Quantum Reed–Muller `[[15,1,3]]` — Level-2 map

This index separates five statements that are often compressed into “the fifteen-qubit code has transversal T.” They have different proofs and different provenance.

1. From punctured Boolean evaluations to a CSS block: the declared qubit order, four coordinate rows, ten-row orthogonal complement, commutation, and encoded dimension.
2. Logical quotient and sector distances: the exact X and Z normalizer quotients, the three-point logical-Z lines, and why d_X=7, d_Z=3, and total d=3 are compatible.
3. Triorthogonality and the logical phase convention: every pair/triple overlap, both logical-coset weight enumerators, and the derivation of physical T as logical T^dagger.
4. Perfect syndromes versus circuit decoding: the two single-error tables, extraction side information, flagged preparation, conversion syndromes, and the boundary around AutDEC/BP-OSD.
5. Distillation, switching, and realization ledger: the exact idealized 15-to-1 accept/error enumerator, protocol costs, and what the trapped-ion and neutral-atom demonstrations do and do not prove.

Reproduction contract

Run from the repository root:

source .venv/bin/activate
python atlas/quantum_reed_muller_15/pedagogy/reproduce.py
make verify-entry E=atlas/quantum_reed_muller_15

reproduce.py loads the shipped constructor and asserts the qubit labels, exact X and Z matrices, ranks, commutation, CSS parameters, exact total distance, both sector distances, logical representatives, syndrome columns, row-space and logical-coset weight enumerators, all pair/triple overlap parities, logical T-versus-T^dagger phase, idealized distillation enumerator, scene/check correspondence, and T3 packet state. The ordinary entry verifier independently checks instances.yaml.

The card has no construction: operation tree because the current replay registry has no Boolean-evaluation/puncturing adapter. The deterministic constructor plus this independent transcript therefore verify the exact presentation, while a future generic adapter remains a capability gap. The broader canonical-presentation problem is tracked by wanted:code-equivalence; operation-level certification beyond the existing triorthogonality predicate is tracked by wanted:transversal-gate-cert.

Provenance split

• Recomputed facts: the ordered nonzero-vector labels; all check rows and ranks; CSS commutation; [[15,1,3]]; d_X=7 and d_Z=3; the thirty-five weight-three logical-Z supports; stabilizer, normalizer, and logical-coset weight enumerators; the pair/triple overlap ledger; physical-T logical phase; the idealized independent-phase-error accept and failure polynomials; and the exact incidence scene.
• Bounded facts: none. The fixed instance is small enough for exact CSS distance enumeration, and no seeded search is used.
• Literature claims: the historical Reed–Muller construction; the tetrahedral 3D color-code, doubled-code, tensor-network, and XP dictionaries; fault tolerance and resource costs of specific distillation, switching, flag-preparation, and AutDEC/BP-OSD protocols; and every hardware demonstration. These depend on a cited circuit, decoder, device, or representation and are not promoted to constructor-verified facts.

No threshold, pseudo-threshold, one-point logical rate, or hardware error rate is quoted in this packet. The exact coefficient thirty-five belongs to the explicitly stated idealized combinatorial model, not to a device or circuit-level threshold.