← 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; RCHM's application to the Hodge conjecture is a separate case, RCHM × Hodge Research Round) 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 (the first batch), 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 now includes standalone pages for all 42 documents — all 38 main-line documents (Round 00–37) plus the four self-designed methodologies — each with full English translations, both the page shell and the source document itself. The complete original material is kept in AMRAL's private research archive and is not offered for public download.

II · Computation and Reproduction: all 42 documents have been independently recomputed by 懸衡 (Claude), in 68 runs (RUN-001 to RUN-068) — what reproduces and to what precision, what is not as the documents state it, and the new readings: see the second layer.

III · Crystallization: this line has released 4 papers; see the third layer.

I · Prior (continued): after the first batch of 42 documents a second batch arrived (2026-09-21 to 09-25), devoted to the root-6217 local minimum — LESR Papers 09C–09U, the Root-6217 handoff and the open-questions list, and the MCC-OE certification line, plus the contemporary BCODR-LUC 01–07 (seven applications of the bidirectional circular-overlap method to root 6217). On 2026-10-07 several of the files missing from it were recovered (MCC-OE-24, BCODR-LUC-06 and the 09V tail snapshot); see the case pages. Layer two has not recomputed this batch, except BCODR-LUC (懸衡 has recomputed its LUC-01 to 05 and 07). Its 'RESEARCH-CLOSED' is an author-defined status word that holds only locally and only at the authors' own research-interval authority; the global bound a_Leb≥0.835 remains NOT CERTIFIED, and every paper lists 'the global Lebesgue problem solved' as not claimed.