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[ 2, 16, 22, 33, 43, 63, 74, 113, 118, 138, 140, 157, 160, 167, 175, 181, 192, 198, 203, 219, 235, 244, 275, 280, 291, 293, 305, 315, 316, 322, 327, 328, 332, 335, 345, 348, 366, 377, 387, 390, 404, 418, 422, 423, 428, 438, 439, 464, 465, 469, 495, 514, 515, 527, 531, 534, 547, 585, 593, 596, 616, 621, 627, 640, 655, 671 ], "witness_provenance": { "found_by": [ "@FarLab", "@MathysRennela", "@MicheldeMessieres" ], "date": "2026-10-09", "found_at_samples": 1000000, "seeds": [ 404 ], "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]]), reproducing that entry's exact published recipe: a(x) support is 6 fixed points plus 4 size-2 orbits under the symmetry j->32j (mod 341, ord=2), b(x)=a(x^4) mod (x^341-1) (Frobenius pairing, shift s=0). Found by screening ~140 candidates built with this exact recipe shape (weight/fixed-point combos (14,6),(14,4),(16,6),(16,8),(16,4),(18,8),(18,6), shifts s in {0,1,7,97,170}) at 20000 trials via a locally-compiled gf2_fast accelerator, then deepening the top 5 to 500000 trials; this candidate (w=14,f=6,s=0) held stable (~9% drop vs ~90%+ collapse on generic random-symmetry candidates tested in the same session).", "origin": "submission", "date": "2026-10-03", "model": "Claude Sonnet 5", "notes": "TIGHTENING (2026-10-03, same day as the d_X correction above): this cyclic (abelian, t=1) generalized-bicycle construction has an exact qubit permutation symmetry -- negate the index mod 341 within each of the two n/2 blocks, then swap the two blocks -- that maps H_X onto H_Z exactly (as a set of check rows), proven by direct computation, not just argued. This means d_X and d_Z are mathematically forced to be equal for this entire family (any cyclic/abelian 2-block generalized-bicycle code), with no search needed: applying this permutation to the already-verified weight-85 X-type witness produces a weight-85 Z-type witness (verified: commutes with H_X, increases rank of H_Z when appended -- a genuine nontrivial logical). d_Z updated from the stale 95 (an un-re-deepened carryover from the original, since-refuted submission) to 85, matching d_X exactly. d (=85) and kd^2/n (=974.633) are unchanged by this -- both sides were already <=85, this only replaces a loose bound with the exact matching one.\n\nCORRECTION (2026-10-03): the CI's verify gate (verify/gate_changed.py, seed=1331609806) refuted the original d=95 claim, finding a weight-85 X-type logical operator (flagged by @vprusso). 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 operator, not a tool artifact. Local greedy reduction (XOR against every row of H_X) does not reduce the weight-85 witness further -- it is a local minimum under single-stabilizer combination. Our own follow-up deep search (24,000,000 trials, gf2_fast, 20 threads, seed=2026) did not find anything lighter than 85 and in fact only reached d_X=94 at that seed -- confirming our own random-trial search has a real blind spot for this specific algebraic structure (the CI's accelerated 'fast pass' found it with a smaller but differently-structured budget; our own gf2_fast.distance_rand_witness calls never did, even at 3x the trial count). d_X updated from 106 to 85 (now the binding side); d updated from 95 to 85; kd^2/n updated from 1217.45 to 974.63. d_Z (95) was NOT independently re-deepened after this correction -- carried over from the original submission's 500k/8M runs, flagged here per our own post-incident protocol (never assume one side is safe just because the other side was refuted).\n\n--- Original submission notes (pre-correction) ---\nDistance confirmed by two independent deep searches with this repository's own gf2_fast accelerator (compiled locally): 500,000 trials/seed=999 found d_X=95 (lighter side), 8,000,000 trials/seed=42 (CI-equivalent budget, 20 threads) found d=96. Reporting the tighter value, d=95, as witness-backed upper bound. Equivalence check: this (m,a,b) triple is distinct from the board leader and all other generalized-bicycle entries at n=682 I am aware of -- a is NOT the leader's a (different support), though both share the same ring Z_341 and the same Frobenius-pairing recipe shape; no isomorphism to an existing entry was found or expected given the different support. 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 <= 64 on 2026-10-09 by @FarLab with the orbit-fold search of PR #2902 (X: 77 -> 64; Z: 77 -> 66; 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" }