0.5,
+        0.5,
+        0.5,
+        0.5
+      ],
+      "W": 8.0,
+      "coverage": [
+        4.5,
+        4.5,
+        4.5,
+        4.5
+      ],
+      "rank_coverage": [
+        9.0,
+        9.0,
+        9.0,
+        9.0
+      ],
+      "B": 6.0,
+      "U_squared": 144.0,
+      "initial_score": 9.0,
+      "final_score": 10.299944744802598,
+      "updates": 3,
+      "proved_update_bound": 504,
+      "final_smallest_singular_values": [
+        0.9571028408875771,
+        0.21019448539453597,
+        0.1940969063480719,
+        0.14373068464311492,
+        0.15588936012656418,
+        0.21693045781865608,
+        0.9674327248572248,
+        0.9659507996688983,
+        0.2897484287531678,
+        0.2169304578186561
+      ],
+      "thresholds": [
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.041666666666666664,
+        0.020833333333333332,
+        0.020833333333333332,
+        0.020833333333333332,
+        0.020833333333333332
+      ],
+      "minimum_actual_to_guaranteed_squared_gain_ratio": 9.792508667678874,
+      "parseval_residual": 4.98192864353311e-16
+    },
+    {
+      "N": 3,
+      "M": 5,
+      "field": "real",
+      "mask_sizes": [
+        1,
+        2,
+        3,
+        4
+      ],
+      "weights": [
+        1.0,
+        0.7,
+        0.8,
+        0.6
+      ],
+      "W": 3.1,
+      "coverage": [
+        1.8,
+        1.2999999999999998,
+        2.1,
+        1.4,
+        0.6
+      ],
+      "rank_coverage": [
+        3.4000000000000004,
+        3.1999999999999997,
+        5.6,
+        4.2,
+        1.7999999999999998
+      ],
+      "B": 2.551470164434615,
+      "U_squared": 40.92,
+      "initial_score": 5.2,
+      "final_score": 5.6435261169272755,
+      "updates": 2,
+      "proved_update_bound": 77,
+      "final_smallest_singular_values": [
+        0.9486832980505138,
+        0.8676345426334842,
+        0.49720247427902503,
+        0.31622776601683794
+      ],
+      "thresholds": [
+        0.09798272520870256,
+        0.06858790764609178,
+        0.07838618016696206,
+        0.058789635125221534
+      ],
+      "minimum_actual_to_guaranteed_squared_gain_ratio": 7.940679403278278,
+      "parseval_residual": 1.57034764384465e-16
+    },
+    {
+      "N": 3,
+      "M": 5,
+      "field": "complex",
+      "mask_sizes": [
+        1,
+        2,
+        3,
+        4
+      ],
+      "weights": [
+        1.0,
+        0.7,
+        0.8,
+        0.6
+      ],
+      "W": 3.1,
+      "coverage": [
+        1.8,
+        1.2999999999999998,
+        2.1,
+        1.4,
+        0.6
+      ],
+      "rank_coverage": [
+        3.4000000000000004,
+        3.1999999999999997,
+        5.6,
+        4.2,
+        1.7999999999999998
+      ],
+      "B": 2.551470164434615,
+      "U_squared": 40.92,
+      "initial_score": 5.199999999999999,
+      "final_score": 5.6435261169272755,
+      "updates": 2,
+      "proved_update_bound": 77,
+      "final_smallest_singular_values": [
+        0.9486832980505138,
+        0.8676345426334842,
+        0.4972024742790249,
+        0.31622776601683783
+      ],
+      "thresholds": [
+        0.09798272520870256,
+        0.06858790764609178,
+        0.07838618016696206,
+        0.058789635125221534
+      ],
+      "minimum_actual_to_guaranteed_squared_gain_ratio": 7.940679403278337,
+      "parseval_residual": 5.719389977854062e-16
+    }
+  ],
+  "cascade": {
+    "depth": 3,
+    "paths_checked": 27,
+    "node_smallest_singular_value": 0.5773502691896258,
+    "proved_endpoint_amplification_bound": 5.196152422706629,
+    "smallest_exact_endpoint_amplification": 2.62272194899462,
+    "largest_exact_endpoint_amplification": 4.892246854163953,
+    "maximum_correct_inverse_order_residual": 6.238781171274309e-16,
+    "maximum_exact_noise_amplification_equality_error": 2.6645352591003757e-15
+  }
+}

research/verification/verify_hierarchical_polar.py

@@ -0,0 +1,172 @@
+"""Numerical examples for weighted construction and square-map hierarchies.
