n
56
Target: the weight-8, unrestricted stabilizer frontier. Start within the known XYZ framework of Baldelli et al., Sections III–V: three pairwise-orthogonal binary check spaces carrying pure X, Z and Y checks. Use five cyclic check groups on eight bit-labeled blocks, with Pauli multiplicities (2,2,1). The aim was a compact finite instance without the short logicals forced by simpler singleton-color patterns. Pair-partition cancellation and symplectic halving have close prior in Lee et al., Sections III–IV. Neither this submission nor its proof claims a new broad construction framework.
Fixed construction: L=7, shifts (1,2,3), eight cyclic blocks and five check groups. The validation scope is the two finite instances. Total historical candidate count and search compute were not recorded reliably.
Let q=(q1,q2,q3) range over binary triples in lexicographic order and s range modulo L. Qubit (q,s) has flat index L*(4*q1+2*q2+q3)+s. For each i modulo L, take one site from every q block at these offsets:
| Check type | Site coordinate s | | --- | --- | | X | i | | X | i+q1+2*q2+3*q3 | | Z | i+q1 | | Z | i+2*q2+3*q3 | | Y | i+2*q2 |
Order the check groups as in the table, then i=0,...,L-1. All sites are reduced modulo L. This reconstructs the ordered checks in codes/56-24-4.json. Each row has Pauli weight eight, each data qubit has degree five, and the 35-check binary symplectic matrix has rank 32, so k=24.
At upstream commit 0433d4ff5ff371976c0c03b02b103770f9fe0ee9 the unchanged official candidate gate passed, found no exact or WL-equivalent board entry, and returned board_advancing=true for the weight-8 / unrestricted stabilizer cell. Its quick refutation pass used 4,740 trials, seed 1356977029. This quick gate is distinct from the deep searches below.
Fresh official deep searches during preparation:
| Seed | Python Pauli RIS | Accelerated doubled-code request | Orbit-fold budget | Lightest Pauli | | --- | ---: | ---: | ---: | ---: | | 2026101701 | 20,000 | 2,000,000 | 50,000 | 4 | | 2026101702 | 20,000 | 2,000,000 | 50,000 | 4 |
The accelerator used eight threads and pair depth eight. Its proposals are mapped back and rescored by the official Python stack. The orbit-fold budget is per folded code under its own official time cap; the runs searched 18 and 19 partitions/frames respectively. The main Python and accelerator calls were not time-truncated. These are distinct methods and budgets, not two million Python Pauli trials. The packaging did not use promote.py.
The JSON reports distance.P.confidence=upper_bound. Its explicit pure-Z witness is [14,21,34,41], namely (010,0),(011,0),(100,-1),(101,-1). It commutes with every check and lies outside the full stabilizer rowspace. The official stabilizer path has no native exact-certification or circuit tier.
Exact finite binary symplectic parameters are additional evidence. The public source and audit package contains the Lean 4.19.0 theorem QLDPC.xyz56_parameters, the delivered independent clean-source rebuild, and the actual deep-search outputs with witnesses. Preparation reran manifest verification, source scans, literal binding, independent candidate computation and byte-identical certificate regeneration. The supplied kernel build was not repeated during preparation. JSON parsing and hashing remain external; signed Pauli/Hilbert-space correspondence, infinite families, LC-CSSness, novelty and experimental fault tolerance are outside the Lean proof. The independent computation checked no nonzero normalizer of weight 1–3 and the weight-four logical.
There is also a full-rowspace finite non-LC-CSS certificate, independent of the official supplied-generator test. For each generator support Q, rank(S)-rank(S restricted outside Q)=1, retaining both symplectic halves. In a CSS image a mixed generator would split into two independent stabilizers inside Q, contradicting that dimension. Every qubit sees all three Pauli axes among these minimal-support generators; local Cliffords cannot send those three axes into only X and Z. This includes arbitrary generator rebasing and qubit permutations. The public package provides a direct replay script.
Literature status remains unknown. The current board and targeted primary-source and exact-parameter searches did not establish an identical published point; absence from this limited search does not prove worldwide novelty or general inequivalence to published codes. No priority or superiority to all CSS codes is claimed. Distance is fixed at four here; no growing-distance claim is made.
The motivating restricted singleton-color incidence pattern permits repeated same-axis syndromes and weight-two normalizers. Simply adding a second group of only one color leaves the analogous obstruction for the other singleton colors. The (2,2,1) choice avoids that specific mechanism. This is a scoped design lesson, not a no-go theorem for arbitrary degree-four stabilizer codes. No other exploratory code or negative-result dataset is submitted in this PR.
Model: GPT-6 Astra. Agent environment: OpenAI Codex. AI assistance covered construction exploration, review, literature checking, official validation and preparation of the submission. Python 3.12.3, the upstream uv.lock, NumPy 2.4.6, Stim 1.16.0, the research extra and the official gf2_fast accelerator were used. Runtime for each deep run was tens of seconds on the preparation host; full historical compute and token usage are unknown. No decoder performance or hardware cost was measured.
Reconstruct the five groups from the table, or use the exact ordered matrix already in codes/56-24-4.json. At the upstream commit above:
uv sync --frozen --python 3.12 --extra research --extra certify make fast uv run python verify/qldpc_verify.py codes/56-24-4.json uv run python verify/validate_candidate.py codes/56-24-4.json uv run python verify/heuristic_distance.py codes/56-24-4.json --trials 20000 --fast-trials 2000000 --fold-trials 50000 --seed 2026101701
Repeat the last command with seed 2026101702. For Lean and the external submission-to-proof binding, follow the commands in the pinned public package. Its original input hash is c598183c0768f29abffb430992543ce3cd7ddbb1337be0701c7db5ad88252c45; only submission metadata and formatting change, not checks or witness.