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 ($\beta\to\infty$, tropical, classical), Forge ($0<\beta<\infty$, real, statistical), Meld ($\beta=it$, 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 $T = \mathrm{diag}(1,, e^{i\pi/4})$ is the BIND opcode at $j=\tfrac{1}{2}$ — 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 $\arg\max_{g\in G} f(g)$ over an abelian group $G$ has a $\mathbb{C}$-MGE quantum counterpart $\mathbb{C}\text{-MGE}(f) = \mathrm{SPLIT}_G \cdot \mathrm{FLIP}(f) \cdot \mathrm{SPLAT}$, 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, $\beta\to\infty$) recovers classical exhaustive search at cost $O(N)$. "Warm-Shor" (Forge ISA, real $\beta$) gives Grover-like $O(\sqrt{N})$ thermal concentration. Full Shor (Meld, $\beta=it$) achieves $O(\log N)$ 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: $\mathbb{E}{\mathbf{e}\sim D\sigma}[e^{i\langle\xi,\mathbf{e}\rangle}] = e^{-|\xi|^2\sigma^2/2}$. 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 $H^k$ complexity ladder (Paper 420) is realised by the trilogy: $H^0$ (tree) = Origami, $H^1$ (loop) = Forge/BPP, $H^2$ (surface) = Meld/BQP. The T-gate is the $H^2$ 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 ($\oplus_\beta$ 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. 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. 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
We introduce the cookie-cutter lifting procedure: a systematic method for determining whether a quantum algorithm requires magic (non-stabiliser resources) or whether its speedup is purely Clifford-level. The procedure writes any classical algorithm as an Origami ISA programme at the classical (β=0) level and identifies whether the quantum lifting requires the TWIST opcode—the ISA marker for non-Clifford C₃-level gates. Applied to Shor's algorithm (N=15, a=7): the ISA trace LABEL→FLIP→FLOP(U_f)→FLIP(QFT) has mana=0 along the pre-measurement branch; for this instance the period r=4 divides the register size 2ⁿ=16 exactly, so the QFT output is a stabiliser state on the zero-measurement branch. The speedup comes from exponential-weight stabiliser generators compressed by coherent cancellation, with no magic required on that branch. Applied to non-abelian hidden shift over D_N: mana=0 before the non-abelian QFT and mana>0 after, for all N≥3; the abelian cases D₁≅ℤ₂ and D₂=ℤ₂×ℤ₂ are the Clifford exceptions. The TWIST opcode fires because non-abelian D_N (N≥3) have 2-dimensional irreducible representations whose Fourier blocks lie outside C₂. We identify the eigenphase spectrum of the NAQFT as a basis-independent replacement for T-count. The results reveal two roads to quantum advantage: Clifford complexity (Shor) and magic (non-abelian hidden shift). Part of the Origami ISA series. Companion papers: 10.5281/zenodo.20955514 (ISA as PROP), 10.5281/zenodo.20956674 (nine normal forms), 10.5281/zenodo.20956685 (three-tier resource theory). Keywords: shor, clifford, lifting, qft, hidden-subgroup, naqft, twist, magic, mana, eigenphase, origami-isa, cookie-cutter, dihedral-group, stabiliser, wigner-function
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 executes algorithms as Pachner moves on simplicial complexes at inverse temperature β→∞ (the discrete, tropical limit). The Meld identifies the β→0 limit, where the same opcodes become the operators of smooth Hodge theory. The Forge ISA is the bridge: the same five opcodes (SPLIT, SPLAT, FLOP, FLIP, TWIST) running at finite inverse temperature β under a Gibbs distribution over H¹-weighted configurations. A Forge ISA programme at inverse temperature β is an Origami ISA programme on a sheaf ℱ over a simplicial complex X, where FLOP corrections are applied with probability proportional to the Gibbs weight exp(−β‖H¹(s)‖), and the Maslov–Gibbs Einsum (MGE) is the unique β-correct implementation. The critical temperature β*(ρ) = (3/8) ln(1/(1−ρ)), where ρ = β₁/|E| is the load factor (first Betti number divided by edge count), is the universal threshold separating the efficient (β < β*) from the stalled (β > β*) regime, empirically confirmed across eight algorithm classes. The Topological Resonance Synthesis (TRS) mandate (five purity conditions) ensures the Forge ISA is thermodynamically correct: β appears only in the Hamiltonian via the MGE; conformal, symplectic, and adiabatic structure are preserved throughout. Standard machine learning optimisers (Adam, dropout, batch normalisation) violate one or more of these conditions. The vorton is the elementary sampler — a single draw from the Gibbs distribution P_β on the state manifold — and the snap event is crystallisation at the harmonic representative of H¹. The Forge ISA is the engine of the TRS trilogy: the Origami ISA (cold reservoir, β→∞), the Forge ISA (the cycle, 0 < β < ∞), and the Meld (hot reservoir, β→0). Keywords instruction set architecture, thermodynamic computation, Gibbs distribution, inverse temperature, Origami ISA, The Meld, TRS trilogy, Maslov–Gibbs Einsum, MGE, simplicial complexes, sheaf theory, H¹ cohomology, Hodge theory, critical temperature, β-ladder, vorton, snap event, Carnot cycle, information geometry, topological resonance synthesis, differentiable programming, tropical geometry, adiabatic computation, Fisher metric
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 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
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