a_Leb Obtains a Finite Two-Sided Bracket for the First Time, While Exact Saturation Remains Unresolved: Candidate-Cover Boundary Compiler and Global Polygon Dictionary

Round 04 (AMRAL-LUC-FC-R04, 2026-09-19) is the final round of the Descent phase of the LUC-FC research line, addressing the last infinite-dimensional domain not yet touched by Rounds 01–03: the candidate universal cover itself. This round first proves an anchor lemma — that every universal cover must (after translation) contain the radius-1/2 disk B_{1/2}(0) — and combines it with any already-proven upper bound Ā (such as Gibbs 2018's 0.8440935944) to obtain a uniform radius bound U⊆B_R(0), R=2Ā=1.6881871888, thereby compressing all near-optimal candidate covers into a single Hausdorff-precompact compact class C_Ā. On top of this, it constructs a finite outer-approximation polygon compiler P(U): after sampling M uniform support directions, every sample is quantized upward to resolution q, which guarantees U⊆P(U) and therefore makes it a monotone-safe (never-undercounts) outer approximation — the outer approximation of a universal cover is necessarily itself universal. This round gives the explicit error of this approximation: the Hausdorff excess ε and the area-inflation amount β, both of whose closed-form expressions are explicit functions of M and q, and which converge to zero as M→∞, q→0. Building on this, this round's core theorem establishes, for the first time, a genuine finite two-sided bracket on the Lebesgue constant itself: A_poly−β ≤ a_Leb ≤ A_poly, where A_poly is the area of the smallest universal polygon in the finite polygon dictionary at a given fixed resolution (M,q). The document itself explicitly distinguishes between two separate propositions — "finite-resolution global convergence" (verdict: YES) and "exact finite saturation" (verdict: OPEN) — and states in its conclusion that this round does not claim to solve the Lebesgue universal covering problem and does not improve on the existing published upper and lower bounds. This round also strictly distinguishes Zeng (2026)'s exact finite-arc Reuleaux-type hierarchy from its own support-polygon outer compiler, making no claim of being stronger, of having a better convergence order, or of priority in discovering the existence of a finite hierarchy — it is described only as an independent finite-representation / cross-verification route. An independent lagged-verification line separately ran 5 rounds of deep review on this round's compiler; all 5 rounds returned a verdict of EXTENDED, COUNTEREXAMPLE: NONE FOUND, noting only that the constants could be tightened further, with no counterexample found. Research direction and methodology were chosen and led by Neo.K; this round's execution was carried out by Aletheia / GPT-5.6 Sol, not Claude.