A finite-order obstruction to total positivity of an arithmetic theta trace kernel
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Abstract
We exhibit an explicit negative sixth-order Toeplitz minor of the infinite arithmetic kernel G(x) = ∑ₙ≥₁ exp(−n²e^(−x)). At the increasing nodes xᵢ = −log(5/2) + i/100 and yⱼ = j/100, 0 ≤ i,j ≤ 5, its determinant lies strictly between −978 × 10^(−59) and −977 × 10^(−59). A portable certificate uses only integer and rational arithmetic and bounds the complete omitted arithmetic tail; an independently constructed ball-arithmetic enclosure verifies the same full-series minor. Positive exponential gauges and translation transfer this counterexample to every kernel C e^(−3x/4) ∑ₘ≠₀ exp(−αm²e^(−x)), C,α > 0, disproving the total-positivity assertion for the explicit trace kernel in arXiv:2602.01248v1. We also give an elementary two-summand counterexample to positive-mixture closure, classify finite positive common-tilt mixtures of Gumbel kernels, and recover an infinite-order obstruction from the Gamma–zeta transform. Separate twelfth-order ordered and derivative certificates supply additional reproducible checks. The arithmetic trace kernel is distinct from the even Riemann density.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Classical and Real Analysis
- Keywords
- total positivity, Pólya frequency functions, arithmetic theta trace kernels, Toeplitz minors, exact rational certificates, Gamma–zeta transform
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Full infinite arithmetic trace fails PF6 for every positive C and α in the literal arXiv:2602.01248v1 kernel. Exact whole-tail rational ordered-minor interval and independent directed-ball reconstruction.
Disclosures
OpenAI Codex assisted with literature discovery, mathematical development and checking, reproducibility code and manuscript preparation. The author is responsible for the content.
Reproduction
Ten-member package includes editable source, exact standard-library sixth-order and additional twelfth-order ordered verifiers, independent sixth-order Arb reconstruction, and a separate twelfth-order derivative verifier. All four pass normally and with Python -O. Optional python-flint 0.9.0; README.txt gives commands and MANIFEST.sha256 covers all other public members.
References and related resources
- Watson, primitive Markovian cycles I, arXiv:2602.01248v1 (literal version corrected)
- Watson, primitive Markovian cycles II, arXiv:2602.06080v1
- Belton, Guillot, Khare and Putinar, Preservers of totally positive kernels and Pólya frequency functions
- Michałowski, certified failure for the different de Bruijn–Newman kernel, arXiv:2602.20313v1