A matrix-valued von Mangoldt measure in the finite Connes-van Suijlekom path

Abstract

A matrix-valued von Mangoldt measure in the finite Connes-van Suijlekom path. Akiva Groskin, 2026. Manuscript (18 pages) and full reproducibility package. Fix a Galerkin level N in the finite Connes-van Suijlekom truncation of the Weil quadratic form (no archimedean cutoff) and vary the prime cutoff u = log c. Differentiating the finite matrix path u → Q_N(u) across a prime-power threshold u = log q returns the von Mangoldt weight exactly, in every matrix entry: the first-derivative jump is -2 Λ(q)/(√q log q) times the all-ones rank-one matrix. This identity is an elementary structural derivative of the path's defining prime sum; the contribution is the isolation and naming of the resulting finite matrix-valued measure and the exact finite geometry around it. The paper proves the event is arithmetically rigid, develops the finite source-to-jet dictionary (confluent Vandermonde, sharp 2N+1 window, universal recurrence), a sharp finite vanishing-moment ceiling at the prime edge (a prime-edge uncertainty principle in the band-limited sense), a coincidence-resolvent generating identity stated by spectral projections, and a rank-one Weyl-function increment; a Krein-string reading is retained only as an explicitly labeled analogy. The paper proves no positivity, no Riemann Hypothesis, and no prime-counting, next-prime, or factoring statement. Reproducibility. The archive contains the LaTeX source with figures and generator script, fourteen independent guard scripts with their JSON artifacts, VERIFICATION.md mapping each guard to the statement it checks, README.md, SHA256SUMS and the CC BY 4.0 licence. All fourteen guards pass from a clean extraction. Nine use only the Python 3 standard library; five import numpy and/or sympy (check_canonical_scale, check_coincidence_readout, check_dirichlet_readout, check_elevations, check_universal_jet). Verify SHA256SUMS before running the guards: six artifacts embed a runtime_seconds field, so regenerated copies differ from the archived hashes even on a faithful reproduction. Versions. This is version 2.2. Relative to version 2.1 the manuscript updates two references that have since appeared in print: Connes and van Suijlekom is now cited as Communications in Mathematical Physics 406 (2025), article 312, and Connes, Consani and Moscovici as a chapter in Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time, EMS Series of Lectures in Mathematics, EMS Press (2026), pages 39-76. No theorem, proof, or numerical result changes; the text is otherwise the corrected version 2 of 2026-07-27. What version 2 changed relative to version 1 is listed in full in README.md inside the archive. Earlier versions remain available in the version history. Companion papers. High-Precision Approximation of Riemann Zeros via the Truncated Weil Form (arXiv:2605.20224) and A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form (arXiv:2607.02828).