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.
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.
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
Every major quantum programming language restricts to unitary operations and Hermitian Hamiltonians. This is understandable: unitarity is tractable, elegant, and sufficient for standard gate-model quantum computing. But real quantum systems are open, dissipative, and non-Hermitian. By refusing to engage with loss, the field is in a bind -- unable to express PT-symmetric computation, Lindblad dynamics, exceptional-point algorithms, or even the classical limit of quantum mechanics. We identify the missing primitive: BIND (the Frobenius comultiplication delta: A -> A tensor A), and the missing parameter: beta (the semiring grade interpolating between tropical, quantum, and PT-symmetric regimes). The Raven ISA is the minimal non-unitary extension of the Heunen et al. phase language that adds these two ingredients. It implements five computational advantages unavailable to unitary languages, and subsumes all existing quantum programming languages as the beta = i special case. It is time to find BIND.
We survey the Origami Instruction Set Architecture (), a universal opcode language whose seven primitives---FLIP, FLOP, SPLIT, SPLAT, TWIST, LABEL, ORBIT---together with an eighth, BIND, describe every physical process at every scale from quantum gravity to metabolism. The algebraic closure of these opcodes is the Fano incidence structure = PGL(3,\GF_2), and their computational complexity hierarchy stratifies into three tiers: H^0 (tropical/classical), H^1 (Clifford/stabiliser), and H^2 (magic/topological). This hierarchy is a physical observable: the H^k tier of a process determines whether it can be simulated classically (H^0), efficiently by stabiliser circuits (H^1), or requires genuine quantum resources (H^2). We survey applications across nuclear and particle physics, quantum field theory, condensed matter, quantum chemistry, biology, mathematics, and economics, citing over forty companion papers in which each domain is developed in detail. We state nine open problems including the uniqueness conjecture and the completeness question beyond qutrits.
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
A manifesto and open standard proposal for Origami: a minimal, hardware-agnostic instruction set architecture (ISA) for quantum computing positioned above physical gate sets and below algorithms. Origami has five opcodes - LABEL (initialisation), ORBIT (symmetry/permutation), TWIST (phase rotation), BIND (entanglement), FLIP (measurement) - stratified by cohomological tier H0/H1/H2. Simulability is a structural property: a programme is classically simulable if and only if it contains no BIND opcode. Every opcode has a canonical realisation on superconducting, trapped-ion, photonic, neutral-atom, and topological qubit hardware. A beta-deformation provides a principled NISQ framework. The H1 layer subsumes ZX-calculus; the H2 layer extends it to universal quantum computation. A call to the quantum computing community to adopt Origami as an open standard enabling a RISC-V-style software compounding effect for the quantum age.
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 complete the TRS trilogy — Origami (β→∞, tropical, classical), Forge (0<β<∞, real, statistical), Meld (β=it, complex, quantum) — by showing that the Meld ISA is the quantum circuit model expressed in the TRS semiring language. Three results follow. Result 1 (ISA completeness). The Clifford group equals the Meld ISA without the BIND opcode. Universal quantum computation equals Meld plus BIND. The T-gate T=diag(1,,eiπ/4) is the BIND opcode at j=21 — the Fano associator obstruction of Paper 407, applied to qubit representations. The magic-typing classification of Papers 365/366 maps exactly onto the 731-ISA opcode hierarchy: Clifford gates occupy Fano-line orbits (SPLIT/FLIP/TWIST at Clifford angles); the T-gate occupies the non-collinear Fano triple (BIND). Result 2 (Quantum algorithm factory). Every classical algorithm of the form argmaxg∈Gf(g) over an abelian group G has a C-MGE quantum counterpart C-MGE(f)=SPLITG⋅FLIP(f)⋅SPLAT, giving exponential speedup via quantum Fourier sampling whenever G is abelian and f has a hidden subgroup concentrating Fourier mass. Shor's algorithm decomposes into three ISA layers: Layer 1 (Origami — classical modular exponentiation), Layer 2 (Meld — QFT as SPLIT-TWIST-FLIP cascade), Layer 3 (Origami — measurement as tropical projection). "Frozen-Shor" (Origami ISA, β→∞) recovers classical exhaustive search at cost O(N). "Warm-Shor" (Forge ISA, real β) gives Grover-like O(N) thermal concentration. Full Shor (Meld, β=it) achieves O(logN) via destructive interference — a resource unavailable in either real ISA. New algorithm candidates follow by identifying the symmetry group of the target problem: equivariant ML, quantum persistent homology (Lloyd-Garnerone-Zanardi 2016), and Boson Sampling (Aaronson-Arkhipov 2011) all fit the five-step factory template. Result 3 (ML-KEM structural no-go). ML-KEM (Module-LWE) is immune because LWE noise exponentially suppresses every Fourier component: Ee∼Dσ[ei⟨ξ,e⟩]=e−∣ξ∣2σ2/2. No Fourier peak exists; SPLAT cannot extract a period. The deeper reason is that worst-case LWE reduces to the dihedral hidden subgroup problem (Regev 2002) — a non-abelian problem for which no efficient Meld program is known. Breaking ML-KEM would require a BIND opcode capable of distinguishing dihedral cosets, a capability not established for the current 731-Meld. The Hk complexity ladder (Paper 420) is realised by the trilogy: H0 (tree) = Origami, H1 (loop) = Forge/BPP, H2 (surface) = Meld/BQP. The T-gate is the H2 generator that lifts computation off the stabiliser submanifold into the curved interior of the Fisher information metric. Series: Paper 258 (Origami ISA) → Paper 419 (Forge ISA) → Paper 443 (⊕β semiring, six pairs) → Paper 453 (multi-sector Forge) → Paper 454 (this paper — Meld ISA, trilogy complete). Related identifiers (Cites) Paper 258 Origami ISA: doi:10.5281/zenodo.19916429 Paper 419 Forge ISA: doi:10.5281/zenodo.20694527 Paper 443 ⊕_β semiring: doi:10.5281/zenodo.20752384 Paper 407 Associamancy: doi:10.5281/zenodo.20702223 Paper 365 Fano Magic Factory: doi:10.5281/zenodo.20541583 Paper 366 CS Magic Composition: doi:10.5281/zenodo.20541665 Paper 445 Kitaev anyons: doi:10.5281/zenodo.20752352 Paper 453 Multi-sector Forge: doi:10.5281/zenodo.20773423 Paper 420 H^k complexity ladder: doi:10.5281/zenodo.20773526