PureOne/EIDOLITH-VII

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 · Download the original research package · Theorem ledger · Verification report

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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

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,

$${\sigma }_{\min }(T_m\cdots T_1)\ge \prod _{\mathrm{\ell }=1}^m{\rho }_{\mathrm{\ell }},\mathrm{\parallel }(T_m\cdots T_1{)}^{-1}\mathrm{\parallel }\le \prod _{\mathrm{\ell }=1}^m{\rho }_{\mathrm{\ell }}^{-1}\mathrm{.}\backslash sigma\_\{\backslash min\}(T\_m\backslash cdots\; T\_1)\backslash ge\backslash prod\_\{\backslash ell=1\}^m\backslash rho\_\backslash ell,\; \backslash qquad\; \backslash |(T\_m\backslash cdots\; T\_1)^\{-1\}\backslash |\backslash le\backslash prod\_\{\backslash ell=1\}^m\backslash rho\_\backslash ell^\{-1\}.$$

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

$$\underset{n\to \mathrm{\infty }}{\lim }\frac{L_n^{\ast }}{n}=\frac{h-D\xi }{\ln 2},\xi =(\theta \chi -\kappa {)}_{+}\le \chi \mathrm{.}\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{L\_n^*\}\{n\}\; =\backslash frac\{h-D\backslash xi\}\{\backslash ln2\},\; \backslash qquad\; \backslash xi=(\backslash theta\backslash chi-\backslash kappa)\_+\backslash le\backslash chi.$$

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:

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.