Toric code — Level-2 map

This index separates proofs, decoder assumptions, operation protocols, and hardware evidence. Each page answers one question completely and states which facts are algebraic, machine-recomputed, or literature-claimed.

1. [Why the square torus gives [[2L²,2,L]]](homology-and-distance.md): cell counts, CSS commutation, the two global check dependencies, homology, and matching upper and lower distance bounds.
2. Why cyclic HGP is exactly the Toric matrix: the square cyclic parity maps, HGP block order, two kernel sectors, exact constructor equality, and the contrast with open Surface factors.
3. Why syndrome is only the boundary of an error chain: endpoint cancellation, homology-class ambiguity, perfect and noisy records, loss and coherent-noise boundaries, and decoder objectives.
4. Which threshold a Toric number refers to: selected primary-source tuples and the reason no scalar belongs to the bare code.
5. Which operations are verified and what hardware demonstrated: logical loops, stacked-block CNOT, proposed Clifford protocols, non-Clifford boundaries, state preparation, and repeated correction.
6. What each representation preserves: chain, symplectic, anyon, HGP, statistical, PEPS, and Majorana dictionaries.

Reproduction contract

Run from the repository root:

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

The same computation can be launched from a Python session:

from runpy import run_path

run_path("atlas/toric/pedagogy/reproduce.py", run_name="__main__")

reproduce.py imports the shipped constructor and asserts the parameter refusal boundary, cell counts, check ranks, row weights, CSS commutation, global dependencies, exact small distances, four logical supports, periodic scene contract, endpoint-syndrome walk, exact cyclic-HGP matrix equality, and the two-block transversal-CNOT symplectic action. It exits nonzero on the first disagreement. The entry verifier independently recomputes every claim in instances.yaml.

The provenance split is deliberate:

• n, k, CSS structure, the committed small distances, logical supports, cyclic-HGP equality, and stacked-block CNOT are machine-verified.
• The all-L distance lower bound is a worked homological proof backed by the primary Toric literature.
• Decoder thresholds, proposed Clifford gadgets, and hardware results do not follow from a check matrix. They remain literature claims with their full model or protocol identity written beside them.