+
+The mathematical proofs are in hierarchical_polar_extension.md.
+"""
+import itertools
+import json
+import math
+from pathlib import Path
+
+import numpy as np
+
+
+def evaluate(phi, family, weights):
+    n = phi.shape[0]
+    moment = np.zeros((n,n), dtype=phi.dtype)
+    smallest=[]
+    score=0.
+    for block,weight in zip(family,weights):
+        left,singular,_=np.linalg.svd(phi[:,block],full_matrices=False)
+        moment += weight*(left*singular)@left.conj().T
+        score += weight*float(singular.sum())
+        smallest.append(float(singular[-1]))
+    return score,moment,smallest
+
+
+def parameters(n,m,family,weights):
+    w_total=float(sum(weights))
+    coverage=np.zeros(m)
+    rank_coverage=np.zeros(m)
+    rank_sum=0.
+    for block,weight in zip(family,weights):
+        rank=min(n,len(block))
+        coverage[list(block)]+=weight
+        rank_coverage[list(block)]+=weight*rank
+        rank_sum+=weight*rank
+    b=math.sqrt(w_total*max(coverage))
+    upper_sq=min(rank_sum*rank_sum,
+                 w_total*float(np.sort(rank_coverage)[-n:].sum()))
+    return b,upper_sq,w_total,coverage,rank_coverage
+
+
+def weighted_example(n,m,family,weights,complex_field,rng):
+    eta=.5
+    phi=np.concatenate([np.eye(n),np.zeros((n,m-n))],axis=1)
+    if complex_field:
+        rotation,_=np.linalg.qr(rng.normal(size=(n,n))+1j*rng.normal(size=(n,n)))
+        phi=rotation@phi
+    b,upper_sq,w_total,coverage,rank_coverage=parameters(n,m,family,weights)
+    thresholds=(1-eta)*np.asarray(weights)/(2*b)
+    initial,_,_=evaluate(phi,family,weights)
+    update_bound=math.floor((upper_sq-initial*initial)/(eta*min(weights)**2)+1e-9)
+    updates=0
+    smallest_gain_ratio=float("inf")
+    for _ in range(update_bound+1):
+        score,moment,smallest=evaluate(phi,family,weights)
+        assert score*score<=upper_sq+1e-8
+        target=int(np.argmin(np.asarray(smallest)/thresholds))
+        if smallest[target]>thresholds[target]:
+            break
+        block=family[target]
+        omega=weights[target]
+        x=phi[:,block]
+        left,singular,right=np.linalg.svd(x,full_matrices=True)
+        rank=int(np.sum(singular>1e-12))
+        s=float(singular[-1])
+        if s<1e-12:
+            u=left[:,rank]
+            local_v=right[rank,:].conj()
+            s=0.
