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Codex project gives exact Thue–Morse matching heights

For the Thue–Morse target in Alexander’s binary avoiding population, the project gives an attained exact maximum matching-path length L(v) for every natural starting vertex v, including zero, through a seven-case formula based on the factorization of ⌊v/2⌋+1 and adjacent digit-parity bits. It also proves a finite digit recurrence.

Sources for this record

What was already known

Alexander’s population model and an earlier Codex classification showed that every start has finite matching height for non-eventually-periodic targets. They did not give this exact all-start Thue–Morse formula.

AI involvement AI-centered

The owner directed the follow-on investigation. The repository credits Codex and parallel research agents with the path-length work, candidate table, digit-state representation and universal proof.

The provenance identifies Codex and parallel agents as developers of the mathematical extension and formal proof under human direction, while the candidate-table origin is only coarsely documented.

Importance Modest

The formula adds a sharp quantitative description for one target in the established avoiding population.

Limitations

The repository reports a cited proof and hosted verification run, but no independent human proof check or exact split of the earlier candidate table between agents and people is documented.

Technical verification Formal (Lean 4 with cited repository toolchain and GitHub Actions build)

The source reports a Lean proof of attained maxima and upper bounds at every starting vertex, with no added axioms and an explicit selected axiom report. The September 25 CI run passed. Finite experiments were discovery aids, not the verification basis.

Formal checking

Compile selected Lean declarations with cited imports; inspect reported axiom footprint and passing CI run

Checker
Lean compiler/kernel in the project's September 25 GitHub Actions run
Scope
FullHeight.height_formula, height_closed_form and height_isMaximum for actual Thue–Morse population paths at every start
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

Default for undocumented human understanding: the detailed trajectory explanation is in the AI-developed project, and no independent human account is identified.

Correction and publication history

2026-10-18 · publication

The formula adds a sharp quantitative description for one target in the established avoiding population.

Read this published revision