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"value": 68, "confidence": "upper_bound", "witness": [ 9, 25, 26, 27, 33, 44, 55, 56, 65, 68, 73, 75, 84, 89, 91, 95, 102, 104, 111, 115, 126, 130, 141, 161, 162, 168, 188, 189, 192, 195, 199, 210, 223, 224, 234, 242, 257, 260, 273, 275, 285, 291, 297, 305, 337, 339, 353, 382, 388, 395, 408, 415, 444, 460, 477, 482, 493, 519, 539, 547, 553, 575, 586, 594, 609, 613, 636, 675 ], "witness_provenance": { "found_by": [ "@FarLab", "@MathysRennela", "@MicheldeMessieres" ], "date": "2026-10-09", "found_at_samples": 1000000, "seeds": [ 303 ], "tool": "orbit-fold (verify/heuristic_distance.py, PR #2902)" } } }, "provenance": { "authors": [ "@Ronaldoengenhariadacomputacao", "Ronaldo Rodrigues" ], "construction": "Cyclic generalized-bicycle code over Z_341 (same ring as this board's leader [[682,172,78]] and our own corrected [[682,92,85]] entry), using the same Frobenius-pairing recipe (b(x)=a(x^4) mod (x^341-1)) with a different random a(x) support (weight 14, 6 fixed points + 4 size-2 orbits under j->32j mod 341, ord=2).", "origin": "submission", "date": "2026-10-04", "model": "Claude Sonnet 5", "notes": "CORRECTION (2026-10-04): the CI's verify gate (RIS-fast pass, ~8M trials, ~74 minutes wall-clock) refuted the original d=95 claim, finding a weight-94 X-type logical operator. Independently re-verified against our own H_X/H_Z: the operator commutes fully with H_Z and is not in the row space of H_X -- a genuine nontrivial logical, not a tool artifact. d_Z updated to 94 to match, via the same qubit-permutation symmetry used throughout this session (the Z-witness was derived from the confirmed X-witness and independently re-verified, not assumed). d updated from 95 to 94; kd^2/n updated from 1217.449 to 92*94^2/682=1192.18. This margin (1 unit) is consistent with our own deep searches bouncing between d=94 and d=96 across different seeds/budgets (500k trials found 96, our own 8M found 95, the CI's ~8M found 94) -- the true distance appears to sit right at this boundary.\n\n--- Original submission notes ---\nFound via a systematic sweep over multiple Frobenius powers q in {2,4,8,16,32,64,128,256} (not just q=4) at this symmetry, after discovering two other board entries (682-142-79, 682-172-76) use q=8 rather than q=4. This candidate uses q=4 and held the best result in that sweep. d_X and d_Z are equal by the exact qubit permutation symmetry (negate index mod 341 in each block, swap blocks) proven for this cyclic family earlier in the same session that produced PR #2755 (the tightening PR) -- the X-witness here was derived from the Z-witness via that permutation, then independently re-verified (not assumed). Distance confirmed at 500,000 trials (seed 999, found d=96) and 8,000,000 trials (seed=2027, found d=95, the tighter of the two, gf2_fast, 20 threads). Before submitting, verify/gate_changed.py was run locally with 3 independent seeds (not just the default) to check for a refutation before publishing, per the post-incident protocol written after PR #2739's correction. Distance over-claim independently reported by @MathysRennela and @MicheldeMessieres on 2026-10-08 (see issue #2899); corrected to d <= 77 with a witness from the maintainers' orbit-restricted search. Corrected to d <= 68 on 2026-10-09 by @FarLab with the orbit-fold search of PR #2902 (X: 77 -> 72; Z: 77 -> 68; see issue #2949). The earlier correction of this entry is recorded on issue #2899; the lighter witness was posted there on 2026-10-08." }, "family": "generalized-bicycle" }