No negative witness. The bounded diagnostic is inconclusive for general positive definiteness of k=C/M. The original matrix has positive numerical eigenvalues at128/256bits, and its one frozen rational test vector has positive form. No complete original certificate was attempted or warranted by the protocol. The original first-Laguerre estimate remains open.
Wall check: Distinct test. Closest PR78–80, NS100/101. This tests a direct real-axis necessary condition for PD, without requiring a complex pole. No new arithmetic lower estimate, NS row, thaw or manuscript version follows. Both original-evidence recovery groups remain OPEN.
The full104-node register and14continuation demands were reviewed at main 0da30a5c92203e45c452df87b68a93d86b7e5fd7; no change since PR80. This was a conclusion/dependency review, not replay of every historical proof.
For t_j=j/4,j=0,…,5 use the complete original M,C and B_ij=(C/M)(t_i-t_j). A rigorous negative rational quadratic form would exclude general quotient PD by definition. It would not refute original Laguerre positivity: quotient PD is a sufficient construction only.
The original matrix’s diagnostic eigenvalues at256bits are approximately
0.000528173, 0.001520214, 0.004401079,
0.012635149, 0.036318801, 0.092077142.
The frozen dyadic vector’s quadratic form is approximately0.001688861, positive at both precisions. Entrywise and normalized-form agreement gates passed. These are numerical observations, not rigorous lower bounds. A positive finite matrix does not establish PD for arbitrary real points and arbitrary matrix sizes. No candidate reached the conditional Arb stage.
Draft PR81 contained the filled proposal before original computation, at commit 1b9efc9. Six arguments only,128/256bits, one rational vector selected from 128bits and reused. No variation of spacing, stencil size or coefficients. The original run completed12evaluations in about3.09seconds.
The Gaussian positive control and the entire pole-free non-PD control passed through the same matrix/witness pipeline. The latter gave a stable negative form, demonstrating the test can catch a failure without a pole. Neither control supplies original arithmetic evidence. All NS/DH hypothesis matches and mismatches are explicit in the proposal and review03.
Complete lattice and real-integration tails are retained as analytic formulas evaluated in ordinary arithmetic. The numerical finite quadrature, roundoff and ratio errors are not enclosed. No midpoint eigensolve is a certificate. Independent replay and final reviews are recorded separately.
Stop this screening round and shelve further quotient falsification screens until independent analytic input appears. Do not infer PD from these finite matrices or enlarge the stencil after this result. PR80’s pole search and this direct six-point test remain separately scoped nulls. Neither closes the original nonlinear theta route. No new positive estimating method or all-height lower bound has been obtained.
See diagnostic record, original values, matrix output, and matched controls.
A separate direct-term, unnormalized implementation with adaptive tanh-sinh quadrature (256bits,36theta terms,R=5) reproduced all six original quotients and the same fixed rational form. The largest observed relative quotient difference is below3.85e-69. This supports stability at these six arguments; it is not an error bound or a proof of finite/global PD. No additional point or stencil was used. See review04 and replay output.