6.6.6 color code — Level-2 map

The phrases "three-colored face code" and "transversal Clifford code" hide several independent arguments. These pages separate constructor-owned algebra from boundary, decoder, protocol, and hardware claims that remain tied to literature.

1. Face incidence, commutation, and parameter count: the modular triangular grid, clipped-face supports, exact n and face formulas, H H^T=0, independent rows, and k=1.
2. Color boundaries, logical strings, and distance: the color/Pauli distinction, the declared boundary logical pair, why its weight is only an upper witness, and the exact-versus-claimed distance ledger.
3. Decoding and syndrome-extraction boundaries: the seven-qubit syndrome table, three-color correlations, projection and Möbius lifting, repeated detectors, and the threshold identity contract.
4. Transversal Clifford gates and non-Clifford completion: exact H, bipartite phase, and pairwise-CNOT Pauli maps; the 2D locality bound; and the additional resources consumed by surgery and magic states.
5. Representations, variants, and realization ledger: colored homology, anyon condensation, folded surface codes, the three-body Ising model, tensor networks, neighboring tilings, and precise experimental claim boundaries.

Reproduction contract

Run from the repository root:

source .venv/bin/activate
python atlas/color_666/pedagogy/reproduce.py
make verify-entry E=atlas/color_666

reproduce.py loads every shipped instance through the ordinary entry API and independently rebuilds the integer-grid incidence data. It asserts site and face formulas, support weights, three face colors, even pairwise overlaps, GF(2) ranks, CSS equality and commutation, the boundary logical pair, exact distances for d=3,5, the distance-seven upper witness, the seven-qubit syndrome table, bipartite phase conjugation, pairwise-CNOT symplectic action, scene/lattice correspondence, tour integrity, and T3 packet state. The ordinary entry verifier separately owns all structured claims in instances.yaml.

The card omits construction: because the replay registry has no exact adapter for "take this modular triangular vertex set, delete face centers, clip six-neighbor supports, and preserve this row/qubit order." The deterministic constructor plus the independent transcript verify the shipped presentation without pretending a generic operation-tree replay exists. A future canonical finite-patch equivalence path is routed to wanted:code-equivalence.

The transcript verifies ideal logical action for this presentation, but it does not promote the structured H, S, or CNOT operations from claimed to verified. A shared gate referee must define equivalence up to stabilizers, logical basis, phases, geometry permutations, and multi-block action; that capability remains wanted:transversal-gate-cert.

Provenance split

• Recomputed facts: all shipped site/face sets and orders; face weights and overlaps; incidence ranks; n, k, and CSS status; exact d=3,5; the distance-seven weight-seven logical witness; the seven-qubit static syndrome table; the H, bipartite-phase, and pairwise-CNOT Pauli maps for the declared logical basis; scene/lattice consistency; and tour structure.
• Bounded facts: none. No seeded search or statistical interval is introduced by this packet.
• Claimed facts: the distance-seven lower bound and family distance theorem; colored-homology and anyon-condensation interpretation; local- Clifford unfolding; decoder performance and circuit claims; logical surgery and magic-state protocols; and every hardware realization.
• Routed gaps: a general distance lower-bound certificate remains wanted:certified-lower-bounds; exact construction/equivalence replay remains wanted:code-equivalence; and generic logical-gate certification remains wanted:transversal-gate-cert. The broader color_code_2d entry is still a typed pending relation target.

No threshold number is promoted. Threshold-like results in the literature differ by tiling, finite boundary, syndrome model, Pauli channel, circuit, decoder, and fitted observable; the decoding page records the full identity tuple required before any such number could enter the Atlas.