Biological unavoidability of finite-alphabet sequences

Which infinite finite-alphabet sequences are biologically unavoidable in Alexander’s labelled-population model? Populations obey finite-root, finite-outdegree, finite real-birthdate-sublevel and incoming-label conditions; eventual periodicity is known to suffice.

Why it matters and what counts as progress

Why it matters: A converse would classify unavoidable finite-alphabet sequences exactly by eventual periodicity in Alexander’s population model.

What counts as progress: Construct an avoiding population for every non-eventually-periodic sequence, or find a further unavoidable sequence within this model.

How the problem developed

2013 · Partial result [1]
Samuel Alexander Proved that every eventually periodic sequence is biologically unavoidable and exhibited avoidable sequences, leaving open whether any non-eventually-periodic sequence could be unavoidable.
18 Oct 2026 · Reported claim [2]
Codex construction project Constructed an avoiding population for each non-eventually-periodic sequence over a finite alphabet; together with Alexander’s theorem this gives the eventual-periodicity classification.

AI contribution

A classification of biologically unavoidable sequences

For every non-eventually-periodic sequence over a finite alphabet, the project constructs an infinite labelled population satisfying Alexander’s finite-root, finite-outdegree, finite real-birthdate-sublevel and incoming-label conditions but with no matching infinite path from any start. With Alexander’s earlier positive theorem, biological unavoidability is equivalent to eventual periodicity.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean 4.32.0-rc1 with pinned mathlib dependencies and endpoint/axiom audit)

Human understandingFew