HEXAGONbeta

Degree Transport and a Smaller Cubic Coefficient for Reset Thresholds

Primarily AI-generated textHuman understanding: no partscs.FL — Formal Languages and Automata Theory

Contributed by SUNGBIN HONG ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

We prove that every synchronizing complete deterministic finite automaton on n≥2n\ge2 states has reset threshold at most 824249923n3+O(n2)\frac{8242}{49923}n^3+O(n^2). The cubic coefficient improves on Shitov's 8257/499238257/49923 by 5/166415/16641. The main ingredient is a degree-transport inequality for graphs of state pairs mergeable by short words. It relates a long gap between consecutive minimum-rank word lengths to a large corank achievable in at most n−1n-1 letters. Combining this relation with a discrete dual of Shitov's counting argument gives the improvement. An appendix makes the quadratic remainder explicit for every n≥2n\ge2. The argument applies to arbitrary finite alphabets and does not prove the quadratic Černý conjecture.

Provenance statement

The research and text are primarily AI-generated. OpenAI Codex (contributor-supplied model label: 6.1 sol) performed the research, including literature searches, exploratory calculations, proof development, programming and primary drafting in October 2026. Google Gemini (Gemini 3.1 Pro) supplied a review used as editorial input, not as a proof certificate. OpenAI Codex (6 astra) handled the November 2026 revision and manuscript completion, including formatting and bibliographic checks, the explicit quadratic remainder and computation reruns. These model labels were reported by the human submitting contributor and have not been independently verified as provider snapshot identifiers. HONG SUNGBIN directed the research request, supplied the Gemini review and requested submission. The submitting contributor claims no personal checking or understanding of any part of the proof; human understanding is none. No external human peer review or proof-assistant verification has been performed. AI review passes and finite diagnostics do not certify the universal theorem or priority. See Section 6 and anc/ai_usage.json.

Tools used

OpenAI
CodexVersion 6.1 sol
Google
GeminiVersion Gemini 3.1 Pro
OpenAI
CodexVersion 6 astra

References

  1. P. Frankl, An extremal problem for two families of sets, European Journal of Combinatorics 3 (1982), 125--127. doi:10.1016/S0195-6698(82)80025-5 (https://doi.org/10.1016/S0195-6698(82)80025-5).DOI
  2. J.-E. Pin, On two combinatorial problems arising from automata theory, in Combinatorial Mathematics , North-Holland Mathematics Studies 75 (1983), 535--548. doi:10.1016/S0304-0208(08)73432-7 (https://doi.org/10.1016/S0304-0208(08)73432-7).DOI
  3. Y. Shitov, An improvement to a recent upper bound for synchronizing words of finite automata, Journal of Automata, Languages and Combinatorics 24 (2019), 367--373. arXiv:1901.06542 (https://arxiv.org/abs/1901.06542); doi:10.25596/jalc-2019-367 (https://doi.org/10.25596/jalc-2019-367).arXivDOI
  4. M. Szykuł a, Improving the upper bound on the length of the shortest reset word, in STACS 2018 , LIPIcs 96 , 56:1--56:13 (2018). doi:10.4230/LIPIcs.STACS.2018.56 (https://doi.org/10.4230/LIPIcs.STACS.2018.56).DOI
  5. M. Szykuł a, Synchronizing automata: Open problems, EPTCS 451 (2026), 33--47. arXiv:2608.24245 (https://arxiv.org/abs/2608.24245); doi:10.4204/EPTCS.451.3 (https://doi.org/10.4204/EPTCS.451.3).arXivDOI

Version history

  1. v1 · 10 pagesSubmitted by SUNGBIN HONGInitial depositCurrentNov 07, 2026