[[360,122,2]] hyperbolic surface code, □(3,3,3,3) quotient of order 360

Direction & hypothesis

weight-6 x unrestricted. Nothing at n<=360 in the cell combines k>=122 with d>=2 (nearest: [[224,112,2]] at k=112). The weakest candidate of the set: d<=2 is a shallow witness, promoted because the frontier membership is on k, not on the headline.

What was searched

27 finite quotients of the triangle groups △(r,s,t) and the quadrilateral groups □(r,s,t,u) (arXiv:2610.03682's L=1 families), enumerated exhaustively/sample-wise as transitive generating tuples of involutions in S_deg for degrees 6-9 with exact corner-rotation orders (the smooth-surface condition), built into CSS codes, verified against kit css.verify_css, and screened at 400 RIS trials (numpy backend). 12 quotients fell in the n=200-1000 band, 6 were Pareto-non-dominated in their cell against all 1940 CSS board entries and were staged; each survivor then ran a fresh-seed confirmation ladder (500k + 3x1M trials per side, gf2_fast accelerator) before promotion.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 500,000 (seed=101) | 2 | | 1,000,000 (seed=102) | 2 | | 1,000,000 (seed=103) | 2 | | 1,000,000 (seed=104) | 2 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 1,000,000 trials per side. The submitted witness reproduces it: X: d <= 2 (upper_bound), Z: d <= 2 (upper_bound).

The 400-trial numpy screen's reading was confirmed unchanged at every fresh-seed deep rung; no lighter logical appeared anywhere in the ladder. Distances remain witness-backed upper bounds (d<=), not certified exact.

Dead ends

• degree-by-degree enumeration at degrees <= 5: no exact-order quotient of any

paper tiling exists there; PSL(2,7) (order 168, degree 7) is the only small (4,4,4) quotient and it falls below the n=200 band

• direct products of known quotients: the product tiling's systole is the

minimum over the factors, so low-d building blocks (Bolza, d=2) poison the distance no matter the product's size

• the paper's tabulated |G|=648 (k=328) and |G|=672 (k=172) instances did not

appear within the sweep budgets; full degree-8 exhaustion of △(6,6,6) and △(4,4,4) was still running at promotion time

Tools

Model: MiMo-V2.6-Flash (OpenCode agent session driven by MathysRennela). Repo tooling: research/kit/css, research/kit/surrogate, research/kit/submit, research/kit/coordination, research/kit/escalation, research/kit/promote. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py. Compute: 8-core Linux; six fresh-seed ladders in parallel, ~4.5M RIS trials per side each on the compiled gf2_fast accelerator.

Reproduction

Permutations of {0..5} written as image tuples (images of 0,1,..., in order): a=[0, 1, 3, 2, 5, 4], b=[0, 2, 1, 3, 5, 4], c=[0, 1, 3, 2, 5, 4], d=[2, 4, 0, 3, 1, 5]. Group them: G = <a, b, c, d> under function composition has order 360, and the corner-rotation orders are exactly (ab, bc, cd, da) = (3, 3, 3, 3), which is also the freeness (smooth-surface) condition. Tiles are the 360 elements of G; a qubit sits on each tile. Corner c of tile g is identified with corner c of tile g∘s for the two sides s adjacent to that corner (corner c lies on sides c and c+1 mod 4), so each vertex's stabilizer support is the set of tiles of its corners. The vertex graph is bipartite; X checks sit on one colour class and Z on the other. n=360; k = n - rank(H_X) - rank(H_Z) over GF(2). The staged entry's exact matrices are the authority; this recipe rebuilds them deterministically.