Facing Round 23's 0/60, Verify Before Escalating: Deep Single-Witness B₇ Refutes the Premature Inference That Joint Witnesses Are Needed, Three of the Four Hardest Cells Close on the Spot

Round 24 (AMRAL-LUC-FC-R24, 2026-09-19) deals with a tempting but unexamined inference left over from the previous round. Under Round 23's shallow search — 12 necessity-marked cells × 5 witnesses at lift depth 16 — all 60 pilot edges scored zero, none reaching strict COMPLETE. Looking at that number alone, it is easy to jump straight to the inference that "the single-witness portfolio is no longer enough, and the next step should be to open a joint multi-witness search tree"; but a joint search pushes the placement dimension from a single witness's 3D (φ,x,y) up to a two-witness 6D (φ₁,x₁,y₁,φ₂,x₂,y₂), and if the two single-witness trees each need L₁ and L₂ terminal placement boxes respectively, the cost of naive Cartesian joint refinement approaches L₁×L₂ rather than L₁+L₂ — starting that work rashly, before there is any evidence of necessity, could be an extremely costly mistake. Round 24 therefore does not escalate directly; instead it first turns around and examines the inference itself: taking the four hardest reference cells flagged by Round 23, it continues deepening the search using only the single witness B₇. The result is the opposite of that tempting inference — cell 0 reaches COMPLETE at depth 30 (18555 nodes, 9278 CERT leaves, 0 pending, the forest identity 18555=2(9278)-1 verified, deterministic replay PASS, though the document itself explicitly labels this only a REFERENCE-COMPLETE-EDGE-PILOT, not a publication-grade interval certificate), and cells 1 and 2 both reach COMPLETE at depth 32; only cell 3 remains PARTIAL at depth 30, its unresolved volume having contracted to 1.49×10⁻⁷ with no plateau evidence. This round also formally proves the Joint Dominance Theorem (the joint value is always no less than any one of its single-witness values, so once a cell already has a strict single-witness edge, a joint proof for that cell is mathematically redundant) and the Pointwise Joint-Necessity Theorem — the latter sets the bar for "joint is truly necessary" far higher than "no single-witness edge has closed yet": joint interaction only becomes truly necessary when every witness in the portfolio has a verified counterexample at the very same exact point q. The four hardest reference cells therefore converge from {0,1,2,3} down to just {3}, though this round states plainly that this is only strategic evidence from four cells — cells 4–11 and the full 77-cell atlas have not yet been systematically rerun at this depth, and the global bound a_Leb≥0.835 remains compute-deferred. Research direction is chosen and led by Neo.K; this round's execution was carried out by Aletheia / ChatGPT, GPT-5.6 Sol.