Strict Incidence Formally Defined, the Witness Portfolio Closure Theorem Proved, yet the 12-Cell Pilot Scores Zero at Depth 16: A Deep Crossover Shows B₇ Overtaking H₁₉, Proof and Scheduling Layers Formally Separated

Round 23 (AMRAL-LUC-FC-R23, 2026-09-19) takes the witness proof-cost Pareto portfolio {B₇, H₁₁, H₁₃, H₁₇, H₁₉} that Round 22 built and upgrades the question asked of it: instead of ranking "which witness is strongest on average," this round asks a theorem-level question cell by cell — for every cell in the necessity-marked base atlas, does at least one witness have its full relevant placement root over that entire cell already closed by a legitimate lift certificate at or above the threshold T? The round formalizes this as Strict Incidence (written I_jK∈{0,1}), and explicitly enumerates what does not create such an edge — a PARTIAL lift tree, a small pending fraction, a small unresolved placement volume, a numerical search minimum above T, the absence of a found counterexample, fast tail contraction, or a witness's success on some other cell — establishing the principle that "a strict edge is certificate-complete, not confidence-complete." On this foundation the round proves the Portfolio Closure Theorem (Theorem 5.1): if every necessity cell has at least one witness reaching I=1, then the portfolio formed from those witnesses maintains the ≥T lower bound over that atlas coverage, because the full portfolio's hull necessarily contains the hull of any single witness already proved. The round also gives a weighted set-cover / facility-location formulation (witness-activation and cell-witness assignment variables, minimizing fixed plus certificate cost, subject to every cell being served by at least one activated witness with I_jK=1), defines budgeted incidence I_jK^(d) and the residual frontier R_d (the set of cells not yet strictly covered by any witness at budget d — R_d being empty is necessary, and on the current finite cell list also sufficient, for strict cover to be feasible at budget d), and states the Partial Edge Non-Composition Rule: two witnesses that each have only a PARTIAL lift tree cannot be combined to claim cell closure, no matter how nearly closed each looks individually, because each faces an independent placement adversary. Because a full same-depth run over all 77×5 cell-witness pairs exceeds a single round's compute budget, this round instead measures the 12 hardest cells (by deepest outer marker) in the necessity atlas against all 5 witnesses — 60 edges — at lift depth 16: the result is 0 of 60 edges COMPLETE, so none of these 12 cells has a strict edge at this depth, the residual frontier covers all 12 cells, and strict weighted set cover is judged INFEASIBLE at this budget — the document states explicitly that this is "the correct result, not algorithm failure," and expressly forbids treating "the witness with the smallest unresolved volume" as I=1, since that would pass off a scheduling heuristic as a theorem. The depth-16 quantitative partial-edge data show that, using unresolved volume alone as a short-horizon scheduler metric, all 12 cells (12/12) point to the cheapest witness, H₁₉ (mean unresolved volume 2.29×10⁻⁴, forcing margin 4.96×10⁻⁵), while B₇'s volume is larger (9.42×10⁻⁴) and its margin looser (about 1.93×10⁻³) — but this is only a routing signal, not a strict cover. The round then pushes the four hardest cells to depth 22 and compares B₇'s and H₁₉'s unresolved volume there: B₇ overtakes H₁₉ in cells 0, 1, and 2, with only cell 3 still favoring H₁₉ — for cell 0, the full depth-22 ranking is B₇ 5.17×10⁻⁵, H₁₉ 5.43×10⁻⁵, H₁₇ 6.27×10⁻⁵, H₁₃ 9.24×10⁻⁵, H₁₁ 1.09×10⁻⁴, so the witness that led at depth 16 has already been overtaken by depth 22. This motivates the round's definition of crossover depth (the smallest depth at which two witnesses' unresolved-volume ranking flips), which the document explicitly calls "a performance quantity, not a theorem threshold" whose job is to tell the scheduler how deep it expects to run: a short horizon favors the cheap H₁₉, a long horizon favors the faster-contracting B₇; witness routing therefore cannot be ordered by any single value such as search forcing, shallow pending count, or orientation period, and should instead use an estimated total cost-to-COMPLETE that folds in unresolved volume, tail contraction, nodes per level, forcing risk, counterexamples, and the expected remaining horizon. Round 23 formally establishes, as a production invariant, that the proof layer (which looks only at whether I_jK is 0 or 1) and the scheduler layer (which looks only at the quantitative signals) must never be mixed. The round also proves portfolio monotonicity (for a fixed cell-witness pair, once I_jK reaches 1 at some budget, it need not revert to 0 at a larger budget as long as only the budget grows and the theorem semantics are unchanged — the strict incidence graph can grow append-only, barring a negative audit or a change in dependencies) and the counterexample effect (if a witness already has a valid counterexample on a cell, I_jK=0 there is not merely "insufficient budget" but means that witness can never be the sole closer for that cell; such an edge is marked DISQUALIFIED and permanently removed from future single-witness set-cover candidacy until the semantics change). The round ships an accompanying AMRAL_LUC_FC_Round_23_LONG_RUN_SPEC.json instructing the local runtime to extend the pilot to the full 77 necessity cells × 5 witnesses at the same budgets, deepen only the residual cells, update the facility-location solver as soon as any COMPLETE edge appears, and hand cells with no strict edge at all to one of three routes — deeper B₇, a joint multi-witness certificate, or a new witness — which is exactly where Round 24 picks up. The 12-cell pilot's residual frontier currently still covers all 12 cells, so Round 23 does not achieve "portfolio closure"; what it delivers is a portfolio capable of a meaningful routing competition but with zero strict complete edges so far, which directly sets the next task. The global bound a_Leb≥0.835 remains compute-deferred, unchanged by this round. Research direction and methodology are due to Neo.K; this round's AI collaborating researcher and primary executor was Aletheia / ChatGPT, GPT-5.6 Sol.