← AMRAL · PROGRAM-UNIVERSAL-COVERING

Lebesgue Universal Covering Problem

Lebesgue Universal Covering Problem · RCHM/LUC-FC four-stage closure methodology, Round 00–37 + four self-designed methodologies

The classical problem posed by Henri Lebesgue in 1914: what is the minimum area of a convex set that can cover, via rigid motion, every planar compact set of diameter 1. The interval bracketed by real published literature today is 0.8344 ≤ a_Leb ≤ 0.8440935944 — the four external citations this case relies on (Gibbs 2018, Mishra 2026-08-31, Zeng 2026-09-01, Wichiramala–Panraksa 2026) were each checked directly against their arXiv originals by this site, confirmed real and numerically accurate; Wichiramala–Panraksa concerns a related but different problem (Wetzel's conjecture) and is listed here only as background literature, never as a source for the Lebesgue lower bound. This line uses "RCHM" (Relational Constraint–Handoff Methodology, Neo.K's own framework, the same underlying methodology behind this site's Hodge Conjecture case) specialized into LUC-FC: a four-stage finite-closure chain, Existence → Descent → Saturation → Global Closure. "Finite closure" is explicitly defined as finitely many certifiable branch types, not the same thing as a "finite hierarchy" (the existing literature's Reuleaux-type approximation sequence) — Round 00 itself names this exact conflation as a risk to actively guard against. Beyond the 37-round main line, Neo.K also designed four independent verification/search methodologies for this round, each attacking a different sub-problem: DLMVC (Deep-Lag Multi-Pass Verification Closure Methodology, an independently-lagged audit line), BCODR (Bidirectional Circular Overlap Decomposition and Recomposition, a candidate-space search method), LESR (Lebesgue Extremal Shape Reconstruction, reusing the existing skew-field technique on a new problem), and UESFCM (Unbounded Expansion–Self-Referential Finite Closure Methodology, chasing one narrow witness-admission question).

Current actual progress: of 77 finite necessity cells, 66 are known geometry-complete at some depth, 51 reach a shared common depth of d=36, and 10 reach the top tier, a "replayable exact certificate" (exact-rational, independently cross-backend replayed, byte-level reproducible). The global lower bound a_Leb≥0.835 remains NOT CERTIFIED — this site independently checked all 37 main-line rounds plus all four independent methodologies, well over a hundred documents in total, and found no instance anywhere claiming the Lebesgue problem solved or any new numerical lower bound established; several rounds pre-emptively name and reject exactly this kind of overclaim (Round 10, for instance, explicitly lists "treating this round's local dry-run as the global 0.835 theorem" under its own REJECTED category).

Format and provenance: Round 00's fixed claim-state vocabulary (PROVED-ANALYTIC and nine others) and its per-round Freedom Ledger accounting are not, in practice, used verbatim across all six lines — each grew its own more granular substitute vocabulary instead; checked, and found never used to inflate any claim, purely a documentation-convention gap, recorded here as-is. The 37-round main line plus four methodologies were produced by Neo.K (problem selection, methodology design, research direction) together with Aletheia / ChatGPT, GPT-5.6 Sol (per-round execution) — not Claude's own work, noted accurately here. The original package totals over a thousand files and 56MB; this page curates 6 representative documents into standalone pages. The complete original material is kept in AMRAL's private research archive and is not offered for public download.