Lazy Proof Cascade Gains an Equivalence Proof, Necessity-Marker Theorem Established: This Round Uncovers and Corrects a Two-Round Sampling-Heuristic Misjudgment

Round 19 (AMRAL-LUC-FC-R19, 2026-09-19) addresses two questions: whether REP is worth computing eagerly at every hot node; and to what degree base refinement should actually proceed before it is right to trigger the B7 lift. The first question is resolved by the Lazy Proof Cascade: the fixed APR→CORE→REP order is changed so that REP is computed only when neither APR nor CORE has closed the cell, and Theorem 2.1 proves that — on the same cell — the Lazy and Eager strategies yield exactly the same CERT/HARD classification, i.e. changing the evaluation order loses no rigor whatsoever; in the fixed-split-tree depth 16→18 A/B measurement (2048 depth-16 pending roots advanced to the same state of 13206 nodes, 4944 pending), REP calls fell from 12659 to 10523 (a 16.87% decrease), and wall time fell from 7.2903s to 6.8687s (a 5.78% decrease); the continuation cost κ was measured for the first time — depth 16→20 accumulates 78.1375 CPU-s (about 38.15 ms per original depth-16 pending root) — which is 2.65 times above Round 18's Tail-Window break-even threshold of 14.41 ms, confirming that this investment has already paid for itself. The second question is resolved and proved by the Necessity-Marker Theorem (Theorem 13.1): once a cell contains a point q* whose verified exact upper-bound area is below the threshold T, no base-only descendant that still contains q* can ever obtain a valid closing lower bound — but the paper also proves that necessity does not mean immediately lifting the entire ancestor into a higher dimension: for the depth-31 cell (along the branch that truly contains the canonical near-minimal configuration q†, where this round's pilot-level run triggers center-sublevel for the first time), running the B7 depth-12 lift directly still leaves 227/4096 (about 5.54%) unresolved; instead, first base-refining that same cell to depth 40 (561 nodes, 1.69 CPU-s) shows that, of the 256 descendants, 85 have exact centers confirmed <0.835, of which 77 remain strictly <0.835 after re-verification with the 20k-support outer bracket — localizing witness necessity from the entire ancestor down to at least 77 explicit descendants. This round also formally logs a self-correction, CORRECTION-R19-001: Round 16 and Round 18 had used the 40-direction sampled inner-center area — flagging 12 of the 10114 pending cells at depth 20 with sampled area<0.835, with a minimum sampled value of 0.83395139 — as a signal that a genuine sub-threshold configuration might exist; Round 19 recomputed all 12 of these candidates using the Mishra official exact support kernel instead, and the result is that 0/12 are truly below the threshold, with the true minimum being about 0.83586503 — the semantics have now been explicitly corrected to state that this kind of sampling can only produce a "worth continuing to search" signal, and can never by itself serve as a proof trigger for necessity. The global bound a_Leb≥0.835 remains compute-deferred, and B7 has also not yet independently closed along this branch. Research direction and methodology are due to Neo.K; this round's AI collaborating researcher and primary executor was Aletheia / GPT-5.6 Sol.