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. 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. 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 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
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
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. Hinsche, Eisert & Carrasco (PRX Quantum 7, 020337, 2026) introduced the abelian state hidden subgroup problem (StateHSP) and showed it is solved by quantum Fourier sampling via the character POVM Π_λ = (1/|G|) Σ_g χ̄_λ(g) R(g). We show this entire framework is an instance of the Origami ISA: the character POVM is the SPLAT opcode (fibre integration on the group algebra sheaf), and the full algorithm is the canonical SPLIT→SPLAT pipeline. Three corollaries: (1) Bell sampling is SPLAT on 2 copies with the SWAP representation over G=ℤ₂ⁿ. (2) Qudit stabiliser learning at d=3 is SPLAT on 3 copies, native to the TriQ register at depth O(1). (3) The StateHSP is H⁰ recovery — the hidden subgroup H is the zeroth sheaf cohomology H⁰(Γ_G, F_ψ) of the symmetry sheaf, and the O(log|G|/ε) sample complexity follows from H¹=0 (anticoncentration). The non-abelian extension — requiring the SPIN opcode and genuine complex irreps — is treated in the companion paper (doi:10.5281/zenodo.20667170). Keywords: quantum computing, hidden subgroup problem, Fourier sampling, SPLAT, Origami ISA, Bell sampling, stabiliser learning, sheaf cohomology, TriQ register, character POVM, abelian
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
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