Rayan (arXiv:2607.08911) introduces a diagrammatic field theory of quantum error correction (QEC) in which syndromes are measured footprints of error configurations. We show that this framework is an instance of the ISA chain complex C^k = direct-sum_{|v|=k} A^{otimes c(v)}, with differential partial satisfying partial^2 = 0 from the Frobenius identity and pentagon relation. Two theorems are proved: Theorem A (fibrewise locality) shows that the complex splits over syndrome fibres, each satisfying partial^2 = 0 independently. Theorem B (Peierls threshold) gives beta*(P) = kappa^{-1} log(1/(2Bn_B(P))), making the ISA threshold computable from the BIND count. The Forge ISA is identified as an interpolating decoder family between the topological (sigma to infinity) and conformal (sigma = 0) limits, with optimal sigma* given by the MGE saddle point. A conjecture is stated that the spectral curve of the decoding problem equals the divisor of beta* snaps in the complex beta-plane.
Rayan (arXiv:2607.08911) introduces a diagrammatic field theory of quantum error correction (QEC) in which syndromes are measured footprints of error configurations. We show that this framework is an instance of the ISA chain complex C^k = direct-sum_{|v|=k} A^{otimes c(v)}, with differential partial satisfying partial^2 = 0 from the Frobenius identity and pentagon relation. Two theorems are proved: Theorem A (fibrewise locality) shows that the complex splits over syndrome fibres, each satisfying partial^2 = 0 independently. Theorem B (Peierls threshold) gives beta*(P) = kappa^{-1} log(1/(2Bn_B(P))), making the ISA threshold computable from the BIND count. The Forge ISA is identified as an interpolating decoder family between the topological (sigma to infinity) and conformal (sigma = 0) limits, with optimal sigma* given by the MGE saddle point. A conjecture is stated that the spectral curve of the decoding problem equals the divisor of beta* snaps in the complex beta-plane.
We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four σ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-σ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply PrL(fail)≤C∣ΩL∣e−cL below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. We show that Kitaev's topological phases — the toric code, the gapless Dirac cone, and the non-Abelian phase — are three computational phases of the Forge instruction-set architecture (ISA), distinguished by opcode mix and inverse-temperature scheduling via the THERMAL compiler directive. The central result is a pair of exact identities. The F-matrix of Ising anyons equals the j = 1/2 recoupling coefficient of the quantum group SU(2)₂ at q = exp(iπ/4). The F-matrix of Fibonacci anyons equals the j = 1/2 coefficient of SU(2)₃ at q = exp(iπ/5). Both identities are verified numerically. In each case the deformation parameter q = exp(iπ/(k+2)) equals exp(iπβ), where β is the Forge ISA inverse temperature, placing topological quantum computation at finite β in the semiring-deformation interpolating family of the Maslov–Gibbs programme. Three consequences follow: (i) The ISA containment hierarchy ZX ⊂ Origami ISA ⊂ 731-ISA maps exactly onto the anyon hierarchy Abelian ⊂ Ising ⊂ Fibonacci. The same algebraic ladder that organises instruction sets organises topological order. (ii) The quantum group deformation parameter q is the Forge ISA inverse temperature. Topological quantum computation is quantum mechanics at finite β. (iii) The logical qubit of the toric code is identified with the generator of H¹(T²; Z₂). Quantum error correction is an H¹ obstruction computation, placing the toric code threshold in the Islands/Flatland/The Deep cohomological framework. Results are verified numerically in experiment x445c (Racah coefficient matching for Ising and Fibonacci anyons). This paper is part of the Origami ISA series. Paper 446 (β-Ladder of Exactly Solvable Models, doi:10.5281/zenodo.20761278) extends the same β-axis to the full taxonomy of integrable quantum spin models. Keywords Kitaev model, toric code, topological quantum computation, anyons, Fibonacci anyons, Ising anyons, F-matrix, quantum groups, Racah coefficients, instruction set architecture, topological order, ZX-calculus, cohomology, quantum error correction, beta deformation, Levin-Wen string nets, honeycomb model, modular tensor categories
Shows that XXZ chain, transverse-field Ising model, Levin-Wen string nets, and Kitaev quantum doubles are a single ISA family parameterised by the beta-deformation. At beta=1/2 the BKT transition is a TWIST failure. Walker-Wang models are identified as H2 (BIND-mandatory). Haah codes require twisted coefficients and lie outside the standard H2 tier (fracton sector). The quantum group parameter q = exp(i*pi*beta) encodes the semiring interpolation between tropical (beta=infinity) and metaplectic (beta=0).
