Certified positivity of a Toeplitz–Cayley matrix family at the linear threshold a=3N

Abstract

Version 2.0 gives a reproducible computer-assisted positivity proof for an explicit full Toeplitz–Cayley matrix family derived from the logarithmic derivative of the completed Riemann xi function. Writing Xi(q)=xi(1/2+q), h=Xi'/Xi, and defining the Toeplitz coefficients by sum_{n>=0} c_n(a) z^n = h(a)+h(a(1-z)/(1+z)), the companion certificates prove T_N^TC(3N) > (1/16000 - 10^-165) I > 0 for every integer N>=653. The proof first establishes L_N^(4,6) > (1/8000) I for an explicit Euler–Maclaurin minorant. It uses the exact reversal-parity split, a rational high-frequency floor above mode 114, a 57-mode even-sector Schur certificate, a corrected 55-mode odd-complement certificate, and a new two-mode certificate for the critical odd modes 1 and 3. Every required finite integer is evaluated by validated matrix balls; exact residual identities and uniform rational bounds cover all larger integers. A separate coefficient and operator-loss ledger transfers the minorant theorem to the analytic matrix defined directly from Xi'/Xi. This release also discloses and repairs a missing finite-versus-infinite moment-tail term in version 1.2 (DOI 10.5281/zenodo.22133134). The corrected bounds preserve that sector theorem, and the original v1.2 files remain preserved in Zenodo's version history. The result is internally certified and reproducible but has not been peer reviewed or independently mathematically validated. It concerns only the single rank-dependent boundary a=3N; it does not prove positivity on the band 3N<=a<=4N, at a sublinear or fixed scale, or the Riemann hypothesis.