A sixth-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(-n^2e^(-x)). At the increasing nodes x_i=-log(5/2)+i/100 and y_j=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(-α m^2e^(-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
An exact negative sixth-order minor of the full infinite arithmetic trace kernel, complete omitted-tail certificates and related finite-mixture/transform obstructions.
Disclosures
OpenAI Codex assisted with literature discovery, mathematical development and checking, reproducibility code and manuscript preparation. The author is responsible for the content. Internal AI-assisted checks and Archive posting do not constitute external peer review or formal proof verification.
Reproduction
8-page manuscript and 10 source-package members. Compile main.tex with Tectonic. Run verify_rational.py and verify_order_twelve.py with standard-library Python; install requirements.txt for the separate python-flint 0.9.0 full-series checks verify_order_six_arb.py and verify_arb.py. Each checker bounds the complete tail within its stated scope.
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