Quantum error correction (QEC) is cohomological encoding made explicit. We show that the three central codes in fault-tolerant quantum computation correspond precisely to three levels of the H^k tower of the Origami instruction-set architecture (ISA): the Steane [[7,1,3]] code is the Fano plane PG(2,2) at the H^1 tier; the surface code encodes logical qubits as generators of H^1(surface;Z_2); and the 15-to-1 magic state distillation factory is an H^1 Clifford circuit that bridges H^1 and~H^2 by purifying the non-Clifford TWIST\ resource. A fourth result is new: dark-magic errors are syndrome-invisible. A unitary with TWIST>0 that commutes with all stabiliser generators corrupts the logical orbit label while leaving every syndrome bit at +1. Standard randomised benchmarking is insensitive to orbit-label decay; we propose orbit-sensitive randomised benchmarking (ORBIT-RB) as the missing diagnostic. The unified picture is captured in a single sentence: a QEC code of depth k specifies which H^k classes are logical and which are trivial; syndrome measurement computes the coboundary ; error correction finds the minimum-weight representative of the observed H^k class.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. The Origami ISA — five opcodes (SPLIT, SPLAT, TWIST, FLIP, FLOP) corresponding to Pachner moves on a triangulated simplicial complex — is already a combinator library, but without a machine-checkable type system. This paper makes the type system explicit, yielding three progressively expressive type systems and one key theorem: associamancy is a compile-time type error. Type System 1 — Simple types. Wire types are irreducible representation labels Rep(G) for a gauge group G. For the Origami ISA: G = SU(2), types are spins j ∈ {0, ½, 1, …}. For the 731 ISA: G = G₂, types are Dynkin labels (p,q). For the financial Pacioli ISA: G = (ℝ₊, ×), types are currencies. The Pentagon identity d² = 0 becomes a typing rule: SPLAT(SPLIT(x)) must return the trivial representation 1. Any circuit producing a non-trivial output is ill-typed — this is the type-level expression of the leakage criterion introduced in Paper 411. Type System 2 — Linear types. Every wire is usable exactly once, encoding the quantum no-cloning theorem. SPLIT is the unique duplication primitive (1 → 4 wires, the Pachner move); FLOP is the unique elimination primitive (trace / Born rule). The category-theoretic model is a symmetric monoidal category extending the ZX-calculus to Rep(G)-typed wires over arbitrary compact groups G. Type System 3 — Dependent types. The output type of SPLIT depends on the input type via the Clebsch-Gordan decomposition: SPLIT_CG : (j : Rep(G)) → Σ(j₁ j₂ j₃ j₄ : Rep(G)). CG(j; j₁, j₂, j₃, j₄) ≠ 0 The fibred-category model (Grothendieck fibration over Rep(G)) captures this: the Schur boundary ν₂ ≠ 0 selects the associative subcategory (Origami ISA, regime 2) from the full category (731 ISA, regime 3). The Schur boundary as a TypeError. The Frobenius-Schur indicator ν₂ ∈ {−1, 0, +1} partitions wire types into two subtypes: AssocWire (ν₂ ≠ 0, Origami ISA) and FullWire (ν₂ = 0 allowed, 731 ISA). The SPIN opcode — the generator of triality, the order-3 outer automorphism of Spin(8) with G₂ = Fix(triality) — has type FullWire → FullWire with the constraint ν₂ = 0. Applying SPIN to an AssocWire raises a TypeError at circuit construction time, before any hardware is consulted. Associamancy (Paper 407) is no longer only a resource-theoretic concept — it is a compile-time type mismatch. The three-level resource hierarchy as a type tower: Level Resource Wire subtype Protection 0 Stabiliser Any Rep(SU(2)) None 1 Standard magic AssocWire (ν₂ ≠ 0) None 2 Associamancy FullWire (ν₂ = 0) Discrete Berry phase (W(G₂), order 12) The Pacioli Combinator Library as the financial instance. The PCL (Paper 306) — typed combinators flow, sequence, choose, fold — is the financial instance of this DSL with G = (ℝ₊, ×). The Pacioli identity (every debit has a credit) is d² = 0 at the financial instance. The same typed circuit compiles to quantum hardware, financial risk models, or nuclear spectroscopy; only the gauge group G and the wire type interpretation change. Python prototype included: ISACircuit class with runtime type checking; spin() raises TypeError if ν₂ ≠ 0; pentagon_check() validates d² = 0 at construction time. Keywords: Origami ISA, typed DSL, linear types, dependent types, Schur boundary, associamancy, no-cloning, Pentagon identity, Clebsch-Gordan, Pacioli combinator library, symmetric monoidal category, ZX-calculus, G₂, triality, type theory, quantum circuits