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"witness_provenance": { "found_by": [ "@dorakingx" ], "date": "2026-09-09", "found_at_samples": 20000, "survived_samples": 5000000, "tool": "gf2_fast RIS ladder (verify/gf2_fast.cpp), both sides searched jointly", "seeds": [ 93001, 93138, 93275, 93412 ] } }, "Z": { "value": 16, "confidence": "upper_bound", "witness": [ 119, 134, 146, 155, 157, 169, 181, 291, 322, 323, 324, 354, 355, 362, 363, 384 ], "witness_provenance": { "found_by": [ "@dorakingx" ], "date": "2026-09-09", "found_at_samples": 4000, "survived_samples": 5000000, "tool": "gf2_fast RIS in the packaging search; the ladder searched both sides jointly", "seeds": [ 7, 93001, 93138, 93275, 93412 ] } } }, "provenance": { "authors": [ "@dorakingx" ], "construction": "Open-boundary planar bivariate-bicycle code (Liang, Eberhardt, Chen, arXiv:2504.08887, Sec. II and Table V) with f = x + x^2 + y^2 and g = 1 + x^2 y + x^2 y^2 on a 15 x 15 grid (research/local2d/planar.py build_open_directional(15, 15), n = 2*15^2 = 450, k = 8), then reduced to n = 410 by 40 qubit removals of three kinds. (26) Restricted r=1 lattice grafting (Sec. III E): a qubit lying in exactly one stabilizer of some type is removed together with that stabilizer; accepted only if k stayed 8 and a bit-packed RIS search found nothing lighter than 16, screened at 20000 trials with a FIXED seed and then confirmed at a deeper rung, a qubit failing the confirm rung being blacklisted. Those in-loop rungs are a filter, not the evidence -- on other members of this family they let a unit of distance through -- so the claim below rests on the separate fresh-seed ladder. Its upper bound for this code equals the unreduced L = 15 code's own ladder bound, which is consistent with the reduction having lost nothing at this size but does not prove it: both are upper bounds and no lower bound exists for either. (13) Weight-1 stabilizer cleanup, which runs no distance search at all because it needs none (Sec. III D step 4, research/local2d/boundary_engine.py _cleanup): a qubit carrying a weight-1 stabilizer cannot appear in any opposite-type check, so multiplying that row into the same-type rows containing it and deleting the qubit preserves k and the distance exactly; the elimination cascades until no weight-1 or zero-degree qubit is left. (1) Capped merge-graft, a new step, searched at r <= 3: when a qubit lies in exactly r stabilizers of one type, pick a pivot row R_p and replace every other R_i by R_p + R_i. Those are row operations on the stabilizer generators and so change no code, only the generating set, and they leave that qubit in R_p alone, where the r=1 graft above applies; every choice of pivot is tried, since each gives a different resulting code. Each merged row is taken only if it still has weight at most 6 and its support still has maximum pairwise distance at most 4 in the layout below -- exactly the quantity the verifier measures as the interaction radius -- so the weight class and the 2D-local single-layer class are preserved by construction; the resulting graft is then accepted under the same k check and the same screen and confirm rungs, and the whole layout radius is recomputed after every accepted move. The first two moves cannot enlarge a check support -- grafting deletes, and the cleanup XOR removes exactly the one fixed qubit from a row -- while the third genuinely can: the merged row may be heavier and wider than either of the two it replaces. The caps bound that growth rather than forbidding it, and the whole layout is re-measured after every accepted move, which is why the code below still has max check weight 6 and radius exactly 4. SINGLE-LAYER integer layout: a surviving qubit whose index in the unreduced code is q = c*225 + i*15 + j (block c in {0,1}, site (i,j)) sits at (i + j, j - i + c), the unit square lattice rotated 45 degrees with the two blocks on its two sublattices; one qubit per site, minimum spacing exactly 1, maximum check diameter exactly 4. The reduction is a randomised search, so the surviving qubit set is recorded by the coordinates in this file rather than re-derived.", "references": [ "arXiv:2504.08887", "arXiv:2308.07915" ], "date": "2026-09-09", "notes": "Checked, not equivalent to any board entry: the gate reports no exact duplicate and no WL-equivalent entry. No board entry has n = 410. It dominates [[450,8,16]] on (n, k, d, w), which therefore leaves the frontier of every cell the two share. Board entries at the unreduced size n = 450 with k = 8: [[450,8,16]], [[450,8,26]]. The construction and both published reduction moves are from arXiv:2504.08887 (Sec. II, Sec. III D step 4 and Sec. III E; this repository's own research/local2d modules cite Table V of that paper for the family's ungrafted distances and Tables II-IV for its qubit-removal layouts such as [[188,8,9]], which come out of the cleanup rather than the graft), but this parameter set, the capped r=2 merge-graft step and this single-layer layout are not, so novelty is left unknown and literature novelty is unverified.", "model": "Claude Opus 5 (Claude Code)", "origin": "submission", "novelty": "unknown" }, "family": "bivariate-bicycle", "locality": { "coordinates": [ [ 0.0, 0.0 ], [ 1.0, 1.0 ], [ 2.0, 2.0 ], [ 3.0, 3.0 ], [ 4.0, 4.0 ], [ 5.0, 5.0 ], [ 6.0, 6.0 ], [ 7.0, 7.0 ], [ 8.0, 8.0 ], [ 9.0, 9.0 ], [ 10.0, 10.0 ], [ 11.0, 11.0 ], [ 12.0, 12.0 ], [ 14.0, 14.0 ], [ 1.0, -1.0 ], [ 2.0, 0.0 ], [ 3.0, 1.0 ], [ 4.0, 2.0 ], [ 5.0, 3.0 ], [ 6.0, 4.0 ], [ 7.0, 5.0 ], [ 8.0, 6.0 ], [ 9.0, 7.0 ], [ 10.0, 8.0 ], [ 11.0, 9.0 ], [ 12.0, 10.0 ], [ 13.0, 11.0 ], [ 14.0, 12.0 ], [ 2.0, -2.0 ], [ 3.0, -1.0 ], [ 4.0, 0.0 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