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129, 140, 149, 156, 167, 172, 189, 195, 227, 232, 238, 239, 240, 241, 264, 268, 275, 276, 305, 335, 341, 346, 349, 378, 380, 381, 388, 389 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-10-16", "found_at_samples": 400, "survived_samples": 400, "tool": "research/kit/surrogate.lightest_logical", "seeds": [ 1 ] } }, "Z": { "value": 42, "confidence": "upper_bound", "witness": [ 0, 1, 2, 9, 10, 12, 41, 44, 49, 55, 85, 114, 115, 122, 126, 149, 150, 151, 152, 158, 163, 201, 218, 223, 234, 241, 250, 261, 303, 305, 306, 321, 325, 328, 329, 333, 342, 353, 365, 373, 375, 380 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-10-16", "found_at_samples": 400, "survived_samples": 400, "tool": "research/kit/surrogate.lightest_logical", "seeds": [ 2 ] } } }, "provenance": { "authors": [ "@vprusso" ], "construction": "Cyclic generalized bicycle at m=195 = 3 x 5 x 13, with ord_2(m)=12, so x^195 - 1 factors over GF(2) into 20 cyclotomic cosets, 1 of size 1, 1 of size 2, 3 of size 4, 15 of size 12. The divisor g is assembled from coset minimal polynomials to degree 25, which fixes k = 2 deg g = 50 by construction, and (a,b) was drawn from the ideal (g) by information-set decoding over sparse supports, keeping weight 12 and weight 12 in the two blocks. Both weights are even, so x+1 divides the gcd; x+1 is x^1 - 1, so the only block logical this forces has weight m and is far above the claim. m is composite, so the quotient fold acts at t=3, t=5, t=13, t=15, t=39 and t=65; a fold over every equal-orbit partition of the qubits returns nothing lighter than 54, clear of the claim. H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T] on n = 2m = 390 qubits at uniform row weight 24, with a = [10, 29, 45, 48, 86, 138, 141, 144, 161, 179, 183, 192] and b = [3, 19, 32, 33, 40, 87, 103, 104, 119, 133, 162, 190].", "references": [], "date": "2026-10-16", "notes": "", "model": "Claude Opus 5" }, "family": "generalized-bicycle" }