Maps the full programme of information geometry onto the H^k ISA ladder. H0 corresponds to flat exponential families and tropical likelihood (A(theta) as tropical potential); H1 to Amari's alpha-connection and TWIST monodromy; H2 to Uhlmann holonomy and BIND. The EM algorithm is a SPLIT/SPLAT cycle; the beta* snap is the maximum Fisher information threshold. Provides a unified ISA syntax for classical and quantum information geometry.
Maps the full programme of information geometry onto the H^k ISA ladder. H0 corresponds to flat exponential families and tropical likelihood (A(theta) as tropical potential); H1 to Amari's alpha-connection and TWIST monodromy; H2 to Uhlmann holonomy and BIND. The EM algorithm is a SPLIT/SPLAT cycle; the beta* snap is the maximum Fisher information threshold. Provides a unified ISA syntax for classical and quantum information geometry.
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
Shows that XXZ chain, transverse-field Ising model, Levin-Wen string nets, and Kitaev quantum doubles are a single ISA family parameterised by the beta-deformation. At beta=1/2 the BKT transition is a TWIST failure. Walker-Wang models are identified as H2 (BIND-mandatory). Haah codes require twisted coefficients and lie outside the standard H2 tier (fracton sector). The quantum group parameter q = exp(i*pi*beta) encodes the semiring interpolation between tropical (beta=infinity) and metaplectic (beta=0).
Barren Plateaus are H1: The Structural Null Space of the Quantum Fisher Information Matrix as Sheaf Cohomology
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. This paper proves that barren plateaus in variational quantum algorithms (VQAs) are exactly the structural null space of the Quantum Fisher Information Matrix (QFIM), and that this null space is the first sheaf cohomology group H¹ of the parameter sheaf of the quantum circuit. The QFIM F(θ) is the Fubini-Study metric on the parameter manifold CP^{2^n−1}. Its rank is bounded by 2(2^n−1) for any p-parameter circuit, regardless of n. When p exceeds this bound, the excess parameters lie in ker(F) — a structural null space that cannot be eliminated by gradient descent. This is the barren plateau: gradients are exactly zero in the null-space directions, not approximately zero due to vanishing. The main result (Theorem 3.1): rank F(θ) ≤ 2(2^n−1) for any p-parameter circuit on n qubits. The structural H¹ dimension is max(0, p − 2(2^n−1)). Experimental validation (x418a/b): the QFIM rank bound is confirmed exactly for random circuits at n=4,6,8 qubits. The FLOP prediction is confirmed: adding a circuit layer (architectural change) escapes the plateau, while parameter adjustment within the same architecture cannot. The Krishnagopal-Bianconi (2022) sparse eigenvector approach to ker(F) computation is identified as computationally efficient alternative to the O(p³) QFIM null-space calculation. This paper sits within the Origami ISA / TRS Framework and connects to the Forge ISA (Paper 419) and the H^k Complexity Ladder (Paper 420). The barren plateau is the H¹ rung of the complexity ladder applied to quantum circuit optimisation. Keywords barren plateaus, variational quantum algorithms, QAOA, VQE, quantum Fisher information matrix, QFIM, sheaf cohomology, H¹ obstruction, Fubini-Study metric, circuit architecture, Origami ISA, TRS framework
We show that every exactly solvable quantum spin model with a phase transition is a different β-schedule of the Origami instruction-set architecture (ISA). The unifying equation is q = exp(iπβ): the quantum group deformation parameter q underlying the XXZ chain, the transverse-field Ising model (TFIM), Levin–Wen string-net models, and Kitaev's quantum doubles is identically the Gibbs ISA inverse temperature. This gives a single β-axis on which every topological phase of matter has a precise address. The main results are: (i) The XXZ Bethe-ansatz two-magnon R-matrix is the FLIP eigenvalue at q = exp(iπβ), with the BKT transition as a TWIST failure at β = 1/2 (exact, by cos(π/2) = 0). (ii) The 1D XY model and the 2D Kitaev toric code share β = 1/4 and quantum group SU(2)₂ — the same ISA programme in different geometries. (iii) For non-Abelian string-net phases in the SU(2)k family, the quantum dimension d{1/2}(β) = 2 cos(πβ) is an exact phase label, verified for k = 1 through 15. (iv) The cohomological chain H⁰/H¹/H² = Islands/Flatland/The Deep corresponds to classical/toric-code/Walker–Wang codes. Fracton phases require twisted cohomology H*(M; L) beyond standard H². The β-ladder (Table I) places every major model on a single axis: β = 0 (Heisenberg SU(2)), β = 1/5 (Fibonacci anyons, SU(2)₃), β = 1/4 (Ising anyons / toric code, SU(2)₂), β = 1/3 (semion / XY model, SU(2)₁), β = 1/2 (BKT transition, TWIST failure). All results are verified numerically in experiments x446a–x446d. This paper belongs to the Origami ISA series. Paper 443 (Planck's Constant in Disguise, doi:10.5281/zenodo.20752384) establishes the semiring deformation; Paper 445 (Kitaev Menagerie, doi:10.5281/zenodo.20752352) treats the Kitaev models as a special case; Paper 447 (Opcode Rosetta Stone, doi:10.5281/zenodo.20761260) extends the ISA dictionary to twelve models. Keywords exactly solvable models, quantum groups, XXZ model, BKT transition, topological phases, string-net models, toric code, Walker-Wang model, fracton phases, instruction set architecture, beta deformation, Yang-Baxter equation, Bethe ansatz, quantum dimension, Luttinger liquid, cohomology, tensor networks, quantum computing