We show that every exactly solvable quantum spin model with a phase transition is a different β-schedule of the Origami instruction-set architecture (ISA). The unifying equation is q = exp(iπβ): the quantum group deformation parameter q underlying the XXZ chain, the transverse-field Ising model (TFIM), Levin–Wen string-net models, and Kitaev's quantum doubles is identically the Gibbs ISA inverse temperature. This gives a single β-axis on which every topological phase of matter has a precise address. The main results are: (i) The XXZ Bethe-ansatz two-magnon R-matrix is the FLIP eigenvalue at q = exp(iπβ), with the BKT transition as a TWIST failure at β = 1/2 (exact, by cos(π/2) = 0). (ii) The 1D XY model and the 2D Kitaev toric code share β = 1/4 and quantum group SU(2)₂ — the same ISA programme in different geometries. (iii) For non-Abelian string-net phases in the SU(2)k family, the quantum dimension d{1/2}(β) = 2 cos(πβ) is an exact phase label, verified for k = 1 through 15. (iv) The cohomological chain H⁰/H¹/H² = Islands/Flatland/The Deep corresponds to classical/toric-code/Walker–Wang codes. Fracton phases require twisted cohomology H*(M; L) beyond standard H². The β-ladder (Table I) places every major model on a single axis: β = 0 (Heisenberg SU(2)), β = 1/5 (Fibonacci anyons, SU(2)₃), β = 1/4 (Ising anyons / toric code, SU(2)₂), β = 1/3 (semion / XY model, SU(2)₁), β = 1/2 (BKT transition, TWIST failure). All results are verified numerically in experiments x446a–x446d. This paper belongs to the Origami ISA series. Paper 443 (Planck's Constant in Disguise, doi:10.5281/zenodo.20752384) establishes the semiring deformation; Paper 445 (Kitaev Menagerie, doi:10.5281/zenodo.20752352) treats the Kitaev models as a special case; Paper 447 (Opcode Rosetta Stone, doi:10.5281/zenodo.20761260) extends the ISA dictionary to twelve models. Keywords exactly solvable models, quantum groups, XXZ model, BKT transition, topological phases, string-net models, toric code, Walker-Wang model, fracton phases, instruction set architecture, beta deformation, Yang-Baxter equation, Bethe ansatz, quantum dimension, Luttinger liquid, cohomology, tensor networks, quantum computing
We show that every exactly solvable quantum spin model with a phase transition is a different β-schedule of the Origami instruction-set architecture (ISA). The unifying equation is q = exp(iπβ): the quantum group deformation parameter q underlying the XXZ chain, the transverse-field Ising model (TFIM), Levin–Wen string-net models, and Kitaev's quantum doubles is identically the Gibbs ISA inverse temperature. This gives a single β-axis on which every topological phase of matter has a precise address. The main results are: (i) The XXZ Bethe-ansatz two-magnon R-matrix is the FLIP eigenvalue at q = exp(iπβ), with the BKT transition as a TWIST failure at β = 1/2 (exact, by cos(π/2) = 0). (ii) The 1D XY model and the 2D Kitaev toric code share β = 1/4 and quantum group SU(2)₂ — the same ISA programme in different geometries. (iii) For non-Abelian string-net phases in the SU(2)k family, the quantum dimension d{1/2}(β) = 2 cos(πβ) is an exact phase label, verified for k = 1 through 15. (iv) The cohomological chain H⁰/H¹/H² = Islands/Flatland/The Deep corresponds to classical/toric-code/Walker–Wang codes. Fracton phases require twisted cohomology H*(M; L) beyond standard H². The β-ladder (Table I) places every major model on a single axis: β = 0 (Heisenberg SU(2)), β = 1/5 (Fibonacci anyons, SU(2)₃), β = 1/4 (Ising anyons / toric code, SU(2)₂), β = 1/3 (semion / XY model, SU(2)₁), β = 1/2 (BKT transition, TWIST failure). All results are verified numerically in experiments x446a–x446d. This paper belongs to the Origami ISA series. Paper 443 (Planck's Constant in Disguise, doi:10.5281/zenodo.20752384) establishes the semiring deformation; Paper 445 (Kitaev Menagerie, doi:10.5281/zenodo.20752352) treats the Kitaev models as a special case; Paper 447 (Opcode Rosetta Stone, doi:10.5281/zenodo.20761260) extends the ISA dictionary to twelve models. Keywords exactly solvable models, quantum groups, XXZ model, BKT transition, topological phases, string-net models, toric code, Walker-Wang model, fracton phases, instruction set architecture, beta deformation, Yang-Baxter equation, Bethe ansatz, quantum dimension, Luttinger liquid, cohomology, tensor networks, quantum computing
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).
