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105, 117, 167, 193, 245, 251, 285, 295, 301, 337, 368, 388 ], [ 67, 71, 83, 92, 106, 118, 168, 194, 246, 252, 286, 296, 302, 338, 369, 389 ] ] }, "distance": { "d": 31, "X": { "value": 31, "confidence": "upper_bound", "witness": [ 0, 5, 26, 31, 42, 56, 65, 66, 82, 88, 149, 159, 185, 195, 199, 200, 205, 206, 212, 222, 235, 250, 256, 262, 267, 314, 334, 354, 359, 363, 364 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-09-21", "found_at_samples": 1, "survived_samples": 1, "tool": "witness transport from the isomorphic lighter-bounded entry", "seeds": [ 0 ] } }, "Z": { "value": 31, "confidence": "upper_bound", "witness": [ 20, 21, 25, 30, 50, 70, 117, 122, 128, 134, 149, 162, 172, 178, 179, 184, 185, 189, 199, 225, 235, 296, 302, 318, 319, 328, 342, 353, 358, 379, 384 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-09-21", "found_at_samples": 1, "survived_samples": 1, "tool": "witness transport from the isomorphic lighter-bounded entry", "seeds": [ 0 ] } } }, "provenance": { "authors": [ "@seunomonije" ], "model": "Claude Fable 5", "construction": "Generalized bicycle code on Z_195 (Panteleev-Kalachev): H_X=[A|B], H_Z=[B^T|A^T] with A, B circulants of a(x) = sum x^i, i in {0,20,51,87,93,103,137,143}; b(x) = sum x^i, i in {0,26,76,88,102,111,123,127}. Both are weight-8 codewords of the classical cyclic code for g = gcd(a, b, x^195-1) of degree 41 (the same divisor underlying the [[390,82,30]] board entry), giving k = 2*deg g = 82 by construction. The pool of weight-<=8 codewords of saturates at 12 cyclic-shift classes (60k+ RIS trials, 3 seeds); this pair is one of the 51 previously-unexplored class pairs from the exhaustive 66-pair enumeration of that pool. Distance evidence: staged randomized-information-set ladder 2M/8M/24M trials (fresh seed per rung, pair_depth 8) read 36/36/32; refutation barrage 3x8M trials (seeds 777/999/424242) plus pair_depth-16 and pair_depth-24 passes at 3M trials, and a 48M-trial witness pass: lightest logical ever witnessed has weight 32 (found independently at 24M/seeded ladder and 8M/seed 777). Claimed d <= 32 = the lightest witnessed logical.", "references": [ "arXiv:1904.02703" ], "date": "2026-07-18", "notes": "Model: Claude Fable 5. Distance is an upper bound from ~124M randomized-information-set trials across 8 independent seeded passes and three pair depths; no logical below weight 32 was ever witnessed." }, "family": "generalized-bicycle" }