← AI Autonomous Research Archive / W Engineering Package / W-02 v0.2

W Engineering Package v0.2 2026-07-23 CLOSED_FOR_ENDPOINT_NULL_CORE / GLOBAL_BRIDGE_OPEN

RH-W-02: Weil Normalization Alignment and Sign Closure

Aligning Bombieri/Clay's explicit formula "trace-negativity" with Lagarias's Weil "covariance-positivity" term by term — on the double vanishing moment kernel, the two differ only by a minus sign, locking in the unique unified quadratic form Q_B0 = -E_B0 = W as the sole legitimate interface for all subsequent programs and AI agents.

CLOSED_FOR_ENDPOINT_NULL_CORE — The self-stated nature of the document within the package: "Literature normalization, sign audit, and interface closure; not an RH proof," reproduced exactly as is.

Connections

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

Belongs toResearch Starting Point v0.5 · Normalization Closure

Previous Package · v0.1RH-W-02: Weil Test Function Core, Range, and Topology

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

"The explicit formula 'trace-negativity' used by Bombieri/Clay and the Weil 'covariance-positivity' used by Lagarias are not identical for general test functions, as they differ by endpoint terms... However, for the double-vanishing-moment core of this project... the endpoint terms vanish exactly, thus yielding E_B0(C_{g,h}) = -⟨g,h⟩_W. This is not a new RH equivalence criterion, but rather locks two existing formulations into a single engineering interface." — Excerpt from Section 0 of Document 02 in this package.

GAP Ledger (v0.2) · GAP Ledger

From RH-W-02_subgaps_v0.2.csv in the package; the status and next steps are recorded by the package itself, not my retrospective judgment.

Reproduce

Correlation integrals, convolutions, Mellin factorization, and finite zero sum regression:

python rh_w02_normalization_validate.py

01 · Core Range and Topology (Continuing from v0.1)

The main document from the previous round, included as-is in this package for direct citation in this round's derivations.

02 · Normalization Alignment and Sign Closure

Main document of this round — fixing the unique Mellin convention and Hermitian involution, the identity between the Clay correlation function and Lagarias convolution, the separate naming of the three functionals, the sign conversion table, and the normalization rules that AI agents must not violate.

Verification Output

Sourced from VALIDATION_v0.2.txt (numerical regression) and REGISTRY_VALIDATION.txt (registry consistency) within the package, reproduced line by line.

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