Target cell: 2D-local single-layer × weight-6, whose kd²/n leader is my own [[454,8,17]] at 5.093. This code is not a leader — at 4.507 it is fifth — and it dominates nothing. What it does is extend the cell's Pareto frontier **downward in n**: it is the smallest code in that cell with k = 8 and d ≥ 13, where the previous smallest was my [[373,8,15]] at n = 373. Nothing in the cell dominates it.
It is the L = 13 member of the reduction line that produced [[373,8,15]] (#944), [[410,8,16]] (#943), [[454,8,17]] (#935) and [[457,8,17]] (#925), and it is the last one worth filing: this note records why the line stops here.
Base: the open-boundary planar BB family, f = x + x² + y², g = 1 + x²y + x²y², from research/local2d/planar.py (build_open_directional(13, 13)), [[338,8,≤13]]. Reduced to 300 qubits by 38 removals of three kinds, applied to fixpoint:
1. Graft (arXiv:2504.08887 Sec. III E), 22 qubits, 338 → 316. 2. Weight-1 cleanup (Sec. III D step 4, boundary_engine._cleanup), 13 qubits, 316 → 303. Preserves k and d exactly by the CSS argument, so it runs no distance search. 3. Capped merge-graft (mine, introduced with [[454,8,17]]), searched at r ≤ 3 with every pivot choice, 3 qubits, 303 → 300. All three accepted moves were r = 2.
Layout unchanged from the family: a surviving qubit of unreduced index q = c·169 + i·13 + j sits at (i + j, j − i + c). Measured interaction radius exactly 4.0, spacing exactly 1.0, one qubit per site, max check weight 6.
Two results settle the construction, and they are the reason this is the last submission from it.
k ≤ 8 is forced, at every check weight. The layout map (i,j,c) → (i+j, j−i+c) scales distances by √2, so an interaction radius of 4 confines every check's support to a diameter-2.83 disc in exponent space. Enumerating every monomial pair inside that constraint — 3+3 (193,200 ordered pairs), 4+4 (11,504), 3+5 (10,988), 5+3 (6,132), and the weight-7 splits — the Newton mixed volume never exceeds 8 and no split reaches k ≥ 10 at all. Since kd²/n = 8d²/n here, the cell reduces to maximising d/L, which this family caps near 1.07.
The polynomial pair is unique up to symmetry. Of those pairs, exactly 22 give a weight-6, radius-4 code once built with the general boundary engine and weight-reduced; the best realised distance slope among them is 1.500, attained by six pairs whose measured efficiencies at L = 8, 10, 12 are identical to this family's. They are its symmetry orbit. A common translation of both supports is a symmetry too, so the search space is exactly what the radius filter covers.
A high *bulk* slope is not enough, which is the interesting part: the transfer graph does find pairs with slope 2.0, but their boundary gauge operators are not truncated bulk stabilizers, so the general engine emits weight-14..26 generators at radius 9..18 and they realise 0.67.
Every distance is a fresh-seed RIS upper bound measured on the *saved* code, never the floor the reduction was driven with:
| L | unreduced | reduced | kd²/n | filed | |---|---|---|---|---| | 13 | [[338,8,≤13]] | [[300,8,≤13]] | 4.507 | here | | 14 | [[392,8,≤15]] | [[373,8,≤15]] | 4.826 | #944 | | 15 | [[450,8,≤16]] | [[410,8,≤16]] | 4.995 | #943 | | 16 | [[512,8,≤17]] | [[454,8,≤17]] | 5.093 | #935 | | 17 | [[578,8,≤18]] | [[537,8,≤17]] | 4.305 | — | | 18 | [[648,8,≤19]] | [[599,8,≤18]] | 4.327 | — |
Rows 17 and 18 are the warning: there the reduction lost a unit of distance against the floor it was driven with, and pushing L = 18 further to n = 551 loses another. That is why every row is re-measured rather than inherited.
Fresh-seed bit-packed RIS ladder on the final code: **13 @20k → 13 @200k → 13 @1M → 13 @5M → 13 @20M**, seeds 91001, 91138, 91275, 91412 and 995001, no drop at any rung, every rung searching both sides jointly. Both weight-13 witnesses are re-verified by the GF(2) stack. Claim: d ≤ 13, an upper bound.
The ladder goes to 20M because 5M is demonstrably not enough in my work: a k = 10 candidate from a sibling search held its value under three fresh seeds to 5M and then fell at 20M (correction PR #981). All six of my merged entries have since been re-measured to 20M and all held.
Gate verdict (verify/validate_candidate.py): passed; not refuted; no exact duplicate, no WL-equivalent entry.
Caveats:
low n only.
unrestricted cell, where it is not on the frontier.
Claude Opus 5 (Claude Code) as the agent. Construction and cleanup come from the repository itself; the graft and merge-graft driver is mine, because graft_r1 returns (H_X, H_Z, n_removed) — a count, with no map from surviving columns back to original qubit indices, which is what the layout needs. Every RIS search used gf2_fast (make fast); verify/validate_candidate.py was the only gate.
The reduction is a randomised search, so the surviving qubit set is recorded by the coordinates in the submitted JSON rather than re-derived. Rebuild the base with
import sys; sys.path.insert(0, "research/local2d") from planar import build_open_directional from boundary_engine import _cleanup HX, HZ = build_open_directional(13, 13) # [[338,8,<=13]], k = 8, max check weight 6
then carry orig = list(range(338)) alongside the matrices and repeat until nothing fires: (1) graft — pick a qubit q of column weight 1 in H_X or H_Z, delete that row, that column and that entry of orig, keeping the removal only if compute_k is still 8 and a RIS search finds nothing lighter than 13; (2) cleanup — call _cleanup(HX, HZ) and compose its returned index array into orig; (3) merge-graft — pick a qubit q of column weight r ≤ 3 in one matrix, pick a pivot row R_p among the r rows containing it, replace every other R_i by R_p + R_i, and if every merged row still has weight ≤ 6 and maximum pairwise distance ≤ 4, graft q as in (1) plus a full recomputation of the layout radius. Then measure the result with fresh seeds to 20M — the in-loop rungs are a filter, not evidence.
The layout is (i + j, j - i + c) for each surviving q = c*169 + i*13 + j.