[[416,36,8]] — non-abelian lifted product over ZSZ(8,4,5)

Credit

This is not an original find: the gap was identified, and the tooling to fill it was written, by @vprusso (using Claude Fable 5.1), in the campaign documented at fieldnotes/2026-09-16-lifted-product-girth-cap.md. @vprusso also wrote research/kit/nonabelian_lp.py (the lifted-product constructor and sampler this submission reuses unmodified) and submitted the neighboring codes/624-52-9.json from the same sweep. That fieldnote explicitly states this exact point was found but left unsubmitted ("left for a later run"); this submission only reruns that already-merged tooling to confirm and package it. No new construction or search method is claimed here.

Direction & hypothesis

fieldnotes/2026-09-16-lifted-product-girth-cap.md (Finding 5) explicitly flags three candidates as found-but-unsubmitted: "Not packaged, ladder-flat at 100k trials: the weight-5 points [[416,36,8]], [[520,44,8]] and [[546,46,8]] ... left for a later run." This is one of those three. The construction is a lifted product of two 2x3 monomial base matrices over F_2[G] for a non-abelian metacyclic group G, giving check weight 5 (row weight 3 of A + row weight 2 of B, and symmetrically for B/A) — a thinner rate (1/13) than the group's published weight-9 form (arXiv:2607.28795, arXiv:2607.27644), landing in a check-weight-5 region of the board that was empty before this campaign's [[624,52,9)] submission.

What was searched

Reproduced the original campaign's sweep using the merged research/kit/nonabelian_lp.py module (constructor + sampler already in the repo — no new code needed): groups = every ZSZ(l1,l2,q) presentation with 31 <= |G| <= 53, plus the small non-abelian direct products in the same order range (small_nonabelian_groups); profile A_w = B_w = [[1,1,1],[1,1,1]] (2x3, all monomial entries, check weight 5).

• 1,500 random codes across 76 groups, staged screening: 400 fast-RIS

trials first, keeping anything at or above efficiency 4.7 or on the running Pareto frontier. 1,307 distinct codes; 38 kept past the first stage.

• This reproduced all three fieldnote-flagged points exactly, plus one not

previously flagged: [[624,54,8]] (submitted separately), which does not dominate or get dominated by the already-board [[624,52,9]] (higher k, lower d — a genuinely separate frontier point at the same n).

Evidence trail

• Screening: d<=8 at 400 fast RIS trials.
• Deep confirmation: d<=8 held at 500,000 fast RIS trials (seed 1234).
• Packaging: submit.make_submission witness search (3,000 trials/side,

numpy) independently found weight-8 witnesses on both sides, confirming d=8 by a different code path than the fast-backend confirmation.

• Gate: verify/validate_candidate.py returned passed: true,

refute.refuted: false, novelty.board_advancing: true for (weight-6, unrestricted), dominated_by: [].

• Claim: witness-backed upper bound, not exact.

Dead ends

None specific to this point — see fieldnotes/2026-09-16-lifted-product-girth-cap.md for the prior campaign's negative results on adjacent profiles (all-weight-2 entries capped by Cayley-graph girth; rate-2/5 profiles capped at d<=5). This run's only new negative information is implicit: nothing at 400 trials beat the fieldnote's three flagged points by more than the [[624,54,8]] bonus point.

Tools

Claude Sonnet 5 (claude.ai chat, computer-use/bash sandbox), single CPU core, no GPU, gf2_fast backend built locally via make fast. Repo tooling: research/kit/nonabelian_lp.py (zsz_params, sample_nonabelian_lp, rebuild), research/kit/search.py (screen, pareto_frontier), research/kit/surrogate.py (distance_rand, fast backend), research/kit/submit.py (make_submission, save_submission), verify/validate_candidate.py. No new constructor code was written; this run only exercises the module the original campaign already merged.

Reproduction

Group ZSZ(8,4,5): relation y x = x^5 y, x^8 = y^4 = 1, element x^a y^b at index 4a+b, |G|=32.

import sys
sys.path.insert(0, "research/kit")
from nonabelian_lp import zsz, lifted_product_base
from submit import make_submission, save_submission

mul = zsz(8, 4, 5)
A = [[[14], [29], [15]], [[5], [23], [0]]]
B = [[[19], [30], [9]], [[23], [25], [9]]]
HX, HZ = lifted_product_base(mul, A, B)   # n=416, k=36

doc = make_submission(
    HX, HZ,
    name="[[416,36,8]] non-abelian lifted product, ZSZ(8,4,5)",
    construction="Lifted product of two 2x3 monomial base matrices A, B over "
                 "F_2[G], G = Z_8 x|_5 Z_4 (metacyclic), entries acting by the "
                 "left (A) and right (B) regular representation.",
    authors=["@msilve160"], family="lifted-product",
    references=["arXiv:2607.28795", "arXiv:2607.27644"],
    confidence="upper_bound", trials=20000, seed=0,
)
save_submission(doc, "codes/416-36-8.json")   # only after human review + PR