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Which sequences are biologically unavoidable?

None: the biologically unavoidable sequences are exactly the eventually periodic ones. For every non-eventually-periodic sequence over a finite alphabet there is an explicit avoiding population; for binary sequences it has two roots, two parents per non-root with complementary labels and at most two children per vertex, and the avoidance argument is a decreasing offset along any matching path. An indicator projection and a finite copy-lift extend this to every finite alphabet, including vertex-gendered witnesses. Combined with Alexander's positive theorem, reproduced with a boundary repair, this settles the last question of his section 6. The paper also characterises which members of the binary family are universal and draws finite-observation and computability corollaries; the ordinal and universal-object questions of the same section are not claimed.

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Occurred: Sep 10, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Which sequences are biologically unavoidable? · Biologically unavoidable sequences

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VibeMathed record: Which sequences are biologically unavoidable?

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Checked here on 22 September 2026. The construction was read in full and verified by hand; it is half a page. For a binary target s, take vertices N with birthdate v, and for v >= 2 the edges v-1 -> v labelled r(v) and v-2 -> v labelled 1-r(v), where r(2j) = s(j) and r(2j+1) = 1-s(j). A path spelling s has v_k >= 2k, so the offset v_k - 2k is a nonincreasing nonnegative integer and stabilises; from then on every step is +2 and the labels force s(k) = s(k+e+1) or s(k) = 1-s(k+e+1), so s is eventually periodic. Finite alphabets reduce to the binary case through an aperiodic indicator and a copy-lift. The Lean endpoint finite_alphabet_real_avoidance was compared with Alexander's Definition 1 clause by clause: finite roots, finite children, finite real sublevel sets, strictly increasing birthdates, infinite vertex set, a parent of each label at every non-root, at most one label per edge, and realisation from any starting vertex. Faithful. 297 lines, no sorry, no axiom, no native_decide; the author's log shows the standard axioms. There is no CI and the site did not rebuild it, so Lean-checked; a GitHub Actions build would lift it. The positive direction is Alexander's published Theorem 10.