slug: chaffin-sloane-gijswijt-merge-refutation bibkey: chaffin2013curling doi: 10.48550/arXiv.1212.6102 url: https://arxiv.org/abs/1212.6102v3 triage: theorem motivation_gids:

• D5/S3/Combinatorics/ChaffinSloaneGijswijtMergeRefutation.result

Chaffin-Linderman-Sloane-Wilks Gijswijt Merge Refutation

Problem

Chaffin, Linderman, Sloane and Wilks, Journal of Integer Sequences 16 (2013), Article 13.4.3 = arXiv:1212.6102v3, Section 5, Theorem 23, state: “Assume the curling number conjecture is true. Let S be an initial sequence not containing a 1, let S^(e) be its ‘extension’ (defined in §1), and let S^(inf) be its infinite continuation. Then S^(inf) = S^(e) G.” Here G is Gijswijt’s sequence A090822, cn(S) is the largest k with S = X Y^k for nonempty Y, the continuation appends cn repeatedly, and the extension is the first prefix whose curling number equals one.

The section closes: “We do not know if the theorem is still true if S is allowed to contain a 1 but does not end with 1.” Section 6 lists this as open question 12: “The question implicit in the last sentence of §5.”

Motivation

The frozen theorem D5/S3/Combinatorics/ChaffinSloaneGijswijtMergeRefutation.result settles that question in the negative. The consequent of the weakened statement is false outright, so the weakened Theorem 23, read as an implication whose hypothesis is the curling number conjecture, is equivalent to the negation of that conjecture: a proof of the weakened theorem would refute it.

Gap

Issue 9218 records the screen carried out before write-up. An independent literature pass opened the journal article, the Chaffin-Sloane preprint arXiv:0912.2382v5, Sloane’s retrospective arXiv:2301.03149v2, both arXiv versions and the journal version of Levi van de Pol’s growth-rate work, the OEIS entries A090822, A093369, A094004, A216955 and A217209 with the OEIS curling-number index, and citing papers including Caveney, Dong and Shallit, arXiv:2608.15670v1. None records an answer to item 12. Major citation-index result pages were not reachable, so this is a bounded negative finding.

Route

Take S = 1 2. It contains a 1 and does not end with 1. Since cn(1 2) = 1 the tail length is zero and the extension is 1 2 itself, so the extension exists without appeal to the curling number conjecture. The continuation is computed term by term: cn(1 2) = 1, cn(1 2 1) = 1, cn(1 2 1 1) = 2, cn(1 2 1 1 2) = 1, cn(1 2 1 1 2 1) = 2 because 1 2 1 1 2 1 = (1 2 1)^2. So the continuation begins 1 2 1 1 2 1 2 2 2 3, while S^(e) G begins 1 2 1 1 2 1 1 2 2 2. They differ at the seventh term.

The published proof breaks at exactly that point. It writes S^(inf) = W(XT)^n n ⋯ with S^(e) = WX and T a prefix of G, and for n = 2 argues that the curling number of the first copy of T is the first term of X, “which is not 1”. Here W is empty, X = 1 2, T = 1 and n = 2, and the first term of X is 1.

Falsifier

A different value for cn(1 2 1 1 2 1), a different tail length for 1 2, or a different eighth term of Gijswijt’s sequence would invalidate the witness. The first eleven terms of G are 1 1 2 1 1 2 2 2 3 1 1; its first 4 occurs at term 220, which the source states independently.

Evidence

The weaker reading, whether the continuation of a starting sequence containing a 1 has a suffix equal to a suffix of G, is a different statement and is not settled here. Sloane states that weaker form as a conjecture for every starting sequence in arXiv:2301.03149v2, Section 4.3: “If true, this implies that if the starting sequence contains no 1s, then the sequence eventually becomes Gijswijt’s sequence [5, Th. 23]. In fact I conjecture that this is true for any starting sequence.”

Measured evidence bearing on the weaker form, for the same starting sequence 1 2: computing 24000 terms of both the continuation and G, every pair of offsets below 6000 gives an aligned common run that ends in a disagreement inside the computed range; the longest such run is 1740 terms, beginning at index 3619 of the continuation and index 30 of G. Runs of this kind lengthen with the computed range, which is consistent with agreement on longer and longer stretches without a shared tail. No suffix equality is thereby excluded beyond the searched offsets and range, so the weaker question stays open.

Whether the starting sequences containing a 1 that do satisfy the displayed equality admit a characterization is also open. Among the 57002 sequences over {1,2,3} of length at most ten that contain a 1 and do not end in 1, 18369 disagree with S^(e) G within 120 appended terms; the remaining 38633 agree over that range, which leaves the infinite equality undecided for them.

Triage

theorem; Tier 1 named external open question, preregistered in issue 9218 before write-up. The admission basis is open-problem-resolution; the conservative classification is proof_shape: bind-only with escape_witness: none. The computational use is a certified-instance with a typed refutes edge from result to claim.

ASSUMED-UNVERIFIED

The literature screen is bounded: the journal article, the Chaffin-Sloane preprint arXiv:0912.2382v5, Sloane’s retrospective arXiv:2301.03149v2, both arXiv versions and the journal version of van de Pol’s growth-rate work, the OEIS entries A090822, A093369, A094004, A216955 and A217209 with the OEIS curling-number index, and citing papers including arXiv:2608.15670v1 were opened; citation-index result pages were not reachable, so no worldwide priority claim is made.

Among the 57002 sequences over {1,2,3} of length at most ten that contain a 1 and do not end in 1, the 38633 that agree with S^(e) G over 120 appended terms are surviving candidates only; finite agreement does not establish the infinite equality for them.

The weaker reading, whether the continuation has a suffix equal to a suffix of G, is not settled here. For S = 1 2, computing 24000 terms of both sequences gives an aligned common run that ends in a disagreement at every pair of offsets below 6000, the longest being 1740 terms; that excludes a shared tail only within the searched offsets and range.