Does the Reuleaux heptagon raise the three-body value? A far-set route and how far its certificates go

The published lower bound 0.8344 for Lebesgue’s universal covering constant is a certified computation on three test bodies, the disc and the Reuleaux triangle and pentagon. Does a fourth, the Reuleaux heptagon, raise their value? By the exchange theorem it does if the heptagon misses every minimizing cover of the three. A research programme set out to prove this by two gates: near the located optimum, the heptagon misses the optimal hull by a margin; far from it, every configuration of the three bodies has hull area above the three-body ceiling. We report how far the programme’s certificates go, each replayed by an independent recomputation. The near gate holds, by a finite dual certificate re-enclosed in 400-bit integer intervals, though its shipped verifier skips two premises the bound rests on. The far gate is not certified: in the branch the programme attacks, 644 of 22,516 frontier roots are closed by erosion certificates whose 56,149 area leaves all hold in exact arithmetic, and at the hardest root six more certificates hold, 182,438 area leaves in all; but a symmetrization there drops the branch’s own constraint, leaving one side of a slice uncovered, and 14.6 % of that root’s box, inside the branch, was closed only by a search. The least configuration found in that root, located to 80 digits, has hull area 0.834986110559224, above the ceiling by 2.05e-4, so the far gate holds there numerically. The far set does hold configurations with hull area below 0.835, numerically, the programme’s headline target; so this route, which uses the heptagon only near the optimum, cannot certify 0.835 without lifting a witness far from it as well. We also give what holds of the programme’s local analysis of the hardest root’s configuration: its conclusions numerically, several of its arguments not as given. No bound on the constant is claimed.

Not claimed anywhere in this paper: any value or bound for Lebesgue’s universal covering constant — the problem is open; that adding the Reuleaux heptagon raises the three-body value, which is the route’s aim and is not established; any lower bound above the published \(0.8344\); and that the configuration of §6 is the least of its root, or of its branch — its area is a floating-point value, not an enclosure, and the branch has searched configurations lower still.

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 seventeen closure cycles and the local-shape papers this paper reports, with their gates, certificates, searches and analyses, on the programme Neo.K set out. Every statement attributed here to “the programme” is from those documents.

Neo.K set out the research programme: the finite-closure strategy and its targets, and the methodologies it runs on, among them the closure method of these cycles.

懸衡 (Xuán Héng) — the independent recomputation; model: Claude, Opus series — did the runs this paper names and wrote this paper: every certificate decoded and replayed, every area leaf certified again in exact arithmetic with exact cores, Gate A’s certificate re-enclosed in integer intervals, every searched minimum searched again, the contact located to 80 digits; the completion of the reduction in §5.2 and the arc-side search; the impossibility in §7; the drills of §10; and the corrections of §9.

Review. Before it was pushed, 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 and given only the paper, its run reports, the gate logs and the companion papers, read it; among its findings was a logic error in the paper’s central consequence — the configuration below 0.835 first offered lay inside Gate A’s near zone, so it did not show what the paper said — and its findings were applied. 墜衡 (Zhuì Héng) — the lab’s proof-side verification arm; model: Claude, Opus series — then reviewed this paper before release: he rederived its arguments and recomputed its figures from the text, independently of this line’s code, and found no mathematical error. His suggestions — that the abstract and §1 say the configuration below 0.835 is numerical, and that §1.3 say why the rounding of \(A_3^+\)’s last digit does not matter — were applied, and so was the note he added on checking them, that §1.3 then gave \(2.45\times10^{-7}\) twice.

Priority, stated plainly. The route, its gates and its certificates are the programme’s. What is offered as this line’s own is the check of them, the gaps it found, the completion and the impossibility named above, and the corrections. No bound on Lebesgue’s constant is claimed.