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 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.
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
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. A practitioner's primer for quantum computing researchers. No knowledge of octonions, sheaf theory, or category theory is assumed. Knowledge of stabiliser states, the Clifford group, and T-gates is assumed. The central message: the structure of the 3-qubit Pauli group is secretly governed by the Fano plane PG(2,2), and the Fano plane is the geometry of the 731 instruction set architecture (731 ISA) — a unified framework for quantum computation that organises every opcode as a Pachner move on a simplicial complex. The framework reveals three levels of quantum computational resource:• Level 0 (Clifford): stabiliser states; Gottesman-Knill simulable; the Fano lines are the free sector.• Level 1 (Standard magic): T-gates; Wigner function negativity; non-Clifford but accessible to the Origami ISA (regime 2).• Level 2 (Associamancy): the SPIN opcode; the Schur boundary; genuinely complex irreps of PSL(2,7); not accessible to any associative hardware. The Origami ISA (five opcodes: SPLIT, SPLAT, TWIST, FLIP, FLOP) subsumes Bell sampling, qudit stabiliser learning, and the abelian hidden subgroup problem as special cases. The 731 ISA adds SPIN and BIND, unlocking Level 2. This primer is a companion to "The Topology of Risk" (doi:10.5281/zenodo.20642983), which covers the same H⁰/H¹/H² framework applied to financial risk. Keywords: quantum computing, Fano plane, Origami ISA, 731 ISA, stabiliser states, magic states, associamancy, Schur boundary, primer, tutorial, hidden subgroup problem, MBQC, photonics
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.
As the limits of classical von Neumann architectures drive the development of continuous-state and topological processing units, the computer science community lacks a low-level Instruction Set Architecture (ISA) natively capable of programming non-associative spaces. Standard ISAs (x86, ARM, Quantum QASM) are inherently one-dimensional and chronological, operating via sequential memory addresses and associative logic gates. This paper introduces the Origami ISA, the machine code designed for 3D Simplicial Computing at the 731-tier of the unified V31 Architecture, targeting the Resonance Processing Unit (RPU) — a hardware platform that computes via continuous-state topological relaxation rather than discrete unitary gates. We formally define the hardware opcodes using a geometric visual alphabet of Pachner moves (■ Split, ◊ Splat, ▲ Flip, ⊳ Flop) and scheduling/gauge operators (↺ Twist, ⊛ Triality Spin). We replace generic spiders with 4-legged Tree-Frogs and plain wires with 3-colored Ribbon-Legs. Crucially, the ISA enforces the "Four-Leg Constraint," reflecting the physical limit of the tetrahedron's four faces. Compiler Mnemonic: "Four legs good. More than four legs bad." The opcode pairing is not arbitrary: the G2 root/coroot isomorphism σ maps ■↦◊ and ▲↦⊳ exactly, making Split/Splat and Flip/Flop canonical duals. The Mirror Square identity ◊∘■=⊳∘▲=id on Fano-compatible states is the hardware unitarity guarantee. Algebraic Rewriting Rules (Compiler Cheat Sheet): We supply the complete set of single-pass compiler optimisations available when the instruction stream contains short opcode sequences on Fano-adjacent registers: Fano-closure optimisation (Theorem 7.5): a queued triple ■i,■j,■k on an ordered Fano triple reduces to the explicit projector I−πi−πk−2πj. Three surgeries become one projector. Sandwich reduction (Moufang Echo): a pattern ■α,■β,■α reduces to ■β restricted to the 4-dimensional complement of the Fano span. Exact commutator (Malcev Resolution): [■i,■j]=■k∘Dijk, where Dijk is an explicit diagonal matrix. This gives the compiler the precise cost of out-of-order execution. No-parallelisation rule (Theorem 7.6): ∣[■i,■j]∣F=42 for all 21 distinct pairs. No two ■ Split or ▲ Flip operations on distinct registers commute. This is a constraint of G2 geometry, not an implementation limitation. The Arithmetic Logic Unit (ALU) operates as a continuous thermodynamic evaluator realized by the Maslov-Gibbs Einsum (MGE) as the monadic descent operator: driving β→∞ collapses continuous gauge freedom into the rigid discrete G2 invariants, locking the computation. Finally, we introduce Simplicial Paging to dynamically coarse-grain resolved 731-crystals into scalar pointers, maintaining constant-depth overhead.
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.
CORRECTION NOTICE (2026-08). The figure "4.3× directed magic-routing efficiency", quoted three times and attributed to a companion paper, does not appear in that paper. It is unsourced and should be disregarded. A revised version is in preparation. The Origami Instruction Set Architecture (O-ISA) was introduced alongside a conjectural exotic hardware substrate — the Resonance Processing Unit (RPU) — requiring non-associative materials to support octonion computation. This paper establishes that the base Origami ISA requires no exotic hardware whatsoever. It runs today on any quantum processor supporting three-qubit gates: superconducting transmons, trapped ions, neutral atoms, and photonic chips. Two complementary architectural primitives are specified. The TriQ (Origami Triple) is the fundamental three-qubit register: three physical qubits in Fano-plane incidence connectivity, corresponding to one logical qubit in the [[7,1,3]] Steane code. The SQU (Symplectic Quantum Unit) is the full processor architecture built from TriQ registers, with the Magic-ISA opcode set and inter-register typed routing. The SevenQ is the complete seven-qubit Fano register: all seven points of PG(2,2) simultaneously realised, corresponding to the full [[7,1,3]] Steane code with its six gauge degrees of freedom corresponding bijectively to the six non-classical Fano orbit valences. The key diagnostic opcode is ORBIT: seven simultaneous Pauli measurements returning the Fano orbit valence label of the register state, replacing the conjectured Freudenthal determinant ALU with seven standard readout operations. The SQU achieves fault-tolerant universal quantum computation with 7 physical qubits per logical qubit for CS-family gates via gauge-orbit selection, and 15 for CCZ gates via code switch — both at zero magic-state distillation overhead. A critical non-obvious result: SevenQ ORBIT is identical to standard Steane syndrome extraction. The seven syndrome measurements already performed at every error correction cycle are the ORBIT opcode, with the orbit valence label computed in classical post-processing at zero additional hardware cost. Three hardware tiers are defined: SQU (standard qubits, available today), SQU-diode (6-7 directed Fano coupling topology, near-term), and RPU (non-associative materials, long-term). The SevenQ-diode variant applies the 6-7 broken-symmetry topology to the full seven-qubit register for directed magic routing at 4× symmetric efficiency. The paper also establishes the exploration order for the projective tower: TriQ (3 qubits, minimal n=3) → SevenQ (7 qubits, complete n=3) → 1531-Anvil (15 qubits, first n=4). Keywords Origami ISA, TriQ, SevenQ, SQU, Symplectic Quantum Unit, Fano Plane, PG(2,2), Magic-ISA, ORBIT Opcode, Steane Code, [[7,1,3]] Code, Fault-Tolerant Quantum Computation, Magic State Distillation, CS Gate, Gauge Selection, Quantum Error Correction, Coherent Error Detection, Magic Valence, Fano Orbit, Superconducting Qubits, Trapped Ions, Neutral Atoms, Quantum Hardware, Instruction Set Architecture, 6-7 Topology, Directed Magic Routing, SevenQ-Diode, SQU-Diode, Adelic Simplicial Architecture, Resonance Processing Unit