A phase-error check needs an explicit basis bridge so the finite parity word can be read in the right ledger. The bridge is exact only for the stated phase basis and mapped word; real stabilizer measurements add gate error, basis miscalibration, leakage, decoder assumptions, and finite-statistics uncertainty.
A phase-flip training case uses only finite labels and a checked bridge token. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The bridge is an explicit finite basis flag
The checked basis flag is phase, and the mapped parity word is 110.
basis=phasem=110
The mapped word reuses the same syndrome table
The mapped word has syndrome 01, so correction index 2 restores mapped word 111.