slug: parry-string-attractor-minimum bibkey: gheeraertromanastipulanti2023attractors doi: null url: https://arxiv.org/abs/2302.13647v2 triage: theorem motivation_gids:

• D5/S1/Words/Powers/WordPower

Original cyclic-morphism string-attractor minimum

Problem

For k >= 2, let c be a length-k vector of natural coefficients with c(0) >= 1 and c(k-1) >= 1. Set d to c with the last entry decreased by one and assume d is lexicographically at least every cyclic rotation, allowing equality. The original substitution sends a < k-1 to c(a) zeros followed by a+1, and sends k-1 to c(k-1) zeros.

Let u be its fixed point starting with zero and U(n) the length of the n-th iterate of that singleton. Conjecture 42 states that for every m >= 1, the prefix of length m has minimum attractor size i+1 when 0 <= i <= k-2 and U(i) <= m < U(i+1), and size k when m >= U(k-1). The minimum ranges over every position subset; each nonempty factor needs an equal occurrence wholly inside the same prefix crossing a selected original position.

Evidence

The preregistration is issue 11754. The exact source is Gheeraert–Romana–Stipulanti, arXiv:2302.13647v2, Conjecture 42, under the original working hypotheses and Definition 2. The unrestricted attractor definition opens Section 4; its one-based positions correspond to Lean zero-based positions by adding one.

Motivation

The existing word-power API reads repeated literal blocks modulo their original length. This makes the source morphism recurrence a natural setting for constructing small hitting sets of original factor occurrences.

Gap

The paper-level endpoint intervals leave residual prefixes below the next canonical interval. Those prefixes require a bounded actual replacement window through the full original coefficient recurrence. Published supplier statements are not accepted mathematical premises.

Route

The Lean result uses the literal morphism and unrestricted minimum. Nesting and U(n) >= n+1 establish fixed-point-prefix semantics. Weak cyclic maximality proves all-level periodic interiors. Canonical endpoint intervals, a finite full-k block scan and exact window transport supply the upper bound; singleton factors supply the unrestricted lower bound.

Falsifier

A valid counterexample must use the original coefficients and letters and one positive prefix length whose unrestricted minimum violates the stated value. A coded-word, primitive-only, assumed-periodicity or finite-box result does not settle this problem. Equal rotations, proper powers, zero middle coefficients, empty residual intervals and r=0 remain in scope.

The original authors retain credit for the conjecture. The mathematical implementation claims no worldwide novelty or independent review status.

Triage

theorem. The complete quantified statement is the target of the retained Lean result; required independent review, registration admission and canonical publication remain separate obligations.

Proved in the implementation: weak cyclic maximality controls every iterate interior, and an actual suffix window transfers the missing residual factors. The unrestricted singleton lower bound makes the constructed upper bounds sharp. The finite-word suppliers are owned by D5/S1/Words/Attractors/FiniteWordAttractors, the coherent-prefix induction and residual scan by D5/S1/Words/Attractors/PeriodicPrefixAttractors, and the literal morphism bridges and sole full result by D5/S1/Words/Attractors/CyclicMorphismAttractorMinimum. The same scan treats equal rotations and proper powers without a primitive-root assumption. No extension beyond the stated coefficient hypotheses is claimed.

ASSUMED-UNVERIFIED

The supplied preregistration and bounded prior-art conclusions do not certify worldwide priority or the current state of every external source. The source version is arXiv:2302.13647v2; later publication variants are outside this claim.