The LUC-FC Research Line Completes Its First Self-Correction While Global Candidate-Cover Optimization Remains Unsolved: Signed Placement Margin and the Fixed-Cover Finite Placement Certificate

Round 03 (AMRAL-LUC-FC-R03, 2026-09-18) addresses the placement continuum that Round 01–02 had not yet touched: under a fixed candidate cover U, the complete optimization over Q∈O(2), t∈R², and u∈S¹. This round first proves a continuous translation dual theorem: for a fixed orientation, the continuous minimax translation problem min_t max_u [a_Q(u)+t·u] is exactly equal to a zero-barycenter probability-measure dual (Theorem 6.1, proved in both directions via subgradient optimality). Combined with the Carathéodory theorem, this yields the exact witness structure: for any optimal placement at a fixed orientation, the active support directions number exactly 2 (which must be an antipodal pair) or 3 (whose convex hull contains the origin). The single most important fact of this round is that it simultaneously completes the first self-correction of the entire 37-round LUC-FC research line, formally recorded as R01-CORRECTION-001: Round 01 Section 22 had loosely listed the active branches into three categories — "one, two, three" — and Round 03 proves, with a short but rigorous argument, that "one-active" is in fact impossible — because a single active direction u would have to satisfy u=0, contradicting that u is a unit vector — and formally corrects the classification to "2-active antipodal" or "3-active and origin-enclosing." The scope of this correction is strictly limited: it only refines a finer sub-classification internal to Round 01, and does not touch or weaken Round 01's main theorem of "at most three active directions" in any way — that main theorem stands exactly as it was. On top of this, the round further establishes a finite support-direction grid (M directions, explicit error η_M) and a finite orientation grid (P samples, explicit error ρ_P), compressing the entire continuous O(2)×R² optimization into a finite primal-dual LP, and gives the complete two-sided bracket G−ρ_P ≤ M_U(K) ≤ G+η_M. Merged with Round 02's finite shape dictionary, this round establishes, for any candidate cover that is "fixed and has a certified support oracle," a complete, replayable, independently verifiable fixed-cover finite certificate — but the candidate cover itself remains an infinite-dimensional optimization variable, and the global Lebesgue optimization problem is left to Round 04. Research direction and methodology are led by Neo.K; this round's execution was carried out by Aletheia / GPT-5.6 Sol, not Claude.