+        else:
+            u=left[:,len(singular)-1]
+            local_v=right[len(singular)-1,:].conj()
+        v=np.zeros(m,dtype=phi.dtype)
+        v[list(block)]=local_v
+        complement=[i for i in range(m) if i not in block]
+        opening=math.sqrt(1-s*s)
+        w=np.zeros(m,dtype=phi.dtype)
+        w[complement]=phi[:,complement].conj().T@u/opening
+        cost=float(np.real(np.vdot(u,moment@u)))
+        assert cost<=b+1e-10
+        assert cost>=omega*s-1e-10
+        angle=math.atan2(omega*opening,max(0.,cost-omega*s))
+        c,h=math.cos(angle),math.sin(angle)
+        diagonal=(c-1)*(np.outer(v,v.conj())+np.outer(w,w.conj()))
+        skew=h*(np.outer(w,v.conj())-np.outer(v,w.conj()))
+        candidates=[phi@(np.eye(m)+diagonal+sign*skew) for sign in [1,-1]]
+        scores=[evaluate(candidate,family,weights)[0] for candidate in candidates]
+        best=int(np.argmax(scores))
+        gain=scores[best]**2-score**2
+        assert gain>=eta*omega*omega-1e-8
+        smallest_gain_ratio=min(smallest_gain_ratio,gain/(eta*omega*omega))
+        phi=candidates[best]
+        updates+=1
+    else:
+        raise AssertionError("proved weighted termination bound exceeded")
+    final,moment,smallest=evaluate(phi,family,weights)
+    assert updates<=update_bound
+    assert min(np.asarray(smallest)/thresholds)>1
+    residual=float(np.linalg.norm(phi@phi.conj().T-np.eye(n)))
+    assert residual<1e-10
+    return {
+        "N":n,"M":m,"field":"complex" if complex_field else "real",
+        "mask_sizes":[len(block) for block in family],"weights":list(weights),
+        "W":w_total,"coverage":coverage.tolist(),"rank_coverage":rank_coverage.tolist(),
+        "B":b,"U_squared":upper_sq,"initial_score":initial,"final_score":final,
+        "updates":updates,"proved_update_bound":update_bound,
+        "final_smallest_singular_values":smallest,"thresholds":thresholds.tolist(),
+        "minimum_actual_to_guaranteed_squared_gain_ratio":smallest_gain_ratio,
+        "parseval_residual":residual,
+    }
+
+
+def cascade_examples(rng):
+    phi=math.sqrt(2/3)*np.array([[1,-.5,-.5],[0,math.sqrt(3)/2,-math.sqrt(3)/2]])
+    blocks=list(itertools.combinations(range(3),2))
+    maps=[phi[:,block].T for block in blocks]
+    rho=1/math.sqrt(3)
+    depth=3
+    bound=rho**(-depth)
+    amp=[]
+    inverse_error=[]
+    noise_equality_error=[]
+    for indices in itertools.product(range(len(maps)),repeat=depth):
+        selected=[maps[i] for i in indices]
+        composite=np.eye(2)
+        for matrix in selected:
+            composite=matrix@composite
+        inverse=np.eye(2)
+        for matrix in selected:
+            inverse=inverse@np.linalg.inv(matrix)
+        inverse_error.append(float(np.linalg.norm(inverse@composite-np.eye(2))))
+        left,singular,_=np.linalg.svd(composite)
+        assert singular[-1]>=rho**depth-1e-12
+        exact_amp=1/float(singular[-1])
+        amp.append(exact_amp)
+        assert exact_amp<=bound+1e-12
+        endpoint_noise=left[:,-1]
+        noise_equality_error.append(abs(float(np.linalg.norm(inverse@endpoint_noise))-exact_amp))
+        state=rng.normal(size=2)
+        estimate=inverse@(composite@state+endpoint_noise)
+        assert np.linalg.norm(estimate-state-inverse@endpoint_noise)<1e-12
+    return {
+        "depth":depth,"paths_checked":len(amp),
+        "node_smallest_singular_value":rho,
+        "proved_endpoint_amplification_bound":bound,
+        "smallest_exact_endpoint_amplification":min(amp),
+        "largest_exact_endpoint_amplification":max(amp),
+        "maximum_correct_inverse_order_residual":max(inverse_error),
+        "maximum_exact_noise_amplification_equality_error":max(noise_equality_error),
+    }
+
+
+def main():
+    rng=np.random.default_rng(20261001)
+    all_mixed=list(itertools.combinations(range(4),2))+list(itertools.combinations(range(4),3))
+    mixed_weights=[1.]*6+[.5]*4
+    varying_ranks=[(0,),(1,2),(0,2,3),(1,2,3,4)]
+    varying_weights=[1.,.7,.8,.6]
+    result={
+        "weighted_examples":[weighted_example(2,4,all_mixed,mixed_weights,field,rng)
+                             for field in [False,True]]+
+                            [weighted_example(3,5,varying_ranks,varying_weights,field,rng)
+                             for field in [False,True]],
+        "cascade":cascade_examples(rng),
+    }
+    rendered=json.dumps(result,indent=2)+"\n"
+    print(rendered)
+    Path(__file__).with_suffix(".json").write_text(rendered)
+
+
+if __name__=="__main__":
+    main()

research/verification/verify_independent_exact.py

@@ -0,0 +1,94 @@
+#!/usr/bin/env python3
+"""Exact, dependency-free verification of the canonical seven-line counterexample.
