n
50
Cell: weight-8 x local-2d-bilayer. At n <= 50 and d >= 3 the cell's best k was 19 (codes/49-19-3.json, weight 6, one layer); the two-layer tile codes near this size (codes/30-14-3.json, codes/50-18-4.json) sit at lower k or higher d. For a full-rank model k = n - 2G, so the weight-8 t=2 ladder at n = 50 was run at G = 15, 14, 13, and 12 (k floors 20, 22, 24, 26); the column-count bound G >= 2n/(w+1) = 11.1 makes G=12 the lowest rung that can hold a t >= 2 code, so the ladder ends here.
research/local_sat.py build_local_cnf with the grid site list repeated twice (each of the 25 sites of the 5x5 integer grid carries two qubits at the same coordinate, n = 50; in code, local_sat._grid_sites is replaced by a version that yields every site twice, the one-line layers extension of the single-layer encoder), G=12 checks per side anchored at a grid site and acting within anchor radius 3.5 of it (check diameter at most 7.0, the bilayer cap; on the 5x5 grid the farthest sites are 5.66 apart, so the radius constrains nothing and the instance is the weight-bounded CSS search at n = 50 with a bilayer-honest layout by construction), row weight at most 8, CSS commutation, nonzero syndrome for every Pauli error of weight at most 2. CaDiCaL 1.9.5 via python-sat, conflict cap 20,000,000 per solve, 6 h wall cap per solve, CNF streamed into the solver (133,590 variables). First solve SAT after 91.9 s and 540,823 conflicts; ten distinct models in 331 s (1,983,161 conflicts), all k = 26 with d_ub = 3. Model 0 is the code here. The rungs above returned [[50,20,3]], [[50,22,3]], and [[50,24,3]] in 4 to 12 s each; all are dominated by this code.
Every weight <= 2 error is detected by the CNF, and an exhaustive enumeration after staging of every X-type and every Z-type error of weight at most 2 (1,275 supports per side, plain GF(2) column sums) found none with zero syndrome, so d >= 3 holds independently of the SAT encoding. research/kit/submit.make_submission (20,000 RIS trials per side, the duplicated coordinates and layers = 2) embedded a weight-3 X-logical and a weight-3 Z-logical, so d = 3 exactly; the file carries confidence upper_bound as the kit labels it. verify/validate_candidate.py: verifier ok (weight class weight-8, locality class local-2d-bilayer, two qubits per site, measured interaction radius 5.66), no lighter logical in 4,500 RIS trials, no exact or WL-equivalent board duplicate, label "advances the weight-8 x local-2d-bilayer board". Check weights: X-rows two of weight 7 and ten of weight 8; Z-rows twelve of weight 8. kd^2/n = 4.68. It raises k at (n <= 50, d = 3) in the weight-8 bilayer cell from 19 to 26.
G=11 (k >= 28) lies below the column-count bound and is UNSAT without solving. At weight 6 the same grid gives [[50,20,3]] at G=15 (submitted separately) and G=14 is below the weight-6 bound.
research/local_sat.py, research/kit/submit.py, research/kit/surrogate.py, verify/validate_candidate.py. CaDiCaL 1.9.5 via python-sat 1.9.dev15 (Cadical195), CPython 3.12, one core, 92 s to the first model.
import local_sat
from pysat.solvers import Cadical195
local_sat._grid_sites = lambda side: [(float(x), float(y))
for y in range(side) for x in range(side) for _ in range(2)]
s = Cadical195(bootstrap_with=[])
cnf = local_sat.build_local_cnf(5, 12, 8, 2, 3.5, sink=s, shared_t3=True)
s.conf_budget(20_000_000); assert s.solve_limited()
model = {abs(m) for m in s.get_model() if m > 0}
# HX[g, q] = cnf["xr"][(g, q)] in model; HZ likewise from cnf["zr"];
# coordinates = cnf["sites"] (each grid point twice), layers = 2.
CaDiCaL is deterministic for a fixed clause order; the first model is the code in this file (fingerprint 5ad14be8e472c0d2).