Support-Handoff and Constant-Width Reduction Both Ruled CLOSED: Establishing the Curvature-Density Domain 𝓡 and a Three-Direction Active Certificate

Round 01 (AMRAL-LUC-FC-R01, 2026-09-18/19) opens the "Descent" stage of the AMRAL LUC-FC research line's attack on the real, historical Lebesgue universal covering problem. This round establishes the first segment of the foundational reduction chain: every planar set of diameter at most one is first exactly reduced, via closed convexification, to a compact convex target (Theorem 3.1: closed convexification preserves diameter; Corollary 3.2), and then exactly reduced, via constant-width completion, to a unit constant-width body — handling the three boundary cases d=1, 0<d<1, and d=0 separately, each with its own independent proof (Theorems 4.1, 4.2). This round proves that support-function containment is the exact handoff on this chain: convex containment, directed Hausdorff distance, and rigid-motion placement can all be rewritten exactly as support-function inequalities, with no approximation loss (Theorems 5.1, 5.2). This round also introduces a new canonical curvature-radius density r=h+h'', establishing a translation-invariant curvature-density domain 𝓡, and proves that it corresponds bijectively to the translation-equivalence classes of unit constant-width bodies (Theorem 12.1) — this becomes the working parameter space for the entire research line going forward. This round further proves that the centered target family is compact under the Hausdorff topology, upgrading the outer worst case from sup to a genuinely attained max, and obtains an abstract, non-constructive finite ε-net existence result. For a fixed orientation, this round proves that the optimal translation configuration is witnessed by at most three "active" support directions — a structural fact that subsequent Rounds 03 and 17 directly reuse and refine. This round explicitly and deliberately rejects two tempting shortcuts, ruling them insufficiently rigorous: testing only "extreme" shapes (e.g., only regular polygons) as though they were representative; and truncated Fourier-series approximation without error bounds. This round does not give a constructive, computable finite shape dictionary (deferred to Round 02) — this round only establishes existence and structure, not yet an algorithm, and does not claim any new numerical bound. Research direction chosen and methodology led by Neo.K; this round's execution was carried out by Aletheia / GPT-5.6 Sol.