Surface code — Level-2 map

This is the index to the worked derivations, not a second intuition essay. Each linked page answers one question completely and states which facts are algebraic, recomputed, bounded, or literature-derived.

  1. Why HGP(rep_L, rep_L) is the unrotated planar patch: the two qubit blocks, check matrices, CSS cancellation, geometry, locality, and k=1.
  2. Why the distance is exactly L: an upper-bound string, a disjoint-representative lower bound, exact enumeration at L=3,4, and the bounded L=5 claim.
  3. Why syndrome is the boundary of an error chain: bulk pairs, boundary condensation, the invisible completed string, detector histories, and what a decoder is deciding.
  4. Why a lattice-surgery seam measures logical parity: the telescoping seam product, merge dimension and distance, and why full fault tolerance uses spacetime decoding rather than bare majority vote.
  5. Which surface-code threshold a number refers to: a model-by-model provenance ledger that keeps code-capacity, phenomenological, circuit-level, and hardware numbers separate.

Reproduction contract

Run from the repository root:

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

The same computation can be launched from a Python session:

from runpy import run_path

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

reproduce.py imports the shipped constructor and asserts every computed statement used across these pages: check matrices, locality, ranks, logical representatives, syndrome walk, exact distances, the seeded L=5 upper bound, resource counts, and the L=3 rectangular surgery merge. It exits nonzero on the first disagreement. The verifier independently reruns the claims in instances.yaml.

The provenance split is deliberate: