Four-qubit code — Level-2 map

Each page below answers one question completely and keeps executable algebra separate from literature claims.

1. How the stabilizers, logical basis, codewords, and distance fit together.
2. Why single-site detection does not imply unknown-site correction.
3. Which Clifford actions are intrinsic and which experimental claims are protocol-scoped.
4. Which neighboring descriptions are presentations, family extensions, or different codes.

Reproduction contract

Run from the repository root:

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

The script imports the shipped constructor. It recomputes CSS ranks and exact distance, enumerates all single-qubit syndromes, verifies the logical symplectic table and codeword eigenvalues, checks the erasure Knill–Laflamme matrices, conjugates the logical basis under physical H, S, and inter-block CNOT, and validates both Scene v2 frames. It exits nonzero at the first disagreement.

The provenance split is deliberate:

• n, k, d, CSS commutation, syndromes, logical representatives, and Clifford conjugations are machine-verified.
• The one-erasure interpretation is theorem-backed and then checked on the exported finite code.
• Distillation performance and hardware outcomes remain claimed by their primary sources. They are not inferred from the static check matrix.