HEXAGONbeta

An explicit upper-bound certificate for the essential minimum of the Zhang–Zagier height

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 give an exact certificate proving C82<0.2534449C_{82}<0.2534449 for the essential minimum of h(α)+h(1−α)h(\alpha)+h(1-\alpha), where hh is the absolute logarithmic Weil height. We use Doche's limit-point method, building on Grinsztajn's factor library and, in particular, Épi's unequal-multiplicity construction. We retain Épi's denominator factors, add Doche's P7P_7, and add 25 numerator factors found by LP-dual-weighted evaluation lattices with LLL. Joint linear-program re-optimisation is followed by exact rational rounding. Two separately written ball-arithmetic checkers certify the bound; a further adversarial computational audit accepts the certificate.

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, two separately written checkers and control certificates are in anc/ and at https://github.com/rpinho/c82-height-certificate (Zenodo 10.5281/zenodo.23483536).

References

  1. Burgos Gil, José; Menares, Ricardo; Qu, Binggang; Sombra, Martín. Closing the gap around the essential minimum of height functions with linear programming. 2026. https://arxiv.org/abs/2601.18978v2. arXiv:2601.18978v2 [math.NT]link
  2. Doche, Christophe. On the spectrum of the Zhang–Zagier height. Math. Comp., vol. 70, no. 233, pp. 419–430. 2001. https://doi.org/10.1090/S0025-5718-00-01183-2DOI
  3. Doche, Christophe. Zhang–Zagier heights of perturbed polynomials. J. Théor. Nombres Bordeaux, vol. 13, no. 1, pp. 103–110. 2001. https://www.numdam.org/item/JTNB_2001__13_1_103_0/link
  4. Épi Team (BakeLab). Épi Results: Zhang–Zagier essential minimum. 2026. https://github.com/BakeLab/Epi-results/tree/main/records/zhang-zagier/zhang_zagier_c82, commit \texttt97fa6a420f417458e11723c8bceb70f1529e447a; accessed 16 October 2026. GitHub repository, Bake AIlink
  5. Flammang, V. On the Zhang–Zagier measure. Int. J. Number Theory, vol. 14, no. 10, pp. 2663–2671. 2018. https://doi.org/10.1142/S1793042118501609DOI
  6. Grinsztajn, Max. Code and certification data for an improved C82 Zhang–Zagier bound. 2026. https://github.com/maaxgrin/zhang-zagier-c82-bound, commit \textttcb88f92434f649a1588b31ae91f67852830c6f77; accessed 16 October 2026. GitHub repositorylink
  7. Tao, Terence; contributors. Optimization constants in mathematics: Essential minimum of the Zhang–Zagier height (constant 82a). https://teorth.github.io/optimizationproblems/constants/82a.html; accessed 16 October 2026. Online tablelink
  8. Zagier, Don. Algebraic numbers close to both 0 and 1. Math. Comp., vol. 61, no. 203, pp. 485–491. 1993. https://doi.org/10.2307/2152970DOI
  9. Zhang, Shouwu. Positive line bundles on arithmetic varieties. J. Amer. Math. Soc., vol. 8, no. 1, pp. 187–221. 1995. https://doi.org/10.1090/S0894-0347-1995-1254133-7DOI

Version history

  1. v1Submitted by Ricardo PinhoInitial depositCurrentOct 21, 2026