A₆(6,5) = 27 — the deficit growth law

Unsealed Prediction

{ "P1_deficit_law": { "claim": "A_6(6,5) = 27", "confidence": "low-moderate; this is a two-point extrapolation dressed as a law", "falsification_bands": "27 exact = law strongly supported; 25-29 = weakly supported; outside [25,29] = REFUTED; >=31 impossible anyway (puncturing gives A_6(6,5) <= A_6(5,4) <= 31 if the in-flight verification completes)", "reasoning": "deficits from Singleton ceiling 36 read delta(3)=0, delta(4)=2, delta(5)=5; increments 2,3 suggest next increment 4, so delta(6)=9 -> 27" }, "P2_balance_principle": { "claim": "at N near the true value, the hard cubes will be exactly those whose row AND column patterns are balanced (max occupancy spread), and lopsided cubes will be margin-infeasible or trivially UNSAT", "computed_structure_at_predicted_N": { "26": { "cubes": 784, "margin_feasible": 347, "patterns": 28 }, "27": { "cubes": 484, "margin_feasible": 222, "patterns": 22 }, "28": { "cubes": 324, "margin_feasible": 156, "patterns": 18 } } }, "P3_core_locality": { "claim": "minimal UNSAT cores of the fixed-cell sub-instances will concentrate on cell-pairs within the hole-bearing rows/columns (the balanced sub-grid), not spread uniformly over the 36 cells", "confidence": "a guess; stated so the core-mining result can refute it" }, "author": "Claude (Opus 5) + David Thompson, TWO SLITS / Euler's shadow program", "sealed": "2026-08-11T08:51:45-0500", "status_at_seal": "A_6(4,3)=34 verified both directions vs literature; A_6(5,4)>=31 verified (5 witnesses); A_6(5,4)<=31 in flight, 19/25 cubes UNSAT, 0 SAT; A_6(6,5) probe QUEUED BUT NOT STARTED on etacompute (N=24,26,28,30) and synapse (N=20,22,25)" }

Verification Notes

REFUTED. The law predicted A₆(6,5) = 27 with a falsification band of [25,29]. The true value is 31 (Sadhu, arXiv:2608.01606), outside the band in the direction the law could not accommodate. This was a two-point extrapolation dressed as a law, and the seal says so in its own confidence field — 'low-moderate'. Recording it as a clean miss is the point of sealing it. What survived: the miss located a real feature. The deficit does not grow the way the law said, it flattens, and that flattening became the cliff question we are still working on. This remains the most scientifically productive artifact in the vault.