This index separates the algebraic definition, the finite-block arithmetic, and the circuit/architecture claims that are often compressed into the label “low-overhead BB memory.”
- From torus polynomials to a CSS chain complex: the cyclic-shift convention, transpose as inverse translation, the swapped CSS pair, exact commutation, and the small toric anchor.
- Gross-code parameter and provenance ledger: the exact matrix shapes, row/column weights, ranks, quotient dimension, bounded distance, and the boundary between data-only and full-system overhead.
- Syndrome circuits and decoding: why static distance, circuit distance, detector construction, BP-OSD, and protocol-qualified thresholds are five different objects.
- Logical operations and their certificates: transversal CNOT, translation automorphisms, fold/automorphism gates, surgery, self-dual variants, and the missing universal-gate claim.
- Realization ledger: what the Kunlun experiment demonstrated and how it differs from neutral-atom, layered, morphing, photonic, distributed, and modular proposals.
- Representation dictionary: the polynomial, group-algebra, homological, Tanner, quotient, anyon, and toric-cylinder views, with the invariant and caveat for each translation.
- Variant and boundary map: ancestry, multivariate and coprime subclasses, pruning, planar tiles, self-duality, qudits, and why a shared bulk tile does not determine open-boundary parameters.
Reproduction contract
Run from the repository root:
source .venv/bin/activate
python atlas/bivariate_bicycle/pedagogy/reproduce.py
make verify-entry E=atlas/bivariate_bicycle
reproduce.py loads the shipped constructor and asserts the gross polynomials, torus relations, matrix shapes, commutation, ranks, quotient dimension, row and column weights, exact first-check supports, translation automorphisms, lattice/check correspondence, seeded distance upper witness, and the exact [[18,2,3]] toric anchor. The ordinary verifier independently checks both instances.yaml records.
The construction: block pins bivariate_polynomial@1 for both polynomial factors and bivariate_bicycle@1 for the CSS block assembly. Instance parameters bind the two cyclic orders and both exponent lists. Replay must reconstruct the exact ordinary-constructor matrix—including row, qubit, and two-block ordering—for both gross and toric_L3.
Provenance split
- Recomputed facts: the two shipped CSS matrices;
n, k, CSS status, check and qubit degrees, gross ranks and quotient codimension, torus periodicity and commutation, translation preservation of the stabilizer spaces, lattice incidences, the gross weight-12 upper witness, and the toric anchor's exact distance. - Bounded facts: the gross distance. Its lower bound is cited to
bravyi2024bicycle; its upper bound is rerun by rand-lowweight-v1(seed=5,iters=2000). - Literature claims: the gross lower-bound certificate, syndrome-circuit depth and threshold estimate, decoder performance, asymptotic limitation, logical-gate protocols, topological/anyon interpretation, and every hardware experiment or architecture proposal.