Beyond the event five-link: curvature layers, a saddle against peak splitting, and a certified ascent of a worm's critical scale

Wetzel’s 30-60-90 triangle, shrunk by 1.0048, is known from a recent preprint to contain a congruent copy of every arc of length one. A research programme searched for the unit arcs that need the most of it; this paper takes the five rounds in which it left polygons for smooth curves and climbed, and states each step at the strength an independent recomputation measured. Each round declared a local optimum, and each was surpassed by enlarging the family — numerically, at high precision: the programme’s best polygon is not a local maximum among curves, since a finite-width curvature layer rises above it; a five-parameter stationary point of that layer, at 60 digits a strict local maximum of its family, is a saddle against splitting its curvature peak in two; and that point is not even stationary in the curvature’s function modes, along which an eight-mode candidate climbs higher still. What is certified, in interval arithmetic, is every gain: the thirteenth round’s curve is 4.0451×10⁻⁹ to 4.2486×10⁻⁹ above the eighth round’s, a split curve at least 1.1924×10⁻⁷ above the thirteenth’s, and the fifteenth round’s candidate 1.3725×10⁻⁷ to 1.3758×10⁻⁷ above it, its least scale at least 0.99891448056696. That curve is a unit arc, so no homothet of the cover smaller than that factor contains it: with the cited preprint, the cover is within 0.11 % of tight in linear scale, about 0.22 % in area. 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; that \(T/1.0048\) covers every unit arc, a cited preprint not checked here; that any curve here is the hardest unit arc for the triangle, or the best of its family — the double-peak family’s and the mode family’s maxima are not sought, and the candidate is not shown to be a local maximum of anything; that an exact stationary point exists near any computed one, or the size of any neighbourhood on which a local maximum holds; anything about the curves as the programme stored them in double precision, rather than as their decimal parameters define them; anything about the programme’s samples, curves or code it does 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 five 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 §12; and the additions marked as the recomputation’s where they occur — Proposition 3.1 and the B2 identification, Proposition 4.1’s measurement, the solution of the thirteenth round’s system and Proposition 5.1, Proposition 6.1, the saddle and the split solutions, the first-order test of §7.1, the certificates of Theorems 5.2, 6.2 and 7.1, Corollary 8.1, and the corrections of §11. Before each of the five 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 text and its evidence, read it independently; its findings — among them chirality cusps first taken at fixed phases in RUN-015, and the gain’s upper end first taken against the wrong bound in RUN-016 — were applied, and the reports record them. The paper was then reviewed by 墜衡 (Zhuì Héng), the lab’s proof-side verification arm, who recomputed the thirteenth round’s solution, the split curve and the candidate from §2’s definitions with code of their own, checked the second-order arguments and the families against the runs’ code, and found two gaps in the curves’ specification and a missing hedge, which are filled here.

Priority, stated plainly. The families, the stationarity systems, the searches, the mode basis and the candidates are the programme’s. What is offered as this line’s own is narrower: the certificates, the saddle and the first-order test that overturn one of the programme’s 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 corollary leans on is Wichiramala and Panraksa’s.