What was already known
Alexander proved eventual periodicity suffices and asked whether any other sequences are unavoidable. His positive theorem remains separate human work.
AI involvement
A human collaborator selected and directed the problem. OpenAI Codex supplied the new construction, proof, formalization and exposition; no human mathematical check is documented.
The explicit provenance assigns the new avoiding-population construction and proof to Codex under human research direction, exceeding mere formalization or editing.
Importance
The new avoiding-population construction supplies the missing converse for finite alphabets, subject to its stated model and priority.
Limitations
The pinned Lean check covers the new negative construction, not Alexander’s positive theorem or later computability claims. The mapping to Alexander’s model was audited by the same AI system, and independent human review and literature priority remain open.
Technical verification
The selected negative theorem has a Lean endpoint for arbitrary finite alphabets and real birthdates, and the release's cited verification record reports successful compilation of the local modules and endpoint audit. The full classification is not an end-to-end Lean theorem because Alexander's positive direction remains prose; Formal here applies to the new negative construction only. No human technical check is documented.
Formal checking
Fresh local-module compilation with pinned dependency artifacts, printed endpoint audit and PASS verification record
- Checker
- Lean compiler/kernel in the project's recorded local verification
- Scope
- BiologicalAvoidance.finite_alphabet_real_avoidance: an admissible real-birthdate avoiding population for every non-eventually-periodic finite-alphabet target
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The detailed proof and audit are AI-produced, and the human collaborator's conceptual explanation of the offset argument is not documented. Few is the evidence default, not a finding of incomprehension.
Correction and publication history
The new avoiding-population construction supplies the missing converse for finite alphabets, subject to its stated model and priority.
Read this published revision