The original six-point matrix has positive numerical eigenvalues at128/256 bits. The one frozen rational test vector has a positive form at both precisions. There is no negative diagnostic candidate and no Arb stage. This is inconclusive for general positive definiteness of k=C/M. The original first-Laguerre target remains open. Stop this screening round; no stencil/spacing enlargement or new manuscript version.
The exact stencil is t_j=j/4,j=0,…,5, B_ij=k(abs(i-j)/4). All six complete original quotient values retain the full M,C lattice/real-tail accounting. Finite quadrature, roundoff and ratio errors remain unbounded diagnostics. Source, control and wrapper hashes are in original.json. Protocol was published in draftPR81, commit1b9efc9, before original evaluation.
Python3.14.6, mpmath1.3.0. From repository root, using the existing .venv:
.venv/bin/python evidence/diag_theta_six_point/matrix_screen.py --output evidence/diag_theta_six_point/controls.json
.venv/bin/python evidence/diag_theta_six_point/original_screen.py --controls evidence/diag_theta_six_point/controls.json --output evidence/diag_theta_six_point/original.json --deadline-utc 2026-10-06T15:59:00Z --preregistered-pr https://github.com/AlexanderEastwood/riemann/pull/81 --authorized-original
.venv/bin/python evidence/diag_theta_six_point/matrix_screen.py --input evidence/diag_theta_six_point/original.json --output evidence/diag_theta_six_point/original-screen.json
These commands document the archived execution. Preserve its outputs; use new output names and a separately admitted deadline for any later replay. The one original run took about3.09seconds,12quotient evaluations total, six per precision. No alternative stencil or extra original argument.
The witness numerators (common denominator2^32) were [-1209425521,3095606301,-4294967296,4294967296,-3095606301,1209425521]. Orientation is chosen before dyadic rounding, so a rounded tie at +/-1 need not have its first component positive. Sign orientation does not change the quadratic form.
At256bits, the computed minimum eigenvalue is approximately 0.00052817273088790117508, and the frozen form approximately 0.00168886096494128373566. These are diagnostic values, not certified lower bounds. The128/256bit entry/form agreement gates passed.
Controls used the identical six-point pipeline. Gaussian minimum eigenvalue approximately1.29538e-5; pole-free non-PD control exp(-t^2)(1+t^2) minimum eigenvalue approximately-0.110628 and fixed rational form approximately -0.415568. These controls validate the finite matrix/witness pipeline, not original theta quadrature or global PD.
.venv/bin/python evidence/diag_theta_six_point/independent_replay.py --approved
The separate direct-summand implementation imports neither the primary kernel nor matrix code. It integrates raw M,C using256bits,36theta terms, R=5 and adaptive tanh-sinh. Exactly the same six real arguments and the same rational vector were used, with no new candidate selection. Runtime about13.61seconds. Largest observed relative quotient discrepancy from the primary256bit values is below3.85e-69; the same form remains positive, with observed relative discrepancy below1.18e-69. These are empirical agreement figures, not complete error enclosures. The input/code hashes, complete analytic truncation majorants and all comparisons are archived. No certificate stage is warranted by this outcome.
For an independent replay, copy this evidence directory to a new local-run directory and execute its copied independent_replay.py there. That script reads its neighboring archives and writes independent.json next to itself. Do not execute it in the historical archive after publication; preserve the executed source and output hashes.