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Research Article · Preprint Mathematics

A finite-order obstruction to total positivity of an arithmetic theta trace kernel

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Classical and Real Analysis
  • Number Theory and Special Functions
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

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; no conclusion about the Riemann hypothesis follows.

Keywords total positivity · Pólya frequency functions · arithmetic theta trace kernels · Toeplitz minors · exact rational certificates · Gamma–zeta transform

Scope of the work

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. No smallest failing order/global PF5, original physical-theta sign, off-critical-line zeta zero or RH consequence. Unseen revisions and conditional companion results are outside this correction.

Reproduction notes

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.

The manuscript

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Author 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.

References and related publications

  1. Watson, primitive Markovian cycles I, arXiv:2602.01248v1 (literal version corrected)
  2. Watson, primitive Markovian cycles II, arXiv:2602.06080v1
  3. Belton, Guillot, Khare and Putinar, Preservers of totally positive kernels and Pólya frequency functions
  4. Michałowski, certified failure for the different de Bruijn–Newman kernel, arXiv:2602.20313v1

A stable scholarly record

Cite this edition

Scott Schubr. (2026). A finite-order obstruction to total positivity of an arithmetic theta trace kernel (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/finite-order-obstruction-total-positivity-arithmetic-theta-trace-kernel/v/1

BibTeX
@techreport{luminary_finite_order_obstruction_total_positivity_arithmetic_theta_trace_kernel_v1,
  title = {A finite-order obstruction to total positivity of an arithmetic theta trace kernel},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/finite-order-obstruction-total-positivity-arithmetic-theta-trace-kernel/v/1}
}

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