Exact Thue–Morse matching heights in the binary avoiding population

In Alexander’s binary avoiding population with the Thue–Morse target, what is the greatest matching-path length L(v) from each natural starting vertex v, including zero?

Why it matters and what counts as progress

Why it matters: An exact all-start formula strengthens the earlier statement that each matching height is finite for non-eventually-periodic targets.

What counts as progress: An attained formula for every v, with its case distinctions justified, or a finite digit recurrence computing all heights exactly.

How the problem developed

September 2026 · Baseline result [1]
Samuel Alexander Established the binary avoiding-population setting and a finite-height classification for non-eventually-periodic targets; this did not provide the exact Thue–Morse height at every start.
25 Sept 2026 · Formal proof [1][2]
Codex and parallel research agents, under the project owner’s direction Reported an exact maximum matching-path length for every Thue–Morse starting vertex, including zero, with a seven-case formula and finite digit recurrence; the repository reports a Lean proof.

AI contribution

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.

AI involvementAI-centered

ImportanceModest

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

Human understandingFew