bibkey: filimonovapuzynina2026abelianperiodicity authors: Arina Filimonova; Svetlana Puzynina year: 2026 title: “On abelian periodicity of purely morphic words” doi: null url: https://arxiv.org/abs/2605.30306v1 claim: “Section 4 asks for an upper bound on the iteration K in the primitive binary rank-one criterion for abelian periodicity.” strata_touched:

• D5/S1/Words/RankOneMorphismIterationBound license: citation-only triage: anchor

Abelian periodicity of purely morphic words

Arina Filimonova and Svetlana Puzynina, On abelian periodicity of purely morphic words, arXiv:2605.30306v1 (28 May 2026), https://arxiv.org/abs/2605.30306v1.

The paper’s Theorem 1 gives a criterion for the primitive binary rank-one branch (the source notation has theta₂ = 0). For a prolongable morphism, the fixed word is abelian periodic exactly when an iterate admits a split f^K(a)=uv, f^K(b)=u'v' whose two cyclic rotations have abelian-equivalent period blocks. The source asks in Section 4 whether the iteration variable K has an upper bound making this criterion algorithmic. The source’s abelian periodicity permits an arbitrary finite preperiod.

The formalization keeps the actual binary images, exact Parikh vectors, the four-word cyclic witness, and arbitrary preperiods. It proves the explicit bound K ≤ 2^(|f(a)|+|f(b)|) for every nonerasing prolongable primitive rank-one morphism and supplies an executable finite checker with the same semantics. The bound is an iteration bound; it is not an incidence-matrix bound and does not restrict to pure periodicity.

Verified locator

URL: https://arxiv.org/abs/2605.30306v1

Locator: arXiv v1, 28 May 2026, Section 4. The open question is: “is there an upper bound on M from Theorem 1 making our criterion algorithmic?” The Lean source uses K for this iteration variable.

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