The S-threshold law for packing-array cube satisfiability

Unsealed Prediction

{ "P1_constant_offset": { "claim": "last-100%-SAT threshold = Smin(N) + 9, invariant in N", "falsification": "if the observed last-100% S at N=28 lies outside 113 +/- 2, the constant-offset law is REFUTED", "observed": { "N20": 9, "N22": 10, "N25": 9 }, "prediction_N28": "Smin=104 -> last 100%-SAT band at S=113", "prediction_N30": "Smin=120 -> last 100%-SAT band at S=129" }, "P2_shrinking_crossover": { "claim": "the 50% crossover offset above Smin DECREASES with N: observed 17, 16, 15 at N=20,22,25", "falsification": "offset outside 13..15 at N=28 refutes the linear-decay reading", "prediction_N28": "crossover at S = 104 + 14 = 118 (offset 14 +/- 1)" }, "P3_universality": { "caveat_stated_honestly": "PA19 S-data was already partially inspected tonight before this seal, so P3 is NOT a clean out-of-sample test and is recorded as weaker evidence", "claim": "the same law with the same functional form holds in a DIFFERENT family (k=8, the PA(19,8,6) table), with only the offset re-fit" }, "derived_from": "6050 labeled cubes at N=20, 4608 at N=22, 2048 at N=25 (family pa(6;6), K=4) \u2014 a sharp phase transition, 100% SAT below threshold, 0% above, window 9-14 wide", "law": "S-threshold law for packing-array cube satisfiability. S = sum_rows C(occ,2) + sum_cols C(occ,2) counts same-line pairs (the strong constraints). Smin(N) = S of the most uniform occupancy split.", "sealed": "2026-08-11T23:16:27-0500", "test": "run pa(6;6) at N=28 (never run; deficiency 8) and read the threshold" }

Verification Notes

BOTH CLAIMS REFUTED. An earlier version of this page reported P1 as CONFIRMED. That was wrong and is corrected here rather than quietly replaced. P1 (constant offset) REFUTED: predicted the last-100%-SAT threshold at Smin(N)+9 = 113 ± 2 for N=28, and this page first reported it confirmed at 112. That score was taken on 313 of the 324 cubes at N=28 — the experiment was not finished. The eleven missing cubes were the hardest, and ten of them lay inside the very band the claim asserts is 100% satisfiable. Completing the census refuted it: unsatisfiable cubes appear at offsets +3, +4, +5, +6 and +7, so the boundary is at +2 (S = 106), seven below the sealed band. The decisive cube was confirmed unsatisfiable by three independent solvers with no disagreement, and the remaining cubes were closed with machine-checkable proofs. P2 (shrinking crossover) REFUTED: predicted the 50% crossover offset would continue decreasing, giving 118 ± 1. Observed 115. Outside the band. WHAT THIS COST THE HEADLINE. The scorecard's finding — that we predict constants well and trends badly — rested on P1 being a confirmed constant. With P1 refuted, constants stand at one for two and that finding is dead. The surviving lesson is narrower and less flattering: we score claims before the evidence is complete, including when scoring ourselves. P3 (universality) was recorded at seal time as NOT a clean out-of-sample test, because PA19 S-data had already been partially inspected that night. It is excluded from the scorecard for that reason, by its own admission.