n
53
Target cell: weight-6 x local-2d-bilayer at mid-small n. On the board, k=20 at d=3 is only reachable at codes/50-20-3.json with check weight 8, and codes/54-20-4.json reaches weight 6 only at n=54 with d=4. The gap is a weight-6, k=20 point below n=54. codes/54-20-4.json (n=54, k=20, d=4, w=6, two-layer layout, interaction radius 5.6569) is one Z stabilizer short of exactly that: drop one Z check, drop the qubit that goes degree-one, and n falls to 53 while k holds at 20.
Bounded check deletion plus degree-one r=1 removal applied to small board codes (campaign label "bounded removal", 2026-09-17). This candidate is one survivor of that screen; MiMo-V2.6-Flash found the parent and reconstructed the code from it. Screening used small random-information-sampling budgets, then the repo's trusted gate at a fresh random seed, and only then packaging.
Trusted gate verify/validate_candidate.py (seed 1267866192): passed true, no lighter logical in 4620 RIS trials, no exact duplicate, no WL equivalent, board_advancing true, cell weight-6 x local-2d-bilayer.
Packaging ladder: the default 20000-trial witness search per side plus a 2000000-trial accelerator pass returned d_X<=3 and d_Z<=4, so d=3 is set by the X side and is a witness-backed upper bound, not an exact distance. Structural verification: 17 X checks, 16 Z checks, max check weight 6, k=20, score kd^2/n = 3.396. Circuit tier verified at d_circ <= 2 (X 2, Z 2) over 3 rounds, generic sequential schedule.
It strictly dominates codes/58-16-3.json (n=58, k=16, w=7) and codes/65-17-3.json (n=65, k=17, w=8): same d=3, smaller n, larger k, lighter checks. It does not displace codes/50-20-3.json (smaller n but w=8) or its own parent codes/54-20-4.json (d=4), and neither displaces it.
Larger candidates from the same screen collapsed under the 2000000-trial accelerator pass: [[288,8,26]] tightened to d<=24 (that point is already on the board), [[312,8,30]] tightened to d<=23 (then strictly dominated), [[480,4,36]] tightened to d<=32 (advances no board point). [[450,8,28]] passed the gate but is dominated by the board's [[434,10,28]]. Two closer relatives, [[39,9,4]] and [[39,10,4]], were refuted by weight-3 logicals; both are claimable only at d=3 and are strictly dominated by the board's incoming [[38,10,3]]. A separate SAT bound campaign over the remaining cells returned zero models (UNSAT at the probed weight) after roughly 43 CPU-hours and was stopped.
MiMo-V2.6-Flash (provenance.model) found and reconstructed the code: it identified codes/54-20-4.json as the parent, applied check deletion plus degree-one removal, and rebuilt H_X, H_Z and the inherited two-layer layout. Reproduction below was written from that reconstruction and re-verified independently. Verification used verify/validate_candidate.py, the submission CLI in cli/qldpc.py with its default 20000-trial witness search and 2000000-trial accelerator, and verify/check_prose.py. Packaging needs --layers 2: the inherited layout puts two qubits on one site under a single-layer assumption and the verifier rejects it. Local CPU only, 10 cores, no paid compute.
Rebuild H_X and H_Z from codes/54-20-4.json using zero-based indices: delete qubit 19, reindex survivors to 0..52, delete Z row 2 (parent row {0, 1, 14, 19, 39, 44}) and keep all 17 X rows, then drop qubit 19 from every remaining row. Keep the parent's layout columns for the surviving qubits (2 layers, interaction radius 5.656854249492381). Expected result: 17 X rows, 16 Z rows, max check weight 6, k = 53 - rank(H_X) - rank(H_Z) = 20. Pass the draft to ./qldpc submit with --authors @mathysrennela --layers 2 --note-file notes/53-20-3.md, keep the witnesses the CLI returns, and re-run verify/validate_candidate.py on the draft before trusting the distance. Do not replace the returned distance with a hand-computed number.