PureOne/bilu-linial-extensive-spectral-failure-v4

Extensive Spectral Failure of Bilu–Linial Signings at Arbitrary Girth

Subtitle: Positive-Density Outliers, Exterior-Power Obstructions, and Finite Moment Certificates
Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Scientific release: v4.0.0 · Date: 2026-09-16
Repository: PureOne/bilu-linial-extensive-spectral-failure-v4
Status: public expert-review research release; not peer reviewed or proof-assistant formalized.

This Hugging Face repository is an AI-friendly research distribution, not a trained model. It packages the v4 manuscript, the complete v3 arbitrary-girth proof dependency, exact verification code, machine-readable claim/evidence tables, retrieval text, provenance metadata, and the original release archive.

Main theorem

For every fixed integer (g\ge 3), the manuscript constructs constants (c_g,\delta_g>0) and infinitely many finite connected simple cubic graphs (F) of girth at least (g) such that every edge signing (\sigma) satisfies

[ #{i: |\lambda_i(A_\sigma)|\ge 2\sqrt2+\delta_g}\ge c_g|V(F)|, ]

where eigenvalues are counted with algebraic multiplicity.

Equivalent/derived certificates include linear negative inertia of (8I-A_\sigma^2), a linear-order exterior-power obstruction, a fixed finite even spectral-moment rejection certificate for each fixed (g), an SOS/localizing dual polynomial, spectral-measure separation, and a Frobenius-distance lower bound.

What is new relative to the bundled v3 dependency

The bundled v3 manuscript proves arbitrary-girth persistence: for every prescribed girth there are infinitely many cubic graphs for which every signing violates the exact two-sided Ramanujan bound. v4 strengthens the obstruction from one unavoidable spectral outlier to a positive linear fraction of unavoidable outlier directions for each fixed girth.

Priority and limits

Zhiqiang Xu's 14 September 2026 preprint has priority for disproving the general Bilu–Linial signing conjecture with a cubic counterexample. This repository does not claim priority for that disproof. The v4 strengthening is a candidate new result whose historical novelty remains provisional pending independent review and a broader literature audit.

This release does not claim:

• a solution of the remaining Ramanujan-base restriction;
• a globally minimum-order counterexample;
• constants (c_g,\delta_g) uniform as (g\to\infty);
• (c_g n) distinct eigenvalue values (the theorem is multiplicity/latent-rank based);
• peer review or formal proof verification.

See docs/PRIORITY_NOTICE.md and docs/STATUS.md before quoting the release.

Closest current literature anchors

• Zhiqiang Xu, A 3-regular counterexample to the Bilu–Linial signing conjecture, arXiv:2609.15591 (priority for the original general disproof).
• Vaibhav Suvagiya, Parity families and a kernel-averaged L-function for near-Ramanujan signings, arXiv:2607.17343.
• Fangfang Lin and Hong Zhou, Improved Bounds for the Bilu–Linial Conjecture via Spectral Recovery from Mixed Determinantal Polynomials, arXiv:2609.15715.

See data/sources.csv for the machine-readable source table.

Researcher quick start

python code/verify_extensive.py
python code/verify_high_girth_v3.py

Expected decisive status: PASS for both verifiers. The code uses exact integer/rational arithmetic for theorem inequalities it audits; it does not enumerate the astronomically large constructed graphs.

For programmatic metadata, start with:

• claims.json — normalized claims and non-claims;
• data/theorems.csv — theorem index;
• evidence_index.json — claim → evidence mapping;
• llms.txt — compact AI/agent navigation;
• llms-full.txt — retrieval corpus;
• research.jsonld — JSON-LD metadata;
• artifact_manifest.json — file inventory and hashes.

Repository map

Path	Purpose
manuscripts/	v4 main paper and complete v3 proof dependency, PDF + LaTeX
code/	exact v4 and v3 verification programs
data/	certificates, claims, theorems, verification results, prior-art table
docs/	construction, transfer ledger, hostile audit, status, priority, reproducibility
corpus/	plain-text manuscript and agent retrieval corpus
releases/	byte-preserved original v4 release ZIP
publishing/	credential-free publisher source for reproducible Hub publication

Machine-readable theorem capsule

OBJECT: finite connected simple cubic graphs with edge signings sigma:E->{-1,+1}
THRESHOLD: 2*sqrt(2)
FIXED PARAMETER: girth target g>=3
EXISTS: c_g>0, delta_g>0, infinitely many graphs F
FOR ALL: edge signings sigma
CERTIFICATE: count{i: |lambda_i(A_sigma(F))| >= 2*sqrt(2)+delta_g} >= c_g*|V(F)|
DEPENDENCY: bundled AMS/HHH v3 arbitrary-girth persistence theorem
OPEN: Ramanujan unsigned-base restriction

Citation

Use CITATION.cff or CITATION.bib. Please cite Xu's 2026 priority disproof separately when discussing the history of the Bilu–Linial conjecture.

License

Research text, metadata, and included scientific source are released under CC BY 4.0, consistent with the source release. See LICENSE.md.

Reproducibility and review

The project deliberately separates analytic proof, exact executable checks, literature/priority statements, and explicit non-claims. Reviewers should begin with REVIEW_GUIDE.md and docs/HOSTILE_AUDIT.md.