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A 256-step certificate for Turán's pure power-sum constant

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Ricardo Pinho ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

We give an exact-input certificate for the upper bound C_42 <= 0.6888863712 in the pure power-sum problem. The construction refines the step profiles of Griego, Röhrig and Plata to 256 middle blocks, without changing their coefficient-cancellation mechanism. It improves the strongest public proposal located in our 22 October 2026 audit, 0.688970, by 0.0000836288. An unmodified public interval checker and a separately written Arb checker verify the strict finite inequalities. The asymptotic implication is proved in the note; no explicit finite threshold, optimality, human peer review or formal verification is claimed.

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, checkers and publication materials. This does not claim a line-by-line human verification of the proof or software. The strict finite inequalities are verified by an unmodified public interval checker and a separately written Arb checker, with mutation/control runs rejected as intended. Repository: https://github.com/rpinho/c42-certificate (commit a8a9bc309bd1992413fa07595bd60c5e3fa1b65b); the certificate is also included in anc/.

Tools used

OpenAI
CodexVersion GPT-6.1 Sol
FLINT
Arb ball arithmetic (python-flint)

References

  1. T. Tao and contributors, Optimization constants in mathematics, 42a , https://teorth.github.io/optimizationproblems/constants/42a.html.link
  2. A. Biró, An upper estimate in Turán's pure power sum problem , Indag. Math. (N.S.) 11 (2000), 499--508.
  3. S. Griego, An improved asymptotic certificate for Turán's pure power sum constant , v1.0.0, https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0.link
  4. A. Röhrig, Verification certificates for C42 and C51 , 9 September 2026, commit ed6387399cf6de02eaa4cf10c43787633496619a, https://github.com/AndreRoehrig/optimization-constants-certificates.link
  5. K. Plata, Proposed six-level bound for Turan's pure power-sum constant , public review draft, https://github.com/krplatz/turan-c42-six-level-review.link

Version history

  1. v1Submitted by Ricardo PinhoInitial depositCurrentOct 22, 2026