Representation ladder

• symplectic-css-checks derived by hgp.build_from_factors -> qeclib.constructions.hgp -> CheckMatrix.from_css hgp-factor-spaces-v1

  For H1 of shape m1×n1 and H2 of shape m2×n2, the executable convention is HX = [H1⊗I_n2 | I_m1⊗H2^T] and HZ = [I_n1⊗H2 | H1^T⊗I_m2]. Expanding these two binary blocks gives the ordered sparse-Pauli baseline used by the verifier; no geometric layout or decoder is implied by the matrix formula.

  References: tillich2009quantum

• typed-factor-product canonical

  Read each Hi as a map from an ni-dimensional bit space to an mi-dimensional check space. Qubits occupy bit1⊗bit2 and check1⊗check2. X checks occupy check1⊗bit2, while Z checks occupy bit1⊗check2. This four-space typing fixes every transpose and tensor identity before a matrix is multiplied.

  References: tillich2009quantum, bravyi2013homological

• product-chain-complex curated from bravyi2013homological

  The same four spaces form the middle portion of a product chain complex. Applying either factor map in the two possible tensor directions creates the two check blocks. The boundary-of-a-boundary cancellation is exactly the CSS commutator cancellation, so the homological and Kronecker-block descriptions are two readings of one ordered construction.

  References: bravyi2013homological, tillich2009quantum

• factor-dimension-ledger derived by constructor.build_from_factors factor ranks and CheckMatrix.k hgp-factor-spaces-v1

  If ri is the binary rank of Hi, the two qubit sectors contain n1 n2 and m1 m2 coordinates. The logical dimension splits into a kernel–kernel contribution (n1-r1)(n2-r2) and a transpose-kernel contribution (m1-r1)(m2-r2). The general constructor helper recomputes both terms and rejects disagreement with the stabilizer rank.

  References: tillich2009quantum

No representation at this rung is recorded for this entry.

• syndrome-circuit-boundary curated from delfosse2021measurement

  Sparse algebraic checks may still connect distant hardware qubits. A realization must add an interaction graph, ancillas, a measurement schedule, fault propagation, and a decoder for repeated records. Distance-preserving circuits and non-local-resource studies solve declared architecture problems; they are not outputs of the bare tensor formula.

  References: delfosse2021measurement, manes2025distance, chandra2025nonlocal

• kernel-cokernel-logicals curated from quintavalle2021reshape

  Factor kernels and transpose kernels supply canonical row- and column-shaped representatives, while quotienting by factor images identifies representatives of the same quantum logical class. These shapes are useful to ReShape and related algorithms; they are not extra logical qubits and do not make an arbitrary correction maximum likelihood.

  References: tillich2009quantum, quintavalle2021reshape

• decoder-regime-split curated from quintavalle2021reshape

  ReShape uses HGP logical shapes under its stated factor and error conditions; erasure decoders are given the lost support; Viderman-style and iterative decoders require declared expansion or update hypotheses; BP-OSD is a generic sparse-matrix benchmark. These inputs and guarantees are not interchangeable, and none yields one decoder for every binary factor pair and every syndrome model.

  References: quintavalle2021reshape, connolly2024erasure, krishna2023viderman, pradhan2025linear, panteleev2021degenerate

• logical-operation-layers curated from patra2025targeted

  A factor symmetry or symplectic matrix first specifies a logical Clifford action. A physical protocol must then realize it by transversal layers, partitions, defects, measurements, or other declared resources, and a fault-tolerance argument must control faults through that protocol. Targeted-Clifford constructions and product-code no-go results constrain different layers, so neither is a construction-wide gate label.

  References: krishna2021gates, quintavalle2023partitioning, patra2025targeted, fu2025nogo

• output-family-boundary curated from kovalev2012improved

  Restricting or coupling the factor matrices creates HGP output subclasses; reducing redundant sectors changes the formula under extra algebraic hypotheses; changing to a multi-factor non-CSS product changes the construction itself; embedding one fixed output in a planar device changes only its realization. Output entries therefore point inward with typed product_of relations rather than being treated as aliases of HGP.

  References: kovalev2012improved, leverrier2020xyz, yang2025spatially, pabla2026spatial