+
+All arithmetic is rational or in Q(sqrt(3)); no numerical tolerance is used.
+"""
+from fractions import Fraction as F
+from itertools import product
+from collections import defaultdict
+
+# A Q(sqrt(3)) value is a pair (rational part, sqrt(3) coefficient).
+def q(a=0,b=0): return (F(a),F(b))
+def add(x,y): return (x[0]+y[0],x[1]+y[1])
+def mul(x,y): return (x[0]*y[0]+3*x[1]*y[1],x[0]*y[1]+x[1]*y[0])
+def scale(x,s): return (x[0]*s,x[1]*s)
+def dot(v,w):
+    out=q()
+    for x,y in zip(v,w): out=add(out,mul(x,y))
+    return out
+
+# Each ring vector below is divided by sqrt(5); the final pole is already unit.
+V=[(q(2),q(),q(1)),(q(1),q(0,1),q(1)),(q(-1),q(0,1),q(1)),
+   (q(-2),q(),q(1)),(q(-1),q(0,-1),q(1)),(q(1),q(0,-1),q(1)),
+   (q(),q(),q(1))]
+weights=[F(5,36)]*6+[F(1,6)]
+normsq=[F(5)]*6+[F(1)]
+assert sum(weights)==1 and all(w>0 for w in weights)
+assert all(dot(v,v)==q(n) for v,n in zip(V,normsq))
+
+# Exact expansion of sum w_i <v_i,x>^4, accounting for sqrt(5) normalization.
+coeff=defaultdict(q)
+for v,w,n in zip(V,weights,normsq):
+    for ix in product(range(3),repeat=4):
+        term=q(1)
+        for i in ix: term=mul(term,v[i])
+        power=tuple(ix.count(i) for i in range(3))
+        coeff[power]=add(coeff[power],scale(term,w/(n*n)))
+expected={(4,0,0):F(1,5),(0,4,0):F(1,5),(0,0,4):F(1,5),
+          (2,2,0):F(2,5),(2,0,2):F(2,5),(0,2,2):F(2,5)}
+assert len(coeff)==15
+assert all(value==q(expected.get(power,0)) for power,value in coeff.items())
+
+# Raw inner products happen to be rational for this construction.
+A=[]
+for v in V:
+    row=[]
+    for z in V:
+        d=dot(v,z)
+        assert d[1]==0 and d[0]!=0
+        row.append(1 if d[0]>0 else -1)
+    A.append(row)
+expectedA=[[2*int(i==j)+2*int((i-j)%6 in (1,5))-1 if i<6 and j<6 else 1
+            for j in range(7)] for i in range(7)]
+assert A==expectedA
+
+# Seven explicit independent Fourier/mean eigenvectors certify the full spectrum.
+eigenpairs=[(4,[2,1,-1,-2,-1,1,0]),(4,[0,1,1,0,-1,-1,0]),
+            (0,[2,-1,-1,2,-1,-1,0]),(0,[0,1,-1,0,1,-1,0]),
+            (-2,[1,-1,1,-1,1,-1,0]),(3,[1,1,1,1,1,1,3]),
+            (-2,[1,1,1,1,1,1,-2])]
+for ev,v in eigenpairs:
+    assert [sum(A[i][j]*v[j] for j in range(7)) for i in range(7)]==[ev*x for x in v]
+# Distinct Fourier frequencies are orthogonal; the two mean/pole eigenvectors
+# are also mutually orthogonal, so their nonzero norms prove independence.
