Arithmetic Matrix and Positive Semi-Definite Certificate Prototype

Arithmetic Matrix / PSD Prototype v0.1 — Establishes M_arith(R) = M∞(R) + M_fin(R) on a real even compactly supported basis, calculating the minimum eigenvalue for each support radius R∈{0.25,...,1.00}. The finite position matrix itself has clear negative directions, but after adding the Archimedean part, the total matrix maintains a positive minimum eigenvalue throughout the currently scanned subspace.

This is not an RH proof, nor is it a strict PSD certificate — The phased status stated by the document in the package, reproduced exactly as is. "numerical_psd=true only indicates that no negative eigenvalues were found under the current discretization." This package is the last one in the entire Batch 01 (48 original engineering packages) to have a detailed page added.

Connections

The relationship with other packages, using the words from its own documents as much as possible, not my interpretation. This is a cross-track connection: this package and the first side-branch prototype together form a complete executable closed loop for the semi-autonomous track's theoretical draft.

Theoretical Source (Semi-Autonomous Track)Off-Axis Positivity Obstructions in the Explicit Formula

Previous PrototypeRegional Phase Shaping Prototype

Side-Branch Prototype Series2 / 2 · Full Series Completed

"The first package establishes c^T M_orb(w) c < 0 (w∈K). This package establishes a finite-dimensional numerical check for c^T M_arith(R) c ≥ 0. The next core engineering problem is to have both packages use the same set of bases, the same support scale, and the same coefficient vector to directly solve: sup_{w∈K} c^T M_orb(w) c < 0 ∧ c^T M_arith(R) c ≥ 0." — Excerpt from Section 7 of the technical notes in this package, "Relationship with the Regional Phase Shaping Package", on how the two prototypes form a complete feasibility check for both the positive and negative sides.

Overview and Usage · README

Implemented normalizations (archimedean principal computation, spectral cross-check), finite place activation conditions, constraints (G(i/2)=G(0)=0), execution methods, output descriptions, important limitations, and main references (Connes-Consani, Connes).

Methodology and Audit Boundaries · METHOD

Logarithmic coordinate setup, correlation matrix field, compactly supported time-domain formula derivation for the Archimedean matrix, spectral cross-check, finite position matrix, constrained projection, distinction between structural vs. numerical parts, and v0.2 planning goals.

Technical Notes

Purpose and three engineering problems, support and prime activation thresholds, full matrix decomposition process, v0.1 actual scan result table (7 R values), grid sensitivity recheck, output field descriptions, relationship with the regional phase shaping package, and parts still uncompleted.

Files

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