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W Engineering Package v0.1 2026-07-23 SPECIFICATION_COMPLETE / ENTRY_ENCLOSURES_OPEN

RH-W-04: Finite-Dimensional Completeness and Rational Negative Certificate

Constructs an explicit nested cutoff–Fourier finite-dimensional dictionary, proving that its Rayleigh–Ritz minimum must converge to the true spectral bottom. It also reduces the rigorous negative certificate to "a rational vector + a set of rational interval matrices + a small verifier using only fractions.Fraction". The demonstration verifier correctly accepts exact negative certificates and correctly rejects floating-point candidates that appear to be negative.

SPECIFICATION_COMPLETE / ENTRY_ENCLOSURES_OPEN — The positioning stated by the document in the package: "Not an RH proof; establishes the finite-dimensional reachability of the Weil quadratic form and machine-verifiable negative certificates," reproduced exactly as is. The rigorous interval envelope of the genuine zeta Weil matrix is not yet complete.

Connections

Regarding its relationship with other packages, I try to use the words from its own documents, not my interpretation.

Belongs toResearch Starting Point v0.7 · Finite-Dimensional Negative Certificate

Previous PackageRH-W-03: Compact Support Separation, Negative Witness Existence, and Dual-Core Architecture

Next PackageRH-W-05: First Genuine Weil Matrix Rational Interval

Batch 014 / 20

"This round rewrites the problem from: 'Finite-dimensional scanning might forever miss the true negative direction,' to: 'The specified cutoff–Fourier dictionary is complete under the form norm; if a negative direction exists, finite dimensions will eventually reach it. The only remaining problem is how to strictly bound the matrix elements.'" — Excerpt from Section 12, "Conclusion," of Document 01 in this package.

GAP Ledger

Sourced from RH-W-04_subgaps_v0.1.csv within the package; the status and next steps are recorded by the package itself, not judgments added by me after the fact.

Reproduce

Pure rational negative certificate verifier using only fractions.Fraction:

python verify_negative_certificate.py

01 · Finite-Dimensional Completeness and Rational Negative Certificates

Main Document — Fixed-support Weil form, cutoff–Fourier complete dictionary, Rayleigh–Ritz reachability theorem, rational witness simplification, component interval negative certificates, firewall from floating-point candidates to rational witnesses, and the minimum credible boundary for a true RH certificate.

02 · Real Weil Matrix Envelope Work Breakdown

Next round work breakdown — Six sub-work packages: endpoint/prime/Archimedean/singular/Hermitian/hash, separation of search and verification layers, and accept/reject rules.

Verification Output

From VALIDATION.txt in the package — Exact demonstration certificates (CERTIFIED_NEGATIVE, exact fractional computation) and floating-point candidates that should be rejected (REJECTED_NOT_RIGOROUS), reproduced line by line.

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