The 8/3-plus stability conjecture for power-free binary morphism lengths
Shallit, Shur and Zorcic ask for the phase transitions in admissible lengths of uniform binary morphisms preserving repetition avoidance. In Section 4 of Power-free complementary binary morphisms, they conjecture that the set of admissible lengths stays constant whenever , with another transition at the strict threshold . Here a morphism is -free if it sends every -free binary input word to an -free output; the plus convention permits exponent exactly . Does this length-set stability hold, and is the strict endpoint different?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Combinatorics on words; repetition-free morphisms
- Posed by
- Jeffrey Shallit, Arseny Shur and Stefan Zorcic, Power-free complementary binary morphisms, arXiv:2310.15064v3 (8 December 2023), Section 4; JCTA 207 (2024), 105910.
- Year posed
- 2023
- Years open
- 3y
- Solved
- 2026-10-04
- Model
- OpenAI Codex
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 6 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
For every extended-real threshold , the admissible positive lengths of -free uniform binary morphisms are exactly all integers except and . Every admitted length has a complementary witness, ; the broader binary class has the same length set by the credited prior impossibility theorem. At the strict threshold , length is impossible, whereas it is possible at . The decisive bound for the prior endpoint family is for every cubefree input. A length-12 seed covers the remaining doubling family through the established finite sufficient criterion. The proof includes finite and infinite admissible inputs with the stated critical-exponent convention. It does not classify every preserving morphism or settle all transitions in the lower interval . Equal zero/one counts in one image are not assumed.
What the AI did
OpenAI Codex produced the central strengthening of the existing boundary-marker argument, the complete length coverage using a new fixed length-12 seed, and the strict length-five endpoint obstruction. It also wrote exact finite certificate programs. A separate AI review checked the uniform proof and independently implemented the seed/endpoint controls. The prior clipped Thue-Morse constructions, cubefreeness theorem, exceptional-length impossibility result and Kobayashi sufficient criterion are explicitly credited. No human expert endorsement is claimed.
Verification
Re-checked by this site on 4 October 2026. The submission's scripts passed (22 seed inputs, 128 length-five maps, four marker cases). A separate brute force written here found no 5-uniform binary morphism surviving the strict 8/3 test on inputs up to length 6, exactly four maps surviving at (8/3)+ on inputs up to length 8 (the paper's endpoint family), and no violation of ce(h(w)) <= max(8/3, ce(w)) for the length-five endpoint map and the length-12 seed over all 1,158 cubefree inputs up to length 13. These are finite checks; the all-length, all-input statement rests on the written proof, which was read here but not line by line. The review in the package is by the same AI agent and is not independent. No Lean formalisation.
Sources
Submitted by ZestyDingo473 on