The tentative Hart–Rowley prefix/suffix equality

Sarah B. Hart and Peter J. Rowley, A note on involution prefixes in Coxeter groups, arXiv:2502.00777v1, section 3, printed page 5, immediately before References, write:

We suspect that the involution length of any element always equals its ‘suffix’ involution length, but have not tested this extensively enough to make a firm conjecture.

This is one explicitly tentative, unnumbered published assertion. The quoted text is licensed CC BY 4.0, https://creativecommons.org/licenses/by/4.0/ ; PDF line wrapping and typographic ligatures are normalized. It is not Conjecture 1.2 or 1.3.

The domain is every Coxeter system whose every nonidentity element has a unique maximum-Coxeter-length involution prefix, and every element of such a system. An involution has order exactly two. A prefix u of w satisfies w = uv and length(w) = length(u) + length(v). The unique ancestor maximizes length over all involution prefixes. Remove it and recurse on v, with zero factors at identity. The suffix procedure removes the analogous right factor. The assertion equates these two actual greedy counts. No finiteness, rank, or Coxeter-element restriction is imposed, and neither count minimizes arbitrary factorizations.

The ancestor decomposition gives a canonical length-additive involution factorization. The question asks whether its number of factors is independent of orientation. The formal claim and its designated negation belong to D5/S3/FiniteGroups/HartRowleyGreedyRefutation; the motivation selector above names the intended resolving result. Its frozen-member validation belongs to the canonical freeze stage.

The source supplies an inverse correspondence between involution suffixes of w and involution prefixes of its inverse. Equality at w would additionally require invariance of the prefix count under inversion. That extra assertion can fail. The two notions of involution length must also be distinguished: a short arbitrary involution factorization does not establish the length-additive, maximum-at-every-step greedy count.

Use the standard adjacent swaps in S4. Establish their Coxeter presentation of type A3, and identify genuine Coxeter length by reduced normal words and a lower bound along every simple-generator step. Check the unique longest involution prefix for every nonidentity permutation, including all competitors. The suffix choices use the same definition with right residuals. Strictly decreasing Coxeter length makes both residual recursions terminate at identity. A single qualifying system and element with unequal counts negates the entire unrestricted equality.

A purported refutation fails if its group is not proved to be a Coxeter system, its length is not proved to be Coxeter length, its ancestor hypothesis holds only on the displayed chains, it replaces maximum length by inclusion maximality or a minimum factor count, or its recursions do not reach identity. For the universal equality itself, any fully qualifying system containing an element with unequal greedy counts is a counterexample.

In one-line image notation and multiplication (uv)(i) = u(v(i)), take w = 2431 in S4. The prefix chain is 2431 = 2143 · 1342 and 1342 = 1324 · 1243, followed by 1243 itself. The successive factor lengths are 2, 1, 1 and the prefix count is 3. The suffix chain is 2431 = 2134 · 1432, followed by 2134; the removed right factor has length 3 and the remaining factor has length 1, giving count 2. Every displayed product is length additive. The group-wide unique-maximum certificate is essential, rather than an assumption inferred from those chains.

The claim is a closed Prop and the designated result has literal type Not claim. Kernel checking of the exact candidate, its axiom closure and nonvacuity is separate from canonical freezing and full repository admission.

Tier 1: a recent specialized side assertion, expressly insufficiently tested by its authors. No extensive prior attack on this exact equality was established. The prospective exact statement is registered in issue https://github.com/the-omega-institute/trureturing/issues/17426 . Bounded source/status qualification found no exact prior resolution; the source note records both the inspected later originals and the coverage limits. The canonical admission audit is tracked in issue https://github.com/the-omega-institute/trureturing/issues/17462; its 22 helper audits remain unfinished (one designated closed result is the sole exception), none is declared_validated, and no audit retry is part of this delivery.

The orientation mechanism is that removing the commuting initial pair (12)(34) leaves two further greedy factors, while the right end admits the longer involution (24) and leaves one. Maximizing the current factor length does not make the eventual factor count invariant under reversal.

Known source conclusions that survive include the Coxeter-element ancestor results and their graph-coloring/path interpretations, and the inverse prefix/suffix correspondence. Conjecture 1.2, asserting the ancestor property for every finite Coxeter group, and Conjecture 1.3, asserting the finite rank bound, are separate open questions in the cited source. This witness satisfies the ancestor hypothesis and both counts obey the A3 rank bound. It does not refute either conjecture or the Coxeter-element conclusions. No rank minimality, classification, adjacent-problem credit or priority is asserted.

Source fidelity and adequacy of the bounded literature and ownership searches require independent assessment. Unindexed, differently worded, unpublished and outside-scope prior resolutions remain unexcluded. The four sibling experiment artifacts are unread, unverified and unused. The author candidate is separate from independent prefreeze fidelity, atom-aware freeze, canonical emission, full immutable-base admission, reviews, required PR-head CI, merge and subsequent independent dev verification. No solved-problem count follows from this dossier or a scoped compile alone.