The hardest worm for the 30-60-90 triangle: fifteen rounds of search, an independent recomputation, and a certified ascent

A recent preprint proves that Wetzel’s 30-60-90 triangle, shrunk by the factor 1.0048, contains a congruent copy of every arc of length one. A research programme then searched, over fifteen rounds, for the unit arcs that need the most of that cover — from grid linear programs over a library of curves, through polygons whose contact branches balance, to smooth curves whose curvature is shaped by layers and by function modes. This paper is the programme’s entry point at the strength an independent recomputation measured: one run for the corpus and one for each round, every figure rebuilt with independently written code, every check drilled with planted faults. Every round’s verdict is partial — its computations reproduce where they can be rebuilt, and specific statements do not hold as written. The certified results are a chain of explicit unit arcs, each certified in interval arithmetic above the one before; the last has critical scale at least 0.99891448056696, so with the cited preprint the cover is within 0.11 % of tight in linear scale. The local-optimality results along the way are numerical, and each was surpassed in the next, wider family. Of the 403 claims the recomputation measured, 14 rest on its own interval certificates and 179 on numerics, 79 were corrected and 13 refuted. The paper proves no bound for Moser’s worm problem.

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 the cover’s tightness measured here is about one triangle, not about the least area; that the triangle \(T/1.0048\) covers every unit arc, which is a cited preprint, not checked here; which unit arc is the hardest for the triangle — every search is a search, and the last candidate is not shown to be a local maximum of anything; any exact stationary point, or the size of any neighbourhood on which a local maximum holds; anything about the programme’s curves, samples or code that its packages do not ship.

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 sixteen runs, checked and audited the inventory, and wrote this paper and its companions: every COMPUTED_HERE figure rebuilt with independently written code, every argument read against its source, the drills of §5, the certificates of §3, and the corrections. The independent readings of §5, the extraction of the inventory’s claims and the reading of its sample were done by separate instances of the same model (model: Claude Opus, version 5.5), each run as a subagent of 懸衡’s Claude Code session and given only its brief and the documents named in it; the reviews were done by 墜衡 (Zhuì Héng).

Priority, stated plainly. The question, the searches, the families and the candidates are the programme’s. What is offered as this line’s own is narrower: the certificates, the findings that overturn one of the programme’s local-optimality verdicts (round 14’s) and correct the inference behind another (round 15’s), an independent check of every step, and the corrections. The covering theorem the question rests on is Wichiramala and Panraksa’s.