This index separates calculations that are easy to conflate when the [[7,1,3]] block is presented as one textbook paragraph.
- From the Hamming matrix to the quantum code: the dual-containing condition, stabilizer commutation, quotient dimension, codewords, logical representatives, and the matching upper/lower distance proof.
- Syndromes, lookup decoding, and fault-tolerant extraction: the complete single-qubit address table, the difference between a decoder and an extraction gadget, and what flags or verified ancillas contribute.
- Why the Clifford gates are transversal: explicit stabilizer and logical-Pauli conjugations for H, phase, and inter-block CNOT, including the logical-S inverse convention.
- Why T is missing and how universality is restored: the codeword-phase and triorthogonality obstructions, followed by separate ledgers for switching, injection, pieceable CCZ, and concatenation.
- What the hardware demonstrations established: a capability ledger that distinguishes encoded preparation, repeated correction, logical gates, code switching, and fixed-distance performance from a growing-distance threshold result.
Reproduction contract
Run from the repository root:
source .venv/bin/activate
python atlas/steane/pedagogy/reproduce.py
make verify-entry E=atlas/steane
reproduce.py loads the shipped constructor and asserts the displayed Hamming matrix, row ranks, self-orthogonality, Hamming/simplex weight enumerators, quotient dimension, exact distance, logical Pauli weights and anticommutation, the seven syndrome addresses, the triangular lattice/check correspondence, and the failed triorthogonality test. The ordinary entry verifier independently checks instances.yaml.
The construction: block now pins hamming_parity@1 at redundancy three and feeds its ordered matrix to hamming_css@1. For the shipped n7 instance the verifier independently reruns that recipe and requires its full stabilizer array—including check and qubit order—to equal the ordinary constructor's output. This certifies the exact Hamming-CSS presentation; the cited history and the triangular color-code interpretation remain separate provenance.
Provenance split
- Recomputed facts: the exact
[[7,1,3]] parameters, CSS commutation, Hamming/simplex dimensions and weight distributions, syndrome-to-qubit table, logical representatives, face supports, logical Clifford conjugations at the binary/phase level, and the transversal-T obstruction. - Bounded facts: none. The only shipped instance is small enough for exact distance enumeration, so no seeded upper-bound search or cited lower bound is used.
- Literature claims: the historical construction, fault-tolerance of particular cat/flag/encoded-ancilla circuits, complete resource costs of code switching, injection, pieceable and concatenated routes, and every hardware demonstration. These depend on a circuit, decoder, platform, or experimental protocol and remain cited claims.