Fix a Galerkin level N in the finite Connes-van Suijlekom truncation of the Weil quadratic form (with no archimedean cutoff), and let u = log c be the prime cutoff, which is retained and varied. Differentiating the finite matrix path u -> QN (u) across a prime-power threshold u = log q returns the von Mangoldt weight exactly, in every matrix entry: the first-derivative jump equals -2 Lambda (q) / (sqrt (q) log q) times the all-ones rank-one matrix. Equivalently, the singular part of the second derivative is a negative semidefinite rank-one matrix-valued von Mangoldt measure, a finite, cutoff-free realization of the prime side of the Weil-Guinand explicit formula on the operator path itself; the identity relating this prime side to the zeros is classical and is not proven here. The event is arithmetically rigid: the entire singular jet at an edge is a universal matrix scaled by the single scalar Lambda (q), with an explicit closed form, so it carries exactly the von Mangoldt weight and nothing more. The surrounding geometry is finite harmonic analysis. The source-to-jet map is a lossless finite dictionary on the squared-integer nodes 0, 1, 4,. . . , N², with an explicit confluent-Vandermonde determinant, a sharp 2N+1 window, and a universal recurrence. A finite prime-edge uncertainty principle bounds the order at which a band-limited normal can be blind to an entering prime, with the centered finite-difference stencil the unique extremizer. Each event is a rank-one negative-semidefinite deflation of the spectral velocity, generated by a single coincidence resolvent, and a Krein-string reading realizes each weight as a boundary-mass increment. The redundancy of the rank-one measure gives a coincidence-averaged readout of the weight with an exact (2N+1) ² variance reduction. Every statement is finite and exact. The event algebra holds across the Selberg class, with a residue-class readout of Lambda (q) across the Dirichlet family; the sign-definite consequences require a positive weight and so hold for the Riemann zeta function. The results are structural: the paper proves no positivity, no Riemann Hypothesis, and no prime-counting, next-prime, or factoring statement. The companion package contains thirteen independent reproducibility guards (exact symbolic, exact modular over prime fields to N up to 1000, and a floating-point check on the assembled operator at the canonical scale N = 200), their JSON artifacts, the figure generator, the manuscript source, and release checksums.
Akiva Groskin (Tue,) studied this question.