Six rounds of a search for the hardest unit arcs in the 30-60-90 triangle, recomputed: the exact critical-scale formula, two chirality conventions, and the event five-link

A recent preprint gives a computer-assisted proof that Wetzel’s 30-60-90 triangle, shrunk by the factor 1.0048, contains a congruent copy of every arc of length one. How tight is that cover — which unit arcs need the most of it? A research programme searched for such arcs over fifteen rounds; this paper takes the first six, from grid linear programs to a stationary point among five-link polylines, and states each result at the strength an independent recomputation measured. We check the programme’s exact formula for the least scale of a rotated polyline in the triangle, and show that the grid programs of its first rounds are relaxations that run low; we apply the programme’s two conventions — rotations only, or rotations and reflections — to every candidate, where they move values by percents; we show that the five-link family nests in the seven-link one, so a reported seven-link deficit was the search’s; we locate the four competing contact branches of the round’s best five-link at exact phases — the method the programme’s own audit proposed and its next round adopted; and we show, numerically and in double precision, that the root of the programme’s nonsmooth KKT system, the event five-link with least scale 0.998903757132509, satisfies the second-order sufficient condition, so that it is a strict local maximum of the congruent least scale among all unit five-links near it, on a neighbourhood whose size is not established. No exact root is certified near the computed one. The paper proves no bound for Moser’s worm problem, and lists the corrections found on the way.

Not claimed anywhere in this paper: any bound for Moser’s worm problem — the least area of a convex set containing a congruent copy of every unit arc is open, and nothing here bears on it directly; that the triangle \(T/1.0048\) covers every unit arc, which is a cited preprint and not checked here; that an exact root of the contact-event system exists near the computed one — that would need an interval Newton or Kantorovich argument, which neither the programme nor this paper makes; that any curve in this paper is the hardest unit arc for the triangle, or the best of its family — every search is a search; that the event five-link is the five-link family’s maximum, or the only root of its system, or that its contact topology is the best one; the size of the neighbourhood on which it is a local maximum; anything about the 27 library curves the programme names without defining, or about the samples and candidates it did not log.

What each author did is stated here, side by side and in no rank; the roles are different in kind, and the paper is worth less if they are blurred.

Aletheia — the web-side research line; model: ChatGPT, GPT series — produced the programme’s fifteen rounds, on the question Neo.K set out; every statement this paper attributes to “the programme” or to “the round” is from those documents, except where the paper marks a reading of them as the recomputation’s.

Neo.K set out the research programme: the question of how tight the 30-60-90 cover is, and the methodology under which its rounds and their independent recomputation were run.

懸衡 (Xuán Héng) — the independent recomputation; model: Claude, Opus series — did the six runs this paper names and wrote this paper: every COMPUTED_HERE figure rebuilt with independently written code; every argument read against its source; the drills of §10; and the additions marked as the recomputation’s where they occur — the measurement of the first round’s grid values in §3 (the second round’s own addendum caught its grid’s), the nesting argument of Proposition 4.1 and the symmetry tie of Proposition 4.2, the families’ higher search values, the grid candidate’s mirror value, the location of the four branch minima at the fourth round’s own curve and the levelling there, the second-order argument of Theorem 6.1 and its transfer in Corollary 6.2, the eight-parameter argument of Theorem 7.2, and the corrections of §9 that the programme did not make itself.

Readings. Before each of the six reports was committed, a separate instance of the same model (model: Claude Opus, version 5.5, run as a read-only subagent of 懸衡’s Claude Code session), briefed only with the report and its evidence, read it independently; its findings were applied, and the reports record them. Two more such readings read this paper before it went to main, and their findings were applied. The paper was then reviewed by 墜衡 (Zhuì Héng), the lab’s proof-side verification arm, who recomputed the event five-link’s four branches, the singular values of \(G\), the pressures and the Lagrangian’s curvature along the null direction from the definitions of §2 and §6 with code of their own, checked the arguments of Theorems 6.1 and 7.2 — the critical cone and the constraint qualification included — and found that the first round’s 98 % test does not hold under the congruent convention, which §8 and §9.1 now say.

Priority, stated plainly. The question, the families, the exact formula, the two conventions, the exact phases as a method and the event system are the programme’s. What is offered as this line’s own is narrower: an independent check of every step, the strength at which each step stands, the arguments listed above, and the corrections. The covering theorem it relies on is Wichiramala and Panraksa’s.