Convergence of the Shieh-Yang-Yu dotted-pattern machine

Michael Yang, Hansen Shieh, and Ashley Yu, Stack-Sorting with Dotted-Pattern-Avoiding Stacks, arXiv:2411.11914v2, Section 6, printed page 11, state:

Conjecture 6.2. All permutations in Sₙ for all n ≥ 1 are eventually mapped to a fixed point of the 21̇-machine after a finite number of iterations through the machine.

The preceding sentence asks whether this machine has any cycle of length at least two. The machine applies the dotted-pattern map first and West's stack-sorting map second.

Issue #8639 preregisters this first-tier external named conjecture with its source, full quantifiers, and literature-search boundaries. The public statement is ∀ n ≥ 1, ∀ w ~ List.range' 1 n, ∃ t : ℕ, (M^[t + 1]) w = (M^[t]) w, where M w = s (r w).

The source proves the fixed-point characterization and enumeration (Lemma 4.3 and Theorem 4.4), but does not prove that every periodic point is fixed. The arXiv API searches recorded in #8639 found no settlement; those searches are bounded and do not establish publication priority.

A valley is an entry strictly smaller than every preceding entry. A valley run starts at a valley and ends immediately before the next valley. The map r reverses each valley run, preserving the order of the runs. The map s implements West's pop/push/flush stack algorithm.

For every word without repeated entries, the reversed word is lexicographically no larger than the reversal of its image under M. The proof uses strong induction on word length and decompositions at minimal and maximal entries. Both component maps preserve the multiset of entries. The reachable orbit is therefore contained in a finite set of permutations. At a reachable word with maximal reversed word, weak monotonicity forces equality of potentials. Injectivity of reversal then forces equality of successive iterates.

A permutation whose orbit under the stated operational machine contains a cycle of length at least two would contradict the conclusion. A mismatch between the formal maps and the source's maps would invalidate the transfer to the source conjecture, independently of kernel checking.

The theorem D5/S1/Words/Patterns/ShiehYangYuMachineConvergence.result proves the fully quantified eventual fixed-point statement. Its five public definitions are IsValley, valleyRuns, r, s, and M. The live monotonicity lemma is private potential_weak; the result is unbounded and uses no finite enumeration or numerical certificate. The Scribe resolution kind is Proved.

theorem. This resolves Conjecture 6.2 for the source's Proposition 3.5 model of the 21̇-machine. Conjecture 6.1 is outside the result.

DEFINITION BOUNDARY — the source defines s_{21̇} through Baril's dotted patterns and PROVES the closed form of Proposition 3.5; this module takes that proved right-hand side (reverse each valley run) as the definition of r; the identification of the dotted-pattern stack with valley-run reversal is the source's theorem and is not formalized here.

No citation index was retrieved — whether a later paper settles Conjecture 6.2 is ASSUMED-UNVERIFIED beyond the arXiv API searches in #8639 and a ChatGPT Pro literature seat that found no citing work.