Lebesgue Universal Covering Finite-Closure Methodology, Round 00 v0.2: Freezing the LUC-FC Existence→Descent→Saturation→Global Closure Four-Stage Closure Chain and the "Finite Closure = Finite Saturated Certifiable Branch Types" Working Definition; This Round Makes No New Mathematical Claim About the Lebesgue Problem

AMRAL-LUC-FC-R00 v0.2 is round zero of AMRAL's new attack line on the real, historical Lebesgue universal covering problem (posed by Lebesgue in 1914: find the minimum-area planar convex set into which every planar compact set of diameter at most one can be embedded by a rigid motion), and it makes no mathematical claim about the problem itself — the document explicitly excludes six classes of claim: proving the problem itself, giving a new upper or lower bound, proving that the optimal covering body has a specific symmetry, proving that finitely many witnesses suffice to determine the global optimum, completing any certificate that still requires large-scale computation, and automatically upgrading the limit of a finite hierarchy into a finite-closure theorem. Its core action is to specialize Neo.K's Relational Constraint–Handoff Methodology (RCHM) into an LUC-FC version for this problem — just as RCHM was previously specialized into MLRSC for this site's Hodge conjecture case — and to freeze a canonical interface that subsequent rounds can accumulate on, rather than restarting from scratch each round. At its core are the four-stage closure chain Existence → Descent → Saturation → Global Closure, and a deliberately narrowed working definition of "finite closure": finite closure means that there exist finitely many saturated, certifiable branch types (continuous parameters may still remain within a branch), not finitely many geometric points; the document lists this definition alongside the finite-hierarchy limit sequences found in the existing literature (such as Zeng 2026's Reuleaux-type hierarchy) as Risk C, requiring that the two never be conflated. The document also mandates a fixed claim-state vocabulary (PROVED-ANALYTIC, VERIFIED-COMPUTATIONAL, VERIFIED-THEOREM-APPLICATION, DERIVED-CANDIDATE, NUMERICAL-EVIDENCE, CONDITIONAL, COMPUTE-DEFERRED, OPEN, REJECTED, CORRECTED) and a per-round degrees-of-freedom ledger (Freedom Ledger), as the research protocol from Round 01 onward. The external literature bracket is cited as the research's starting point and was independently verified by checking arXiv directly: Gibbs's (2018, arXiv:1810.10089) upper bound of 0.8440935944, Mishra's (2026-08-31, arXiv:2608.30538) lower bound of 0.8344, and Zeng's (2026-09-01, arXiv:2609.01284) independently certified Reuleaux-type hierarchy. The document itself states its attribution explicitly: problem selection, methodology, and research direction are by Neo.K, and per-round active research execution is by "Aletheia / GPT-5.6 Sol" — that is, Human-Directed + Semi-Autonomous AI Mathematical Research, a process output driven by GPT-series models, not a research goal decided by AI alone, and not Claude's own work.