[[126,14,7]] — bivariate bicycle code on Z_7 x Z_9, check weight 6

Direction & hypothesis

Target: the unrestricted x weight-6 cell at n in 111..220, where the board's frontier for k >= 12 at d >= 8 is thin (frontier map of 2026-10-17). A (126, 14, 7) point at check weight 6 needed d >= 7 to clear every board entry with n' <= 126, k' >= 14, w' <= 6; the gate's novelty check reports it advances the cell on d, k, n (board-relative novelty from verify/validate_candidate.py; literature novelty unverified).

What was searched

Designed-common-zero bivariate bicycle sweep: for every torus Z_l x Z_m with l*m in 56..110 (n in 112..220), a point (alpha, beta) of mu_l' x mu_m' (l', m' the odd parts) with a Frobenius orbit of size t >= 2 (t >= 3 or 4 depending on the run) is fixed first and A, B are drawn only among the 2- and 4-term polynomials that vanish there (so k >= 2t by construction; this code's point is index [3, 0] in Z_7 x Z_9). Funnel for this job: 200454 candidates built, 151921 distinct with k >= 10 screened at 400 gf2_fast RIS trials, 400 promoted to 20,000 trials, 400 to 200,000 trials, 18 survived with the reading still above the board's weight-6 frontier (any reading below the needed d is final, since the search returns an upper bound).

Evidence trail

Witness-backed upper bound d <= 7: the lightest logical operator found is embedded in the submission (distance.X / distance.Z witnesses) and is checked by the verifier. Confirmation ladder on the box (gf2_fast, research/kit/surrogate.distance_rand_witness, 24 threads):

| trials per side | lightest logical found | |---|---| | 400 / 20,000 / 200,000 funnel readings | 7 | | 4,000,000 fast RIS trials per side, fresh seed (55143) | 7 (X) | | 4,000,000 fast RIS trials per side, fresh seed (63062) | 7 (X) | | 4,000,000 fast RIS trials per side, fresh seed (70981) | 7 (X) |

Ladder verdict: corroborated (min over fresh seeds = 7, claimed 7). The claim is an upper bound; no exactness is asserted.

Dead ends

• Random (undesigned) bivariate bicycle supports in this band: 250,000 weight-6 draws gave only

204 codes with k >= 10 and none above the frontier at 400 trials; 230,000 weight-8 draws gave 3,529 with k >= 10 and 5 survivors at 200,000 trials.

• Random 2BGA supports on 150 non-abelian groups of order 56..110 (43,800 weight-6 draws): 146 with

k >= 10, none above the frontier at 400 trials.

• Designed-common-zero weight-6 bicycle codes (45,176 draws, 7,306 with k >= 10): all readings at or

below the frontier; the [[150,16,8]] cell is reproduced many times but never beaten.

Tools

Model: Claude Fable 5.1 (Claude Code agent, campaign agent A03). Brev n2d-highcpu-64 box, gf2_fast RIS (make fast) with 60 threads; research/kit modules bb / group_algebra / coset / nonabelian_lp / gf2poly / search / surrogate / submit / prescreen; gate = verify/validate_candidate.py via research/kit/coordination.gate_and_record. Approximate compute: 12 CPU-box minutes for the sweep plus the ladder above.

Reproduction

bivariate bicycle on Z_7 x Z_9: A = x^0y^0 + x^0y^4, B = x^0y^0 + x^4y^2 + x^4y^1 + x^0y^2.

import sys; sys.path[:0] = ['research/kit', 'verify']
from bb import build_bb
HX, HZ = build_bb(7, 9, [(0, 0), (0, 4)], [(0, 0), (4, 2), (4, 1), (0, 2)])

Witness search: research/kit/surrogate.lightest_logical (8,000 NumPy trials, seeds in the witness_provenance block) tightened by gf2_fast RIS at 200,000 trials; ladder seeds are listed in the table above.