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
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
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 Eastin-Knill theorem states that no quantum error correcting code can have a universal set of transversal gates. This is widely interpreted as requiring magic-state distillation (MSD) as an unavoidable overhead for fault-tolerant non-Clifford computation. This paper demonstrates three distinct routes to CS-family non-Clifford gates on the SQU architecture that circumvent the practical impact of Eastin-Knill without violating it. Route 1 — Gauge-orbit CS transversality: The [[7,1,3]] Steane code has 6 gauge degrees of freedom corresponding bijectively to the 6 non-classical Fano orbit valences. Preparing the gauge subsystem in orbit valence L ∈ {1,2,3} via the DISTIL opcode enables transversal application of CS_{ij} — the qubit pair associated with orbit L — across all 7 physical qubit triplets simultaneously. This implements a logical CS gate fault-tolerantly at 7 physical qubits, versus 15 for the [[15,1,3]] Reed-Muller approach, with zero magic-state distillation and zero code switching. Route 2 — CONVERT as free valence switching: The CONVERT opcode is a free Clifford gate that switches orbit valence within the primary family {1,2,3} or secondary family {4,5,6}. Any CS-type gate is reachable from any other via one SWAP gate, at zero distillation cost. The gauge-orbit correspondence makes CS-family universality a matter of gauge selection, not resource consumption. Route 3 — Syndrome-blind valence verification: The CS₀₁, CS₀₂, and CS₁₂ states are indistinguishable by stabiliser syndrome (all produce syndrome (+1)⁷) but are perfectly distinguished by the ORBIT opcode. This enables fault-tolerant gate verification at zero additional syndrome cost — the ORBIT measurement is performed on the same 7 qubits already measured for error correction. Together these routes reduce the physical qubit overhead for CS-family computation from 15 to 7, eliminate MSD entirely for CS gates, and provide built-in coherent error detection. The Eastin-Knill theorem is respected: the CS gate is not transversal in the code sense. What changes is the architecture — the gauge structure of the Steane code is the resource, not ancilla magic states. Keywords Eastin-Knill Theorem, Transversal Gates, Magic State Distillation, Fault-Tolerant Quantum Computation, CS Gate, Controlled-S Gate, Steane Code, [[7,1,3]] Code, Gauge Selection, Fano Orbit Valence, Orbit Valence, ORBIT Opcode, DISTIL Opcode, CONVERT Opcode, Magic-ISA, SQU Architecture, TriQ, Quantum Error Correction, Syndrome Blindness, Non-Clifford Gates, Gauge Subsystem, W(5,2), Fano Plane, PG(2,2), Origami ISA, Zero Overhead, Coherent Error Detection
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. Magic-state distillation (MSD) is the dominant overhead cost in fault-tolerant quantum computing: each logical T or CCZ gate currently requires 100–1,000 physical qubits dedicated to distillation, consuming the majority of a quantum processor's resources. This paper introduces the Origami Instruction Set Architecture (Origami ISA) as a compiler layer for fault-tolerant quantum error correction — operating on existing lattice surgery hardware without modification. The five opcodes — FLIP, FLOP, SPLIT, SPLAT, TWIST — operate at the logical level and compute exact symbolic (rational or algebraic) amplitudes via the thermion library, enabling algebraic verification of gate correctness before any physical circuit is generated. Three contributions: 1. Frobenius cancellation: The Frobenius axiom (SPLIT(j) ∘ SPLAT(j)= id) is an automatic compiler optimisation that detects and removes redundant merge-boundary open/close pairs in lattice surgery protocols, reducing physical gate count by 29–67% on standard protocols. 2. Regime ladder: Every quantum error-correcting code is classified by its F-move amplitudes. Rung-0 codes (stabiliser codes, surface codes) require MSD for non-Clifford gates. Rung-1 codes ([[15,1,3]] Reed-Muller, colour codes) support transversal non-Clifford gates with zero MSD overhead. The ladder predicts which hardware can eliminate MSD and which cannot. 3. Zero-MSD CCZ protocol: The SPLIT;FLOP;FLOP;CCZ;FLOP⁻¹;FLOP⁻¹;SPLAT protocol implements a universal CCZ gate on the combined [[7,1,3]] + 1531-Anvil hardware without any magic-state distillation. Machine-verified correctness via the Frobenius axiom and Pentagon identity, proved by exact symbolic arithmetic. The Pentagon identity holds exactly for all five codes examined — proving by exact symbolic arithmetic rather than numerical simulation. Keywords Origami ISA, Fault-Tolerant Quantum Computing, Magic State Distillation, Lattice Surgery, Frobenius Axiom, Pentagon Identity, Regime Ladder, CCZ Gate, Transversal Gates, Reed-Muller Code, [[15,1,3]] Code, Steane Code, [[7,1,3]] Code, 1531-Anvil, FLIP Opcode, FLOP Opcode, SPLIT Opcode, SPLAT Opcode, Quantum Compiler, Diagrammatic Calculus, Exact Arithmetic, Thermion Library, F-Move Amplitude, Colour Code, Surface Code, Quantum Error Correction, Zero Overhead
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.
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 articulate a structural parallel between the MIP*=RE theorem and the Origami Instruction Set Architecture (ISA). The algebraic rigidity of entangled-prover verification — exact-or-fail Pauli basis tests, low-degree tests, Mermin Magic Square constraints — corresponds precisely to the Frobenius axiom SPLIT∘SPLAT=1, the Pentagon identity, and the seven-point Fano plane PG(2,2) in the ISA framework. Six numerical experiments confirm the parallels: Pentagon unitarity holds exactly for all j ∈ {1/2,...,15/2}; Frobenius fidelity degrades as (1-p)^{2(2j+1)} under depolarising noise; the GHZ stabiliser group forms a perfect quantum Fano strategy satisfying all seven constraints simultaneously while no classical strategy can; G₂ non-associativity produces a sharp excluded-volume step function above j=7/2. This version adds a synthesis remark showing that the numbers 3 and 7 — the two dimensions where the cross product exists, gyroscopes are stable, and choreographic orbits can close — are not two separate coincidences but the same theorem: the Fano plane PG(2,F₂) has 7 points and 3 points per line because these are the imaginary unit counts of the octonions O and quaternions H respectively, forced by the unique projective plane of order q=2 over the field F₂. The broken Fano (6/7 architecture) and the magic/non-associativity conjecture are restated in this unified language. The practical implication is a conjecture that fault-tolerant quantum computing overhead is governed by the regime of the code's F-move amplitudes, with MSD-free computation possible at rung 1 and higher. Keywords MIP* Equals RE, Interactive Proof Systems, Quantum Entanglement, Pentagon Identity, Frobenius Axiom, Fano Plane, PG(2,2), GHZ Stabiliser, Origami ISA, Wigner 6j Symbol, Fault-Tolerant Quantum Computing, Magic State Distillation, Regime Ladder, Complexity Theory, Quantum Prover, Local Pauli Test, Low-Degree Test, G₂ Non-Associativity, Excluded Volume, Depolarising Noise, Rung-1, Quantum Verification, Adelic Simplicial Architecture
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.