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
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 seven opcodes of the Origami instruction-set architecture (ISA) — FLOP, SPLIT, TWIST, LABEL, BIND, FLIP, SPLAT — play structurally identical roles across every exactly solvable quantum model with a phase transition. This paper establishes a universal dictionary: FLOP is always the fermionisation map (Jordan–Wigner / slave-boson / Majorana representation); SPLIT is always the diagonalisation (Bogoliubov / Bethe rapidity / stabiliser expansion); TWIST is always the gap check (Berry phase / Chern number / spectral gap); LABEL is always the sector projection (gauge fixing / stabiliser projection / Bethe vacuum). BIND — the three-body frog vertex — is the unique opcode whose presence or absence determines whether a phase supports Abelian or non-Abelian anyons. Two theorems are proved. The BIND = Non-Abelian theorem: a gapped topological phase has non-Abelian anyonic order if and only if its minimal ISA programme contains BIND. The Universal Phase Boundary theorem: for any model in the SU(2)q family at q = exp(iπβ), the quantum phase transition is a TWIST failure at β = 1/2, where the quantum dimension d{1/2}(β) = 2cos(πβ) = 0 exactly. A master table across twelve models — toric code, Kitaev honeycomb, TFIM, XXZ chain, string nets, quantum doubles, SYK, MBQC, ZX-calculus, and others — makes the dictionary explicit. The Origami ISA is the universal language in which condensed-matter physicists, quantum-information theorists, and representation theorists describe the same computation under different names. Keywords instruction set architecture, exactly solvable models, topological order, non-Abelian anyons, quantum phase transitions, Origami ISA, Jordan–Wigner transformation, Bogoliubov transformation, Bethe ansatz, toric code, Kitaev honeycomb, XXZ chain, string-net condensation, quantum doubles, SYK model, MBQC, ZX-calculus, Frobenius algebra, quantum groups, Berry phase, BIND theorem, β-ladder, tensor networks, categorical quantum mechanics
Heunen, Lemonnier, McNally, and Rice (POPL 2026) introduce the phase language: two primitives, Ph(theta) and 'if let p then s', that together are universal for unitary quantum computation. Their language is elegant and expressive, yet by design it restricts to unitary operations, to a fixed real-phase semiring, and to a single computational tier. We show that the phase language is the grade-beta=i special case of the Origami instruction-set architecture (ISA), a beta-graded Frobenius monad defined over the beta-semiring. Three main results are proved. Theorem 1 (restricted embedding): the phase-only and empty-pattern fragment of the phase language embeds faithfully into the H^1-tier subcategory of Origami_{beta=i} generated by ORBIT, TWIST, and FLIP. Theorem 2 (strict extension): the Raven ISA (complex beta) is strictly more expressive than the phase language; any computation using a non-unitary TWIST serves as witness. Theorem 3 (Frobenius monad): the quadruple {ORBIT, FLIP, BIND, MERGE} forms a Frobenius monad in the PROP semantics, satisfying the Frobenius condition and giving a categorical foundation for the full ISA. The graded structure enables a three-tier computational hierarchy: tropical (beta -> infinity), quantum (beta = i), and PT-symmetric (beta in C), with the phase language occupying exactly the middle tier.
We construct exactly solvable models for a wide class of symmetry-enriched topological (SET) phases. Our construction applies to two-dimensional (2D) bosonic SET phases with finite unitary on-site symmetry group G and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: If we couple them to a dynamical lattice gauge field with gauge group G, the resulting gauge theories are equivalent to string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the e and m type anyons. We further illustrate our construction with a number of additional examples.
The Z2 topological order in Z2 spin liquid and in the exactly solvable Kitaev toric code model is the simplest topological order for 2 + 1D bosonic systems. More general 2 + 1D bosonic topologically ordered states can be constructed via exactly solvable string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly solvable models and fermionic topologically ordered states.