n
64
Target cell: weight-8 x unrestricted (max check weight N = 8, no layout). Bars at n = 64 in this cell, computed from the board: k >= 19 with d >= 6, k >= 23 with d >= 4, k >= 35 with d >= 2; the largest k on any n <= 64 entry is 34 ([[64,34,2]], d = 2). Hypothesis: the family's w_X = w_Z = 2 corner at ell = 3 reaches k = 34 with enough distance to clear the d >= 4 bar, which [[64,34,2]] cannot answer — the two would then differ on d alone, and that is exactly the pattern the gate flags for a matched-depth audit.
Two sweeps of the family at ell in {3,4}: 71 configurations earlier, then every (wX, wZ) pair with 60 random multiplier/shift variants each (13,920 builds, exact k from GF(2) ranks), best-k variant per bar-clearing pair screened at 50,000 RIS trials (seed 7) — 20 configurations at ell = 3. The candidate is the paper-default (2, 2) point (a = c = [0, 1], b = d = 0), k = 34, unimproved by 80 random variants. At ell = 3 no bar-clearing pair (k >= 19) screens above d = 4, and the pairs that do reach d = 8 top out at k = 18 (the paper's own [[64,18,8]]), one short of the k = 19 that bar needs.
Witness ladders (upper bounds, min over X/Z sides; flat across three fresh seeds at every rung):
| trials/side | seed 101 | seed 202 | seed 303 | | ---: | ---: | ---: | ---: | | 100,000 | 4 | 4 | 4 | | 500,000 | 4 | 4 | 4 | | 2,000,000 | 4 | 4 | 4 |
CI-depth confirmation, two fresh seeds at 8,000,000 trials each (the frontier budget of verify/gate_changed.py): d <= 4 on both, flat.
Matched-depth peer audit (the gate flagged this as a d-only gain over [[64,34,2]] w=8): both entries measured at 2,000,000 trials, seeds 51 and 52, pair depth 64, same instrument —
64-34-4: claim d<=4 seed 51: d<=4 (X) seed 52: d<=4 (X) holds 64-34-2: claim d<=2 seed 51: d<=2 (X) seed 52: d<=2 (X) holds DECISION: credible -- both claims held at matched depth, the gain survives
Final claim: d <= 4, witness-backed upper bound (`confidence: upper_bound`), not certified exact. kd^2/n <= 8.5 at the witnessed bound. The gate (verify/validate_candidate.py) returned passed: true with label `advances the weight-8 x unrestricted board on d, k, n; its gain over [[64,34,2]] is d-only: distance is the suspect axis ...` — resolved by the audit above; verdict JSON staged beside the submission JSON. The candidate stays board-advancing down to d = 3 (the bar there is k >= 25); only a refutation to d = 2 would erase the gain, and no rung at any budget found anything below 4.
pair with k >= 19 screened at d <= 4, and every pair screening at d = 8 has k <= 18 ([[64,18,8]] itself, and its asymmetric neighbours 17 and 16), so the k >= 19 with d >= 6 opening was not reached.
structural ceiling of the (2, 2) point over 80 random variants.
check weight 32 and breaks the n <= 700 cap for w > 8.
Tools: an affine-Frobenius constructor written for this campaign (local staging output, not committed — the reproduction recipe below is self-contained), the research kit's surrogate distance search with the gf2_fast accelerator, the trusted gate verify/validate_candidate.py, research/audits/leader_audit.py pair for the peer audit, and verify/gate_changed.py for the deep budget. Approx compute: ~15 min of screening + ~5 min of ladders and confirmations (n = 64 is cheap).
Rebuild from the paper's Eq. (4) with ell = 3 (N = 8, n = 64), w_X = w_Z = 2, a_0 = alpha^0, a_1 = alpha^1, c_0 = alpha^0, c_1 = alpha^1 (F_8 under x^3 + x + 1), b = d = 0: P_X[u][j] = a_u * l_j, P_Z[v][j] = c_v * l_j^2, lifted to size-8 dyadic permutation matrices (row r of block (u, j) writes its 1 in column j*8 + (psi(p) XOR r)). k = 64 - rank(HX) - rank(HZ) = 34; HX HZ^T = 0 by Theorem 2. Row weight 8, column weight 2.