For a CSS face matrix H, a Z-error support e_Z produces X-check syndrome s_X=H e_Z^T; an X error produces the identical formula in the other sector because H_X=H_Z. The symmetry of the static matrix does not make all decoders or extraction circuits identical.
For the distance-three patch, the three face rows are
{0,1,2,3}, {1,2,4,5}, and {2,3,5,6}.
A single Z error therefore produces:
| qubit | X-check syndrome (f0 f1 f2) |
|---|---|
| 0 | 100 |
| 1 | 110 |
| 2 | 111 |
| 3 | 101 |
| 4 | 010 |
| 5 | 011 |
| 6 | 001 |
All seven nonzero three-bit syndromes occur once. This is the Hamming-code view of the smallest patch. The same table diagnoses single X errors after exchanging check sectors. For larger patches, bulk qubits touch three faces while boundary qubits touch fewer; syndromes no longer form a single finite lookup table that scales with d.
A bulk error creates defects in three colored sectors at once. Treating each sector as statistically independent throws away the fact that they share one physical cause. The random-three-body Ising mapping of Katzgraber, Bombín, and Martín-Delgado keeps the joint constraint at the maximum-likelihood level, but a statistical partition function is not automatically an efficient decoder implementation.
Delfosse's projection decoder maps color-pair restrictions into surface-code problems. Matching those projected defects is an intermediate step. The results must be lifted to one physical color-code correction whose full syndrome is correct. On a triangular patch, boundary connected components enter both projection and lifting; reusing a closed-surface rule can create a correction with the wrong boundary class.
Sahay and Brown instead combine the three boundary sectors in a matching graph embedded on a Möbius strip. The nonorientable seam is bookkeeping for how a path changes color sector when it crosses the folded boundary. It is not a claim that the physical qubits live on a Möbius strip.
The idealized hexagonal-versus-square-octagonal comparison of Ohzeki and the faulty-syndrome spacetime gauge model of Andrist et al. belong to different statistical ensembles. Their estimates cannot be merged merely because both papers use color-code language.
A static syndrome is one bit per face check. A repeated circuit produces a time history of noisy outcomes. A detector is typically a parity between outcomes whose ideal value should agree, possibly combined with flag or ancilla information. One circuit fault may create several correlated detectors and data errors.
Consequently a circuit-level decoder must retain:
Flag circuits in Chamberland et al., superdense and middle-out circuits in Gidney and Jones, and detector-based matching in Lee, Li, and Bartlett therefore define different inference problems even when their final stabilizer group is the same.
Before a threshold-like number can appear, record all of:
| field | required identity |
|---|---|
| family | tiling, finite boundary, and scaled instances |
| observable | fitted logical failure quantity and scaling variable |
| extraction | ideal syndrome, phenomenological rounds, or named circuit |
| noise | channel, locations, correlations, and probability normalization |
| decoder | algorithm, lifting/reconciliation, postprocessing, and version |
| source | figure/table/section and fit procedure |
| claim type | threshold, pseudo-threshold, one-point rate, bound, or hardware result |
The 2026-08-02 decoder sweep found several legitimate but incompatible models. T102 introduces no threshold number because no single tuple belongs to the bare check matrix. This is a positive epistemic decision: omitting a number prevents a code-capacity transition, a faulty-syndrome result, and a hardware scaling observation from being rendered as estimates of one quantity.