[[546,236,5]] — the class 28 periodic identification of the {7,7} PSL(2,13) hyperbolic surface code: same n, k and check weight as the filed entry, distance 5 instead of 4

Direction & hypothesis

weight-8 x unrestricted, family topological, from arXiv:2610.10948 (Mahmoud and Rayan). The paper's Table 2 tabulates this member of the family as [[546,236,5]]; the board's codes/546-236-4.json carries d <= 4, and its own note records why: five of the ~1500 generator classes were measured, all five read 4, and the d = 5 claim "is neither claimed nor refuted". That is an open question with a published answer attached, and the construction pins n, k and check weight, so a class that does reach 5 changes exactly one axis. The hypothesis was simply that the filed entry sampled an unlucky slice of the class set.

What was searched

For {7,7} at q = 13 the relations ord(x) = ord(y) = 7, ord(xy) = 2 with y = J x J^-1 for J in PGL(2,13) admit many generator pairs; deduplicating them on the check-support signature gives 1638 distinct classes, all n = 546, k = 236, w = 7 (enumeration ~3.5 min, cached).

• Screen: classes in enumeration order at 4,000 gf2_fast trials per

side, seed 7, stopping at the first class reading >= 5. Classes 0-27 read 3 or 4 (histogram at the hit: {"3": 14, "4": 14, "5": 1}); class 28 read 5 on both sides, validated, after 62 s of screening.

• Confirmation: two fresh deep rungs at 400,000 trials per side

(gf2_fast, seeds 108 and 209) and one numpy rung at 4,000 trials per side, all reading 5 on both sides.

• Matched-depth audit against the peer it would beat only on d

(research/audits/leader_audit.py pair, 1,000,000 trials per side, seeds 51 and 52, pair depth 64 — the budget of the peer's own ladder, four times its packaged witness search — 21m37s): **credible, both claims held**. This class read 5 on both seeds, the peer read 4 on both seeds.

Evidence trail

| budget (trials/side) | rung | lightest logical | | ---: | --- | --- | | 4,000 gf2_fast, seed 7 | class 28, screen | 5 both sides, validated | | 400,000 gf2_fast, seed 108 | class 28 | 5 both sides, validated | | 400,000 gf2_fast, seed 209 | class 28 | 5 both sides, validated | | 4,000 numpy, seed 7 | class 28 | 5 both sides | | 1,000,000 gf2_fast, seeds 51/52, depth 64 | class 28 vs 546-236-4 | 5 (holds) vs peer 4 (holds) |

Claim, precisely: a witness-backed upper bound d <= 5 — X witness [119, 132, 230, 446, 451], Z witness carried in the entry — not certified exact. Four rungs at two backends agree on 5 for both sides. kd^2/n = 10.81 against 6.92 for codes/546-236-4.json.

Two flags came with the gate, and both are answered by the same measurement:

• d_only_peers: [[546,236,4]] — the gain over that entry is on d alone,

which is exactly the axis that inflates, so the pair audit above is the precondition for packaging rather than an afterthought.

• wl_equivalent_of: 546-236-4.json — same Weisfeiler-Leman

signature. WL equality is a flag, not a verdict: two codes that really are the same up to a qubit relabelling have one distance, and the matched-depth audit read 5 for this class and 4 for the peer on every seed, so the equivalence is refuted by measurement rather than argued. (The other branch of that audit — the peer coming down — would have been redirect, meaning the right contribution was a distance revision to codes/546-236-4.json instead of a new entry.)

Dead ends

• Classes 0-27 never reached 5: 14 read 3 and 14 read 4 at screen depth,

so the class sampled by the filed entry (class 0, all five measured classes) is representative of only part of the family. Within-family spread here is the opposite of {6,6}, where all 30 measured classes at (6,11) agreed and the paper itself reports no within-group variation for that tiling.

• Cheap screens can look conclusive and are not: a class reading 4 at

4,000 trials has *not* been shown to have d = 4 — it has been shown to have a weight-4 logical, which is what matters for stopping the scan here, and nothing more. The 1614 unmeasured classes are unmeasured, not rejected.

• The rest of the paper's slice stays out of reach: its headline

instances ([[51330,17112,10]], [[17220,7382,8]], [[25944,12974,8]] ...) are 1.2x-51x over the blocklength cap, and the modular-compiler half of the paper yields at most the Layer-3 modular flag.

Tools

Model: MiMo-V2.6-Flash (the opencode agent that ran the enumeration, the screen, the audits and this write-up, under @mathysrennela's account). Repo tooling: research/hyperbolic_psl.py (construction and class enumeration), research/kit/css.py, research/kit/surrogate.py (distance_rand_witness, lightest_logical), the gf2_fast accelerator (make fast), research/audits/leader_audit.py for the matched-depth audit, and the untouched verify/ stack for the gate. Compute: ~3.5 min enumeration, ~1 min screening, ~18 min of confirmation rungs, ~22 min of audit. The screen was a throwaway script written for this run, so it is described rather than cited: enumerate the classes, measure each with `distance_rand_witness(backend="fast", trials=4000, threads=4)` on both sides, stop at the first reading >= 5, then re-measure a hit twice at 400000 trials on fresh seeds.

Reproduction

1. Enumerate: `uv run --extra research python research/hyperbolic_psl.py --list 13 7` gives the 1638 classes in ~3.5 min, and ... 13 7 --build 28 --outdir /tmp writes class 28's Hx.npy / Hz.npy (the submitted matrices are in the entry itself either way). 2. Re-measure: lightest_logical(HX, HZ, trials=40000, seed=7) on both sides for a quick reading, or `distance_rand_witness(HX, HZ, trials=400000, backend="fast", threads=4)` for the rung the ladder records. 3. Re-run the audit: `uv run --frozen python research/audits/leader_audit.py pair codes/546-236-5.json --trials 1000000 --seeds 51 52 --pair-depth 64`. 4. uv run python verify/qldpc_verify.py codes/546-236-5.json re-runs the trusted gate against the filed entry.