The candidate asks for circular avoidance. For a word of length n, it forbids adjacent abelian-equivalent h-blocks only when 2h <= n. Peltomäki and Whiteland define the stronger cyclic condition by forbidding every such pair with h < n in the periodic word w^omega.

Their Theorem 1.2 proves A_infinity(4)=2. Its witnesses are phi^r(01), where phi is Keränen's 85-uniform abelian-square-free morphism, so arbitrarily long four-letter words satisfy the stronger cyclic condition. These lengths form a sparse sequence and give no cofinite length interval.

Section 6 reports that their computer experiments found no counterexample to A(4)=2 among lengths below 150. Any cyclic witness located in that experiment is also a circular witness. The article provides no witness table, code, or certificate for the bounded search, so this record preserves the authors' wording and evidence level. The exact mathematical theorem remains the arbitrarily-long result.

For every n in 1,...,36, the replay artifact contains an explicit word w_n over {0,1,2,3}. It tests every start s modulo n and every integer h with 1 <= h <= floor(n/2). The test compares the four-component Parikh vectors of w_n[s:s+h] and w_n[s+h:s+2h], with indices read modulo n.

The witness table has SHA-256 `53a8546a80f5700a254e23bfdbb005539a4b596848401919f92f8046b1f30054` under the artifact's canonical JSON encoding. The bounded result covers each integer length in the stated range. It supplies no construction for lengths above 36 and leaves eventual existence open.

A restricted-growth word fixes the first letter as 0 and permits each later letter to be at most one more than the largest earlier letter, capped at 3. Exactly one restricted-growth word represents each orbit under permutation of the four letter names.

Complete depth-first enumeration pruned a branch whenever its newest suffix was a linear abelian square. At full length it applied the circular verifier. The resulting orbit counts for n=1,...,16 are `[1,1,1,3,5,14,21,16,12,20,66,177,208,210,405,160]`.

Weighting every orbit by the number of injections of its used letters into a labeled four-letter alphabet gives `[4,12,24,72,120,336,504,384,288,480,1584,4248,4992,5040,9720,3840]`. A direct enumeration of all 4^n labeled words independently agrees for n=1,...,8. Two full replays produced the same deterministic count digest `33442591f6a555df5e58ad8d5eb444f0e2499e36f3b9a7c440af0a7ec69421e0`.

The exact formulation is Wilson and Wood's conjecture that the anagram-free chromatic number of cycles is at most four with finitely many exceptions. A color word around C_n is an anagram-free coloring exactly when every circular factor of even length at most n has unequal Parikh vectors in its two halves.

The audit checked Wilson and Wood's journal PDF and arXiv:1607.01117, Peltomäki and Whiteland arXiv:2006.06307v1 and v2, the open journal manuscript, Crossref metadata, Semantic Scholar and OpenAlex citation lists, the 2023 survey arXiv:2207.09937v2, and the later citing papers arXiv:2008.08125, arXiv:2112.05347, and arXiv:2509.20773. The later citations use the 2020 article as general background and do not discuss the cycle conjecture. Exact web searches used the phrases `anagram-free chromatic number cycles`, `circular abelian-square-free`, `four-letter word circularly`, and the article title.

TheoremDB orient resolved problem 2820 exactly and returned zero attached research records. Problem-directory searches for `circular abelian square`, `abelian square free`, and `cyclic abelian` found only this target. Research-memory search returned no overlapping attempt. Citation indexes can omit manuscripts and unindexed work. This dated audit is an inconclusive literature search and supplies no mathematical evidence of openness.

For each target length n, the search began with 0. It extended only restricted-growth words, so every alphabet-permutation orbit remained represented. After each appended letter, it rejected any suffix whose two halves had equal four-letter counts. At depth n it tested every circular factor under the candidate convention and stopped at the first witness.

The runs for n=1,...,36 visited 7,302,877 nodes, made 4,629,394 linear-suffix prunes, and rejected 847,487 full words at the seam. Their summed measured runtime was 478.900 seconds. Stopping after a witness gives an exact positive decision for each covered n and supplies no count of all witnesses above the separately enumerated range n<=16.

The replay built the complete insertion graph on every alphabet-permutation orbit through length 16. An edge inserts one of four letters at one of n cyclic gaps, canonically relabels the result, and applies the exact circular verifier.

