Five-qubit code — Level-2 map

Each page below pays one convention or proof debt completely.

1. How cyclic checks, rank, the logical basis, non-CSS structure, and exact distance fit together.
2. Why all sixteen syndrome alternatives fit exactly and what each decoder input means.
3. Which operations are intrinsic and what the representative experiments actually establish.
4. Which descriptions are equivalent representations, broader classes, or different code families.

Reproduction contract

Run from the repository root:

source .venv/bin/activate
python atlas/five_one_three/pedagogy/reproduce.py
make verify-entry E=atlas/five_one_three

The script imports the shipped constructor. It verifies the exact symplectic rows and rank, all pairwise commutators, the absence of a CSS split, the canonical logical pair and its minimum-weight equivalents, the complete single-qubit syndrome bijection, exact distance, the perfect-packing count, and the stabilizer-incidence scene. It exits nonzero on the first mismatch.

The provenance split is deliberate:

• n, k, d, non-CSS status, stabilizer rank, logical representatives, syndrome injectivity, perfect packing, and transversal logical Paulis are recomputed exactly.
• Perfect, MDS, and Frobenius class membership is source-backed and checked against the executable parameter and cyclic witnesses.
• Flag extraction, circuit-history matching, automorphism Clifford circuits, pieceable gates, distillation, and hardware outcomes remain claims of their cited primary protocols. No threshold is inferred from the static matrix.