Reformulates the Origami ISA as a coloured PROP (product and permutation category), giving the resource theory of magic a precise circuit-syntax foundation. Preparation morphisms are ket-morphisms, measurement morphisms are bra-morphisms, and dark magic corresponds to programmes that rewrite to the identity. Extends the ZX calculus to the C3 level and connects to Wakeham's props construction.
Reformulates the Origami ISA as a coloured PROP (product and permutation category), giving the resource theory of magic a precise circuit-syntax foundation. Preparation morphisms are ket-morphisms, measurement morphisms are bra-morphisms, and dark magic corresponds to programmes that rewrite to the identity. Extends the ZX calculus to the C3 level and connects to Wakeham's props construction.
WITHDRAWN BY THE AUTHOR (2026-08-02). This record should not be cited. Fileshave been restricted. --- Why --- The paper proposes reading quantum resource theory syntactically, as a colouredproduct-and-permutation category (coloured PROP), with the Clifford-hierarchylevels C1/C2/C3 as colours and resource cost as the minimum C3-gate count in anormal form. Two of its load-bearing inputs have since been withdrawn, and oneof them carries an error that this paper inherits verbatim. 1. THE TOTAL-VARIATION GRADING IS BASED ON A SIGN ERROR. The paper defines"dark magic" as states that are non-stabiliser yet have zero Wigner negativityand total variation TV < 1, and states (Sec. 6) that TV(p) = 1 if and only if pis a free state. This is incorrect. For any quasi-probability distributionsumming to 1, the l1 norm satisfies Sum |W| >= 1 always, with equality exactlywhen W is non-negative. Direct computation: the stabiliser state |+> givesTV = 1.0000, and the magic state T|+> gives TV = 1.2071. The inequality runsthe other way, so the condition "TV < 1" describes the empty set and the"dark magic" tier as defined has no members. The source of this definition(a companion paper on magic resource theory) has been withdrawn for the samereason. Separately, TV as defined is exp(mana), a monotone reparameterisationof the established Wigner-negativity monotone of Veitch, Mari, Ferrie, Emersonand others, and was not compared against it. 2. THE WORKED EXAMPLES NO LONGER EXIST. Every worked example in the paper isone of the "9 magic orbit classes" of a companion paper that has since beenwithdrawn because its own experiment scripts did not reproduce its claims(printing FAILED for the orbit-inheritance theorem, and fidelity 0.000000against a claimed value of 1/4, with 8 classes printed where 9 were claimed).Two further companion papers cited for the rewrite-rule and normal-formresults have also been withdrawn. 3. NO COMPUTATIONS WERE PERFORMED. There is no experiment directory associatedwith this paper. --- What was not wrong --- The organising idea -- that a resource theory can be presented syntactically,with resource tier readable from a diagram -- is reasonable and is not thereason for withdrawal. But the specific cost measure proposed (minimum C3-gatecount) is T-count, a standard and well-studied quantity (Amy, Maslov and Mosca;Gosset, Kliuchnikov, Mosca and Russo, and subsequent work), and gradedcategorical presentations of Clifford+T are the subject of the ZX-calculuscompleteness literature (Jeandel, Perdrix and Vilmart 2018 and successors).Neither literature is engaged here. A future treatment would need to start fromthose. Readers wanting the correct treatment of the underlying physics should consultVeitch et al. on Wigner negativity and mana, Howard and Campbell on robustnessof magic, and the ZX-calculus completeness results cited above.
Standard quantum resource theory uses Wigner negativity N>0 as its resource indicator for magic states. We show this is incomplete: there exist states outside the stabiliser polytope with N=0, which we call dark magic. These states are undetectable by the standard monotone yet are non-stabiliser by every other measure. The Origami ISA rewrite system R=R3,R4 (completeness proved in companion Paper 469) partitions the single-phase 3-qubit fragment F into exactly 9 normal-form classes in three tiers: Tier Class TVN 0 Stabiliser (1 class) =1=0 1 Dark magic (6 classes) <1=0 2 Genuine magic (2 classes) <1>0 Theorem 1 (TV monotone). The total variation TV(ψ)=∑u∣Wψ(u)∣ of the discrete Wigner function is a complete resource monotone on F: TV(ψ)=1 if and only if ψ is a stabiliser state. Wigner negativity N=21(TV−1) vanishes on all of Tier 1, so it is merely sufficient (Tier 2) but not complete. Theorem 2 (Three-tier completeness). The tier of any state in F is decidable at compile time: apply R3 (T-absorption counit) — if the T gate is absorbed the state is Tier 0; otherwise compute TV from the 9-entry precomputed table. Corollary (Distillation failure mode). A magic-state distillation protocol using N as its sole convergence criterion may silently converge to a dark-magic state (Tier 1, N=0, TV<1) and declare success. The fix: require TV=1 as the termination criterion. Dark magic occupies the tropical boundary of the MGE β-ladder: Tier 1 states are quantum in structure (TV<1) but classical in Wigner measure (N=0), exactly at the Maslov dequantisation saddle point β→0. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 469 (completeness theorem) Keywords: quantum resource theory, magic states, dark magic, Wigner negativity, total variation, stabiliser states, Clifford hierarchy, Origami ISA, rewrite systems, normal forms, magic state distillation, tropical geometry, Maslov dequantisation, coloured PROP
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. The resource theory of quantum magic (Veitch et al. 2014) identifies non-stabiliser states as resources and T-gates as the elementary injection operation. Subsequent work established that the magic sector itself splits into two sub-tiers: dark magic (Wigner-negative but TV = 1, Clifford-simulable under reframing) and genuine magic (TV < 1), where TV is the total variation of the discrete Wigner function. We identify a strictly richer resource — associamancy — that sits above both sub-tiers and that standard magic theory cannot see, cannot create from below, and cannot measure without the SPIN opcode of the 731 ISA. The boundary is algebraically precise. A quantum state possesses associamancy if and only if its hidden symmetry group contains an irreducible representation with Frobenius-Schur indicator ν₂ = 0 — the condition that the representation is genuinely complex, realisable over ℂ but not over ℝ or ℍ. We call this the Schur boundary. Three equivalent characterisations of the Schur boundary are established: (1) Algebraic: ν₂(V_λ) = 0 for some irrep V_λ of the hidden symmetry group. (2) Cohomological: the obstruction class [α] ≠ 0 ∈ H²(G, U(1)) is non-trivial. (3) Number-theoretic: for G = PSL(2,q), associamancy holds if and only if q ≡ 3 (mod 4) — the same condition that makes the quadratic Gauss sum τ(q) = i√q purely imaginary. The Schur entropy S_Schur(|ψ⟩) = −∑_{λ:ν₂=0} p_λ log p_λ is a valid resource monotone, independent of both Wigner negativity and TV: a state can have TV ≪ 1 (genuine magic) and zero Schur entropy, and a state can have S_Schur = log 2 at either dark or genuine magic level. PSL(2,7) is the minimal group exhibiting associamancy (order 168). The Freudenthal magic square organises associamancy into a ladder G₂ ⊂ F₄ ⊂ E₆ ⊂ E₇ ⊂ E₈, each rung strictly richer than the last. Keywords: quantum resource theory, magic states, associamancy, Frobenius-Schur indicator, Schur boundary, non-associative, octonions, PSL(2,7), Freudenthal magic square, G₂, 731 ISA, dark magic, total variation, 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 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