The length-eight layer has 16 valid orbits. Exhausting all 32 labeled insertion choices per representative produces zero valid edges to the 12 length-nine orbits. Thus every length-eight circular abelian-square-free word, under every alphabet relabeling, fails every one-letter insertion. A proof that grows a word by one letter at every step cannot cross from length 8 to length 9.

The same replay tested the selected DFS witness at each n through 36. For every n=8,...,36, all four fixed-seam appends fail. At n in `{8,17,18,19,21,24,25,26,27,28,29,33,34,35,36}`, all 4n cyclic-gap insertions fail. The later failures apply to the selected witnesses. At n=8 the conclusion covers every witness orbit.

The scan used the 85-uniform Keränen image printed in Peltomäki and Whiteland, Section 3: `phi(0)=0120232123203231301020103101213121021232021013010203212320231210212320232132303132120`. The images phi(1), phi(2), and phi(3) were obtained by adding 1, 2, and 3 modulo 4 to every symbol. Iterating twice on 0 produced a 7,225-letter prefix with SHA-256 `4a86ca2cd0821b5f67313b1ae8a58e88d0db1cef4bdf06bc8e025d24645c249a`.

For each n=36,...,100, the scan checked factors from left to right. It stopped at the first circular witness. When no witness appeared, it exhausted all 7,226-n windows. Witnesses occurred at `{36,39,40,41,44,46,47,48,50,54,55,58,60,63,66,67,70,79,81,87,89,90,95,100}`. Every explicit witness was replayed by the packet artifact. The other 41 lengths have no witness among factors of this finite prefix. That prefix-specific absence says nothing about all four-letter words.

The computation tested 296,943 windows in 455.913 seconds. A separate full replay tested the same 296,943 windows and matched every stable row, including witnesses and final failure certificates. A follow-up scan of the first 20,000 windows of phi^3(0) at 17 missing lengths found no new witness before its stated per-length budget. The repeated failure suggests that short factors of this morphic word have a persistent seam profile.

The encoding used integer variables x_i in {0,1,2,3}, fixed x_0=0 and x_1=1 for alphabet symmetry, and added one constraint for every cyclic start s and h with 2h<=n. Each constraint was a four-way disjunction saying that some letter count differs between the adjacent h-blocks.

Z3 5.0.0 returned `unknown` at the 60,000 ms timeout for each of n=36,40,50,60. The measured solver times were 60.047, 60.062, 60.150, and 60.236 seconds. Existing independently checked witnesses show that these are solver timeouts. They carry no satisfiability or nonexistence conclusion.

Reusable residue: the direct pseudo-Boolean expansion created 180,081 Boolean variables at n=36 and 847,598 at n=60 according to solver statistics. A later encoding should exploit prefix Parikh differences, conflict-specific lazy cuts, or a finite automaton instead of expanding every color count eagerly.

A linear abelian-square-free word can fail the circular condition only through a factor that crosses the chosen seam. Write P_i for the prefix Parikh vector, extended by P_{i+n}=P_i+P_n. A circular violation at start s and half-length h is exactly `P_s - 2 P_{s+h} + P_{s+2h} = 0`. This makes the seam obstruction a boundary-profile question.

A next experiment should enumerate several structurally varied witnesses per length. For each witness, store all nonzero boundary second differences by h and the shortest edit that changes them. Cluster the profiles, search two-block and three-block splices of Keränen images, and seek a finite transition graph whose accepted path lengths contain a cofinite set. Every transition needs an exact verifier and an interval or residue-class coverage certificate.

A successful finite system would separate the proof into a local seam lemma, a semigroup or automaton argument covering all large lengths, and a finite checked list of gaps. The selected-witness insertion failures and sparse phi^2 window successes give concrete regression cases.

The program uses exact `Counter` equality. Its circular verifier loops over every cyclic start and every half-length h with 2h <= n. The enumeration uses restricted-growth representatives under alphabet permutation, with a direct 4^n labeled cross-check through n=8. It builds every one-letter insertion edge between the complete orbit layers through n=16. A second pass tests all four letters at the fixed seam and all 4n pairs of a cyclic gap and inserted letter for each stored witness through n=36.

Join `source_lines` with newline, append one final newline, save as `casf4_replay.py`, then run the recorded command. Eight runs produced byte-identical standard output. The last three reconstructed the program directly from the packet's stored `source_lines`; the final run used Python isolated mode. No network, random number generator, floating-point arithmetic, or external service is used.