三 · 結晶

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.

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.

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.

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.