+for i,(_,v) in enumerate(eigenpairs):
+    assert sum(x*x for x in v)>0
+    for _,z in eigenpairs[:i]: assert sum(x*y for x,y in zip(v,z))==0
+
+# Independent conference-exclusion certificate.
+residuals=[]
+for omit in range(7):
+    ids=[i for i in range(7) if i!=omit]
+    S=[[A[i][j]-int(i==j) for j in ids] for i in ids]
+    R=[[sum(S[i][h]*S[h][j] for h in range(6))-5*int(i==j)
+        for j in range(6)] for i in range(6)]
+    residuals.append(sum(x*x for row in R for x in row))
+assert residuals==[64]*6+[96]
+
+# Explicit switching/permutation equivalence with the rational orbit example.
+B=[[2*int(i==j)-1 for j in range(3)] for i in range(3)]
+old=[[B[i%3][j%3] if i<6 and j<6 else
+      (1 if (i==6 and j==6) or (i==6 and j<3) or (j==6 and i<3) else -1)
+      for j in range(7)] for i in range(7)]
+switch=[1,1,1,-1,-1,-1,1]
+order=[0,4,2,3,1,5,6]
+assert A==[[switch[i]*old[i][j]*switch[j] for j in order] for i in order]
+
+# Algebraic check of optimizer b=(26-5 sqrt(13))/39 is represented below in Q(sqrt(13)).
+def mul13(x,y): return (x[0]*y[0]+13*x[1]*y[1],x[0]*y[1]+x[1]*y[0])
+b=(F(26,39),F(-5,39)); b2=mul13(b,b)
+assert (39*b2[0]-52*b[0]+9,39*b2[1]-52*b[1])==(0,0)
+print('PASS: positive normalized weights; all 15 exact quartic coefficients; no orthogonality.')
+print('PASS: canonical sign matrix, seven orthogonal eigenvectors, spectrum [4,4,3,0,0,-2,-2].')
+print('PASS: conference residuals',residuals,'and explicit switching/permutation equivalence.')
+print('PASS: exact scalar optimizer polynomial 39 b^2 - 52 b + 9 = 0.')

research/verification/verify_information_codec.json

@@ -0,0 +1,38 @@
+[
+  {
+    "depth": 4,
+    "states_checked": 16,
+    "robust_boundary_checks": 80,
+    "observation_entropy_bits_exact": 3,
+    "conditional_entropy_bits_exact": 1,
+    "code_bits": 3,
+    "excess_bits_over_lower_bound": 2,
+    "natural_metadata_rate_exact": "1/4",
+    "compensated_metadata_bits": 0,
+    "arithmetic": "Fraction; no tolerance"
+  },
+  {
+    "depth": 8,
+    "states_checked": 256,
+    "robust_boundary_checks": 1280,
+    "observation_entropy_bits_exact": 6,
+    "conditional_entropy_bits_exact": 2,
+    "code_bits": 4,
+    "excess_bits_over_lower_bound": 2,
+    "natural_metadata_rate_exact": "1/4",
+    "compensated_metadata_bits": 0,
+    "arithmetic": "Fraction; no tolerance"
+  },
+  {
+    "depth": 12,
+    "states_checked": 4096,
+    "robust_boundary_checks": 20480,
+    "observation_entropy_bits_exact": 9,
+    "conditional_entropy_bits_exact": 3,
+    "code_bits": 5,
+    "excess_bits_over_lower_bound": 2,
+    "natural_metadata_rate_exact": "1/4",
+    "compensated_metadata_bits": 0,
+    "arithmetic": "Fraction; no tolerance"
+  }
+]