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both sides searched jointly", "seeds": [ 777001, 777138, 777275, 777412, 777549 ] } }, "Z": { "value": 15, "confidence": "upper_bound", "witness": [ 36, 50, 62, 81, 87, 95, 101, 113, 238, 268, 275, 297, 303, 304, 333 ], "witness_provenance": { "found_by": [ "@dorakingx" ], "date": "2026-09-12", "found_at_samples": 4000, "survived_samples": 20000000, "tool": "gf2_fast RIS in the packaging search; the ladder searched both sides jointly", "seeds": [ 7, 777001, 777138, 777275, 777412, 777549 ] } } }, "provenance": { "authors": [ "@dorakingx" ], "construction": "Reduction of the board's codes/392-8-15.json ([[392,8,15]]) by 20 qubits at unchanged k, unchanged check-weight class and unchanged locality class, by one move applied to a fixpoint: graft a qubit away using a low-weight element of the stabilizer ROW SPACE. Let S be an element of the row space of H_X with support T and let a be in T. Replace one generator of the subset summing to S by S itself -- the span is unchanged, that generator being S plus the rest of the subset -- and add S into every other X row meeting a, so that a survives only in S. Then apply the CNOT fan-out from a to T\\{a}, which acts on the check matrices as H_X[:, b] ^= H_X[:, a] and H_Z[:, a] ^= H_Z[:, b] for each b in T\\{a}. Only S still meets a, so the first substitution turns S into the weight-1 stabilizer X_a and leaves every other X row alone; the second zeroes column a of H_Z, because commutation forces every Z row to meet T an even number of times. Qubit a is then disentangled and is deleted: n -> n-1, k unchanged, and the Z rows only LOSE an index, so their weights and support diameters cannot rise. The weight of S separates three regimes, and the difference is entirely in the clearing step R ^= S. |S| = 1: nothing is added anywhere, so the weights, the radius AND THE DISTANCE are all preserved exactly, and no distance search is needed -- this is the row-space form of the weight-1 stabilizer cleanup of Liang, Eberhardt and Chen (arXiv:2504.08887 Sec. III D step 4), which the implementation in research/local2d/boundary_engine.py applies only to literal weight-1 generator ROWS. |S| = 2: R loses a and toggles one other index, so its weight changes by 0 or -2 and can never rise. |S| >= 3: R can grow, so every touched row is checked against the code's weight class and its locality radius, measured in the source's own layout. This generalises the capped merge-graft I introduced with [[454,8,17]], which only ever built S from the generators a single qubit happens to lie in. Here the accepted grafts were |S| = 1: 4, |S| = 2: 5, |S| = 3: 5, |S| = 4: 1, |S| = 5: 1, |S| = 6: 4. Low-weight row-space elements are enumerated as sums of at most three checks with a connected overlap pattern. Every graft with |S| >= 2 was accepted only if k was unchanged and a bit-packed RIS search found nothing lighter than 15, screened once and confirmed twice with independent seeds; those in-loop rungs are a filter, not the evidence. THE LAYOUT IS NEW, and it is the other half of the result. The source sits on a spacing-1 square lattice with two qubits per site; this code sits on ONE layer, by the map (site (i,j), slot c) -> (i+j, j-i+c). The site part is a 45-degree rotation scaled by sqrt(2) and the slot separates co-sited qubits by one unit, so the map is injective -- a collision needs i+j = i'+j' and j-i = j'-i'+1, whose sum gives 2j = 2j'+1 -- and every site carries exactly one qubit at spacing exactly 1. Whether it fits the single-layer radius cap of 4 is a constraint problem, not a search: a check pair at site displacement (a,b) lands at (a+b, b-a+dc) with dc = c_u - c_v, and the dependence on dc is NOT symmetric in its sign. A displacement of (+-2,+-2) lands at (+-4, dc), so both dc = +-1 exceed the cap and the two qubits must share a slot; but (2,-2) lands at (0,-4+dc) and (2,-1) at (1,-3+dc), each of which only ONE sign of dc breaks. Mixing equalities with implications is 2-SAT over one boolean per site (does this site swap its two slots), solved here with the implication graph and Tarjan's SCC. It is satisfiable for this code and the realised radius is exactly 4.0000 with minimum site spacing exactly 1.0000. It is UNSATISFIABLE -- a proof, not a failed search -- for every other reduction of this source I tried, which is why the reduction above is capped at the source's own check diameter of 2.8284 rather than at the class cap: an |S| >= 3 graft buys a qubit with check diameter, and check diameter is exactly what pays for a single layer. The surviving qubits' indices into the source numbering are [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14]... ", "references": [ "arXiv:2504.08887" ], "date": "2026-09-12", "notes": "Derived from codes/392-8-15.json, not an independent construction; the source's authors are @mathysrennela and the reduction is mine. The gate reports no exact duplicate and no WL-equivalent entry. It dominates [[373,8,15]], [[392,8,15]] on (n, k, d, w), which therefore leave the frontier of every cell they share with it. The measured interaction radius is 4, against 2.82843 for the source: unlike the graft and the cleanup, the merge-graft can widen a check, and it is capped by the locality class rather than by the source's own radius, so the reduced code is less local even though it stays in the same cell. Distance is a witness-backed upper bound: the deepest null result is 20000000 fresh-seed RIS trials. Literature novelty is unverified.", "model": "Claude Opus 5 (Claude Code)", "origin": "submission", "novelty": "unknown" }, "family": "bivariate-bicycle", "locality": { "coordinates": [ [ 1.0, 1.0 ], [ 2.0, 3.0 ], [ 3.0, 3.0 ], [ 4.0, 5.0 ], [ 5.0, 5.0 ], [ 6.0, 7.0 ], [ 7.0, 7.0 ], [ 8.0, 9.0 ], [ 9.0, 9.0 ], [ 10.0, 11.0 ], [ 11.0, 11.0 ], [ 1.0, 0.0 ], [ 2.0, 0.0 ], [ 3.0, 2.0 ], [ 4.0, 2.0 ], [ 5.0, 4.0 ], [ 6.0, 4.0 ], [ 7.0, 6.0 ], [ 8.0, 6.0 ], [ 9.0, 8.0 ], [ 10.0, 8.0 ], [ 11.0, 10.0 ], [ 12.0, 10.0 ], [ 13.0, 12.0 ], [ 2.0, -1.0 ], [ 3.0, -1.0 ], [ 4.0, 1.0 ], [ 5.0, 1.0 ], [ 6.0, 3.0 ], [ 7.0, 3.0 ], [ 8.0, 5.0 ], [ 9.0, 5.0 ], [ 10.0, 7.0 ], [ 11.0, 7.0 ], [ 12.0, 9.0 ], [ 13.0, 9.0 ], [ 14.0, 11.0 ], [ 15.0, 11.0 ], [ 3.0, -2.0 ], [ 4.0, -2.0 ], [ 5.0, 0.0 ], [ 6.0, 0.0 ], [ 7.0, 2.0 ], [ 8.0, 2.0 ], [ 9.0, 4.0 ], [ 10.0, 4.0 ], [ 11.0, 6.0 ], [ 12.0, 6.0 ], [ 13.0, 8.0 ], [ 14.0, 8.0 ], [ 15.0, 10.0 ], [ 16.0, 10.0 ], [ 4.0, -3.0 ], [ 5.0, -3.0 ], [ 6.0, -1.0 ], [ 7.0, 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