Certifying a worm's critical scale over the whole rotation circle: a phase-contact map, interval enclosures, and the gaps of a semi-validated certificate

The critical scale of a curve in a container is the least, over every rotation, of the scale at which it fits; a scan over rotations finds candidates for that least but cannot exclude a narrow dip between its samples. This paper takes four rounds of a research programme on the unit arcs that need the most of Wetzel’s 30-60-90 triangle, shrunk by 1.0048 — rounds that built, step by step, towards a certificate for one curve, a mirror-symmetric curve with a finite-width tanh curvature layer — and states what each established at the strength an independent recomputation measured. In interval arithmetic the recomputation certifies, first, that the curve’s least scale over the whole rotation circle exceeds that of the programme’s earlier best polygon, the event five-link, by more than 1.050233666×10⁻⁵, and then that the least is reached at the rotation 3π/2 alone and lies in [0.998914339066915, 0.998914339102351], with the competing cusp at 2π/3 at least 1.630111×10⁻⁹ above it. It also fixes, at 40 digits, a smooth minimum 5.44×10⁻⁹ rad beside a contact switch that every sampled guard of the programme missed, and shows that the programme’s “semi-validated” certificate leaves the 2×10⁻¹⁰ rad beside each switch enclosed by no derivative interval, so that three of its exclusions rest on intervals that do not contain the derivative they bound. 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 the triangle \(T/1.0048\) covers every unit arc, which is a cited preprint not checked here; anything about the two curves as the programme stored them in double precision, rather than as its ninth round defines them by decimal constants (the tanh-layer curve’s constants differ from their doubles by at most \(1.01\times10^{-16}\), and the transfer was not carried out); that the tanh-layer curve is the best of its family, or a local maximum among curves; anything about the event five-link’s least over all rotations beyond an upper bound — and so any upper bound for the gain over it; the number of stationary points on the curve’s wing intervals as an interval enclosure — it rests on a refined scan; that the programme’s own boxes and root boxes, in its eleventh and twelfth rounds, are valid enclosures.

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 four 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 — the certificates of Theorems 4.1 and 6.1, the interval redo of the round’s monotone bound, Proposition 5.1, the smooth minimum of §5.3 (first seen in RUN-009, fixed and placed in RUN-011), the measurement of the eleventh round’s neighbourhoods in §7.2, the replay of §8, and the corrections of §11. Before each of the four reports was committed, and before this paper was, 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 boxes three times wider than first reported in RUN-010’s certificate, a sliver of \(10^{-41}\) rad left unchecked beside each cusp in RUN-011’s, and the single omission behind RUN-012’s fourteen failures — 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 with code of their own the value at \(3\pi/2\), the event five-link’s branches, the smooth minimum and the certificates’ constants, and read both certificates’ code line by line; their suggestions were applied.

Priority, stated plainly. The curves, the support identity at \(3\pi/2\), the monotone bound, the phase-contact map, the exact switch formulas and the targets the certificates meet are the programme’s. What is offered as this line’s own is narrower: the certificates of §4 and §6, the findings of §5.3 and §7, an independent check of every step, and the corrections. The covering theorem the question rests on is Wichiramala and Panraksa’s.