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An explicit upper-bound certificate for the Schur–Siegel–Smyth trace constant

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.OC — Optimization and Control

Contributed by Ricardo Pinho ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

We document an explicit certificate proving C86<1.8117251C_{86}<1.8117251 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers. The construction uses the primal problem of Orloski and Talebizadeh Sardari, with their four-polynomial set enlarged to eleven. A discretised solution is rounded to a rational step density, then normalised to a probability measure. Exact rational calculations and rigorous ball-arithmetic inequalities verify Smith's sufficient criterion for an arithmetic measure. An independently written agent checker accepts the certificate and rejects four corrupted controls. This note records the proof, data provenance, reproduction results and limitations of the previously released certificate; it makes no optimality claim.

Provenance statement

Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared the certificate, checking software, and publication materials. This does not claim a line-by-line human verification of the proof or software. The certificate and an independently written checker are in anc/ and at https://github.com/rpinho/c86-trace-certificate (Zenodo 10.5281/zenodo.23486957).

References

  1. Niedbala Giraudin, David. A new lower bound for the Schur–Siegel–Smyth trace problem. 2026. arXiv:2609.29298v1 (https://arxiv.org/abs/2609.29298v1), 24 September 2026; checked 19 October 2026arXiv
  2. Orloski, Bryce. Upper bounds on the Schur–Siegel–Smyth trace problem. 2025. Seminar abstract announcing 1.81618; checked 19 October 2026. Gordon seminar, Hebrew University of Jerusalem (https://mathematics.huji.ac.il/node/3189563), 2 January 2025link
  3. Orloski, Bryce; Talebizadeh Sardari, Naser; Smith, Alexander. New lower bounds for the Schur–Siegel–Smyth trace problem. Mathematics of Computation, vol. 94, no. 354, pp. 2005–2040. 2025. Published online in 2024; doi:10.1090/mcom/4004 (https://doi.org/10.1090/mcom/4004); arXiv:2401.03252v3 (https://arxiv.org/abs/2401.03252v3)arXivDOI
  4. Orloski, Bryce Joseph; Talebizadeh Sardari, Naser. A quantitative converse of Fekete's theorem. 2024. arXiv:2304.10021v2 (https://arxiv.org/abs/2304.10021v2), 25 March 2024; checked 19 October 2026arXiv
  5. Pennsylvania State University, Department of Mathematics. Ph.D. graduates: Bryce Orloski, 2025. 2025. Official dissertation title and year; checked 19 October 2026. Thesis contents not inspected. Official departmental Ph.D. list (https://science.psu.edu/math/graduate/phd-alumni)link
  6. Pinho, Ricardo. C86: improved upper bound 1.8117251. 2026. Merged 18 October 2026; checked 19 October 2026. GitHub pull request 227 (https://github.com/teorth/optimizationproblems/pull/227)link
  7. Pinho, Ricardo. A certificate for C86≤1.8117251C_{86}\le1.8117251 (Schur–Siegel–Smyth trace constant). 2026. GitHub repository, commit 116a58ff99681226ce6299d4a02eeff1e7d3c5c7 (MIT); checked 19 October 2026. GitHub: rpinho/c86-trace-certificate (https://github.com/rpinho/c86-trace-certificate/tree/116a58ff99681226ce6299d4a02eeff1e7d3c5c7)link
  8. Smith, Alexander. Algebraic integers with conjugates in a prescribed distribution. Annals of Mathematics, vol. 200, no. 1, pp. 71–122. 2024. doi:10.4007/annals.2024.200.1.2 (https://doi.org/10.4007/annals.2024.200.1.2); arXiv:2111.12660v2 (https://arxiv.org/abs/2111.12660v2), Theorem 1.5 and Proposition 5.15arXivDOI
  9. Tao, Terence; contributors. Optimization constants: 86a, Schur–Siegel–Smyth trace constant. 2026. Live page and raw source checked 19 October 2026. Online table, entry 86a (https://teorth.github.io/optimizationproblems/constants/86a.html)link

Version history

  1. v1Submitted by Ricardo PinhoInitial depositCurrentOct 21, 2026