The algebra says a center strip is a product of local gauge generators. A fault-tolerant protocol must additionally say when those factors are measured, which outcomes are multiplied, how measurement faults are detected, and what history reaches the decoder.
All X-type gauge generators commute with one another. In an X-gauge round, measure the c vertical pairs crossing a fixed adjacent-row gap and record outcomes m^X_{i,j} in {+1,-1}. Their product is the eigenvalue of the X-center strip:
Likewise, in a Z-gauge round the product of the r horizontal-pair outcomes across one adjacent-column gap gives
reproduce.py verifies the operator identities by XORing the corresponding symplectic gauge rows and comparing the exact constructor stabilizer rows. This proves the noiseless product rule. It does not certify a measurement circuit.
An X gauge edge and a Z gauge edge that meet once anticommute. Measuring the Z edge after the X edge may randomize the earlier X gauge eigenvalue. The center product survives because it commutes with every gauge generator, but the individual factor history depends on time order.
A circuit-level detector therefore needs at least:
Multiplying one X-round outcome with a later Z-round outcome has no schedule-independent meaning. Even comparing the same gauge factor at two times is safe only after the intervening measurements and detector convention are specified.
| regime | data presented to the decoder | statement licensed |
|---|---|---|
| perfect static gauge readout | exact center parities derived from exact pair outcomes | two finite classical repetition decisions modulo gauge |
| repeated low-weight extraction | a time history of pair outcomes, repetitions, and possibly flags | performance of that named extraction circuit and recovery rule |
| encoded-ancilla Steane extraction | encoded-ancilla verification plus blockwise extraction outcomes | comparison with other complete ancilla/extraction protocols |
| period-four Floquet descendant | constant-weight spacetime detectors from the scheduled dynamical code | performance and logical behavior of the descendant, not the static center |
The first row is derived on the distance/decoding page. The other rows are literature claims because their circuit, noise, and postprocessing layers are not exported by this static constructor.
Low gauge weight limits how many data qubits one ancilla interaction chain touches, but "weight two" is not itself fault tolerance. A fault can occur on the ancilla, a two-qubit gate, preparation, measurement, idling data, or a classical record. Repeating gauge measurements supplies time redundancy, and the decoder must infer a protected center history despite local gauge values that may legitimately change.
Cross, DiVincenzo, and Terhal (cross2009comparative) provide the fixed-block repeated-syndrome/majority baseline. Its performance belongs to the complete small-block circuit and stochastic model. It is not a universal property of the 3 × 3 check matrix.
Steane-style extraction couples an encoded ancilla block transversally rather than measuring every gauge pair with an individual bare ancilla. The ancilla preparation, verification, coupling direction, measured block, accepted histories, and recovery rule define a different syndrome object.
Escobar-Arrieta and Gutiérrez (escobar2025improved) compare this route with a Shor-style extraction protocol. The comparison cannot be transferred to an unspecified "Bacon–Shor decoder" because changing extraction changes both the fault locations and the available side information.
gidney2023less obtains threshold behavior from local lattice-surgery circuits arranged in a self-concatenating construction. The fitted family is the circuit construction, not a direct sequence of increasingly large static rectangles under an unspecified stabilizer measurement.
alam2025board instead defines a period-four measurement schedule with constant-weight spacetime detectors and applies matching under a uniform circuit-level model. Time-dependent checks and logical representatives are part of that code definition. The scheduled descendant is therefore recorded as an R3 view, not as a schedule exported by this static entry.
This packet quotes no threshold-like number. Before any future number enters, the full identity tuple must include:
The 2026-08-02 decoder sweep found no extraction-independent threshold for plain unspecified static stabilizer products. That dated negative result does not say no Bacon–Shor-derived protocol has a threshold; it says the protocol identity cannot be erased.
aliferis2007subsystem: low-weight gauge extraction and subsystem fault-tolerant gadgets.cross2009comparative: repeated-syndrome fixed-block circuit baseline.napp2013optimal: perfect-readout biased-noise decoding and rectangle optimization, with model-dependent optima.escobar2025improved: complete encoded-ancilla versus Shor-style extraction comparison.gidney2023less: self-concatenated local-surgery threshold construction.alam2025board: scheduled period-four detectors and MWPM for the Floquet descendant.Every bibkey resolves in registry/refs.yaml; the fetch-verification details and dated searches are retained in docs/lit/bacon_shor/.