The preregistered bounded original quotient-pole locator found no stable approximate root satisfying its threshold. General quotient PD remains undecided; the original signed theta target remains open. No zero-free region, negative Laguerre value, RH/G2 implication or manuscript version.
See outcome and scope and frozen protocol.
Python3.14.6, mpmath1.3.0; pinned requirements-replay.txt. Existing .venv
was used; no dependency installation. From repository root:
.venv/bin/python evidence/diag_theta_pole_locator/locator.py --output evidence/diag_theta_pole_locator/controls.json
.venv/bin/python evidence/diag_theta_pole_locator/locator.py --original --original-approved --budget-seconds 4200 --output evidence/diag_theta_pole_locator/original.json
pyright evidence/diag_theta_pole_locator/theta_eval.py evidence/diag_theta_pole_locator/locator.py
These commands describe the archived execution; do not overwrite the historical outputs for a replay. Use new output filenames. The original run was admitted only after matched controls, protocol review and draftPR80. The full90-minute cap included preparation/controls; 4200seconds was the conservative remaining cap supplied to the original runner. Actual original runtime48.46seconds,171calls:81grid,10replayed corners,80refinement evaluations. No domain extension or later root scan.
Analytic lattice and real-tail majorants are supplied for M,C,M’. Their numeric evaluations use ordinary mpmath, not interval rounding. Composite Gauss-Legendre integration and roundoff errors are not rigorously enclosed. 128/256bits,24/32theta terms,48/96nodes per segment,R=4/4.5 are diagnostic replays. Four-corner sampled phase is not an argument-principle certificate. Positive/negative synthetic outcomes are algorithm checks only.
Original output retains every evaluated point, derivative, normalization, majorant, selection score, stop and replay comparison. The smallest final relative residual0.03914 is far above2^-40; it is not a regional lower bound.
A separately implemented direct-term theta sum (36 terms), 256-bit adaptive tanh-sinh quadrature to R=5, and a breakpoint at Re(z) replayed exactly the four saved final endpoints. No extra corner, root search or scan. Runtime 65.94 seconds. The replay also independently differentiates phi and includes the normalization derivative.
.venv/bin/python evidence/diag_theta_pole_locator/independent_replay.py --approved --input evidence/diag_theta_pole_locator/original.json --output evidence/diag_theta_pole_locator/independent.json --bits 256 --max-endpoints 4
Largest relative differences from the 256-bit Gauss-Legendre endpoints: M:8.374e-29; C:1.404e-26; normalized M’:1.048e-26 (rounded upward for display, not interval bounds). This supports stability of these four samples and does not establish their full working-precision accuracy. Both methods have unbounded finite quadrature/roundoff error. Tanh-sinh’s returned integration error is a heuristic, not a certificate. The input archive hash is retained.