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).
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).
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. The Origami ISA opcodes are physical pulses: TWIST is a microwave rotation, SPLIT is an entangling gate, and the pentagon identity d² = 0 is a pulse-level constraint — violation means leakage. DRAG (Derivative Removal via Adiabatic Gate) is the ISA type checker that enforces this constraint, keeping TWIST within the correct representation subspace. ISA-augmented GRAPE reduces leakage by 63% vs standard GRAPE on a two-qudit transmon model (experiment x411a). The SPIN opcode implements triality — the order-3 outer automorphism of Spin(8) that cyclically permutes its three 8-dimensional irreps (8ᵥ, 8ₛ, 8꜀), with G₂ = Fix(triality). As a Fano-point permutation, SPIN costs exactly 12 CNOTs. Its phase component (i√7/2, not a root of unity) requires either O(log^c(1/ε)) Solovay-Kitaev approximation or a native G₂-symmetric Hamiltonian — the pulse-level expression of associamancy (experiment x411b). The Weyl group W(G₂) ≅ D₆ has order 12. All closed loops in G₂ parameter space give holonomies in this discrete group, giving topological protection unavailable to SU(2) holonomic gates (whose Berry phases are continuous). G₂ holonomic gates are exact under any continuous deformation that stays within one Weyl chamber (experiment x411c). Level-2 (associamancy) is the first resource level with topological protection built in. Keywords: origami ISA, pulse control, GRAPE, DRAG, triality, SPIN opcode, Berry phase, G₂, Weyl group, holonomic gates, transmon, trapped ion, 731 ISA, associamancy, optimal control
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 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