The Origami ISA as a Structural Model for MIP* Verification: Pentagon Identity, Frobenius Axiom, and Fano Geometry as Analogues of Entangled-Prover Rigidity Tests
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. 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
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
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 Fano plane PG(2,2) — seven points, seven lines, three points per line — appears three times in quantum error correction: as the stabiliser geometry of the 3-qubit GHZ state, as the parity-check matrix of the 7-qubit Steane ⟦7,1,3⟧ code, and as the constraint graph of a device-independent self-testing protocol. We prove that these three appearances are facets of a single geometric object. The central result is a three-way identification: the non-trivial elements of the GHZ stabiliser group (Z₂)³\{0} are canonically bijective with the 7 points of PG(2,2) = GF(2)³\{0}, which are the columns of the [7,4,3] Hamming parity-check matrix. The 3-qubit GHZ stabiliser group generates the 7-qubit Steane code's parity-check matrix. We prove an orbit decomposition of 3-qubit phase space: PG(5,2)\PG(2,2) = ⊔_{L=0}^{6} O_L, seven orbits of eight phase-space points (one per Fano line), giving a new Clifford-invariant magic label {p_L} that refines Wigner negativity into seven independent channels. The Fano game — measuring the seven GHZ stabilisers — achieves quantum maximum w=1, classical bound w≤5/7, and self-testing robustness C=7/8. The value 7/8 = |PG(2,2)|/2³ is a Fano-geometric invariant. Three practical tools follow: a device-independent GHZ certification protocol, the orbit-label magic invariant for magic state distillation, and a hardware coupling specification for a Fano-structured quantum processor (6 of 7 Fano lines at full coupling, one at r≈0.18, matching the FMO efficiency η≈0.1825). Keywords: quantum error correction, Fano plane, Steane code, GHZ state, magic states, self-testing, Wigner function, orbit decomposition, stabiliser codes, device-independent, PSL(2,7), quantum computing
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. NOTICE (2026-08-04). This record uses a magic measure that has been shown to be incorrectly signed, and it should not be cited for quantitative magic claims. The "total variation" quantity TV = sum |W(u)| over the discrete Wigner function was used with the condition TV < 1 characterising non-stabiliser states. The inequality runs the other way. For any quasi-probability distribution summing to 1, the l1 norm satisfies sum |W| >= 1 always, with equality exactly when W is non-negative. Direct computation: the stabiliser state |+> gives TV = 1.0000 and the magic state T|+> gives TV = 1.2071. The condition "TV < 1" therefore describes the empty set, and any tier defined by it has no members. Separately, TV as defined is exp(mana) -- a monotone reparameterisation of the established Wigner-negativity monotone of Veitch, Mari, Ferrie and Emerson -- and was never compared against it. Readers should consult Veitch et al. on mana and Wigner negativity, Howard and Campbell on robustness of magic, Bravyi et al. on stabiliser extent, and Leone, Oliviero and Hamma on stabiliser Renyi entropy. NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. We recast the three foundational quantum information protocols — quantum teleportation, superdense coding, and the CHSH Bell test — in the language of the Origami Instruction Set Architecture (ISA), and then lift each to its TriQ (3-qubit GHZ) and 6-731 (broken-Fano-line) extensions. In the ISA reading, teleportation is SPLIT;SPLAT with a classical wire, superdense coding is FLIP;FLOP on a Bell pair encoding log2∣G∣ bits via the group G=Z22, and the CHSH violation is a hybrid H¹/H² computation: the Bell pair is a stabiliser (H¹) resource but the Tsirelson-optimal measurement applies a T-gate rotation (H², TWIST=1). We prove that the Tsirelson bound cos2(π/8) equals the total variation of any dark-magic (Tier 1) state in the ISA magic-type classification — the Tsirelson bound is the Wigner mass of the dark-magic sector. Lifting to the TriQ register introduces the ORBIT opcode, raises superdense coding capacity to 3 bits per qubit (log2∣Z23∣), and replaces the CHSH game with the Fano Line Verification Game whose win conditions are the ±1 values of the 6j symbol of Z23. The 6-731 extension (one Fano line weakened to coupling η) makes teleportation directed, reduces superdense capacity to 3−H(η) bits per qubit, and interpolates the Fano game win probability continuously between classical and quantum bounds. Remarkably, the same asymmetry parameter η is independently measurable by three methods — thermodynamic Carnot efficiency (Paper 325), information-theoretic channel capacity, and game-theoretic win probability — whose agreement constitutes a Pentagon-identity self-verification test for the hardware. We close with the Hk Bell tower: for each cohomological depth k there is a Bell-type inequality whose maximum quantum violation requires an Hk resource, providing a game-theoretic ladder that extends beyond CHSH through the Fano game (H²) to a conjectured G2-correlation game (H³) accessible only to the 731-ISA SPIN opcode.Appendix B provides side-by-side quantum circuit and ISA spider diagrams for all three baseline protocols. Keywords quantum teleportation, superdense coding, CHSH inequality, Bell test, Tsirelson bound, Origami ISA, H^k cohomology, stabiliser states, magic states, Fano plane, GHZ state, TriQ register, 6-731 architecture, ORBIT opcode, TWIST opcode, topological heat engine, dark magic, Wigner mass, Bell hierarchy, Mermin inequality, Pentagon equation, quantum nonlocality, quantum channel capacity, G2 correlations
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