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The ISA Completeness Theorem: Every Physical Transformation is an Origami ISA Programme

Ian Richard C Buckley

Published Jul 6, 2026• 0 citations
Publisher

Abstract

Proves the ISA completeness theorem: every transformation on a representation-theoretic state space can be expressed as a finite sequence of the seven Origami ISA opcodes (FLIP, FLOP, SPLIT, SPLAT, TWIST, LABEL, ORBIT). The proof uses the Solovay-Kitaev theorem for the H1 sector and the Brauer-Weyl theorem for the H0 sector. The H2 sector requires the BIND opcode and the Uhlmann holonomy theorem.

DOI
10.5281/zenodo.21224121

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Proves the ISA completeness theorem: every transformation on a representation-theoretic state space can be expressed as a finite sequence of the seven Origami ISA opcodes (FLIP, FLOP, SPLIT, SPLAT, TWIST, LABEL, ORBIT). The proof uses the Solovay-Kitaev theorem for the H1 sector and the Brauer-Weyl theorem for the H0 sector. The H2 sector requires the BIND opcode and the Uhlmann holonomy theorem.
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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.
Establishes that ISA perturbation theory is the Dyson series: the n-th order correction equals n extra TWIST insertions in the ORBIT loop. The coupling constant lambda = J/zeta_eff is the ratio of two opcode amplitudes; the perturbativity criterion lambda < 1 equals finite programme depth. Derives an ISA analogue of the Solovay-Kitaev theorem bounding programme depth. Identifies coupling constants across physics from gravity (alpha_G ~ 1e-38) through QED (alpha ~ 1/137) to cuprate pseudogap (lambda ~ 21-28, non-perturbative).
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.