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The Three Forms of the Maslov-Gibbs Einsum: Tropical, Clifford, and Metaplectic

Ian Richard C Buckley

Published Jul 6, 2026• 0 citations
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Abstract

Shows that the Maslov-Gibbs Einsum (MGE) takes three canonical forms corresponding to the H0/H1/H2 complexity tiers: tropical (min-plus at beta = infinity), Clifford/geometric-Kitaev (at beta = beta*), and metaplectic/Maslov-index (at beta = 0 or imaginary). Establishes the Maslov duality theorem connecting H0 and H2 forms via Legendre transform. Introduces the Penrose empire analogy for H2 non-locality and identifies an H3 frontier.

DOI
10.5281/zenodo.21219718

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The Three Forms of the Maslov-Gibbs Einsum: H⁰ Tropical, H¹ Clifford, H² Metaplectic

Ian Richard C Buckley

The Maslov-Gibbs Einsum (MGE) is not a single algorithm but a family of tensor contraction schemes indexed by the cohomological level $H^k$ of the physical system. This paper identifies three distinct forms: $H^0$ MGE (classical/tropical): contraction over the min-plus tropical semiring, corresponding to Maslov dequantisation ($\hbar \to 0$). Finds the dominant classical path; computational class P. $H^1$ MGE (Clifford/stabiliser): contraction over the stabiliser calculus of the Clifford group $\mathsf{C}_2$. Captures quantum interference within the stabiliser sector. Classically simulable in polynomial time by the Gottesman-Knill theorem, but exponentially more expressive than $H^0$: captures non-local correlations, entanglement, and quantum error correction. $H^2$ MGE (metaplectic/magic): contraction with Maslov index phase corrections. Each gate composition $g \circ h$ acquires a $\mathbb{Z}_2$ phase factor $(-1)^{\tau(L, gL, ghL)}$ from the Maslov index of three Lagrangian subspaces. The TWIST opcode in the Origami ISA counts how many times this correction fires; TWIST $> 0$ is necessary and sufficient for $H^2$ MGE. Computational class BQP-hard. The central observation is that Viktor Maslov named both ends of this ladder: Maslov dequantisation ($H^0$, 1990s) and the Maslov index ($H^2$, 1965) are not coincidentally related — they are the two limits of the same quantisation programme, with the Clifford/Berry-phase level $H^1$ interpolating between them. The name "Maslov-Gibbs Einsum" was more prophetic than intended: Maslov's index governs the $H^2$ correction as well as the $H^0$ tropical limit. The Origami ISA compiler selects the MGE form automatically: TWIST $= 0$ implies $H^1$ MGE suffices; TWIST $> 0$ requires $H^2$ MGE. This makes the Schur multiplier $\mathbb{Z}_2$ a compile-time certificate for quantum advantage. The paper also introduces the Penrose empire problem as a non-quantum illustration of why $H^0$ local consistency checking cannot detect $H^1$ and $H^2$ obstructions, and discusses the $H^3$ frontier (Dijkgraaf-Witten state sums, Fibonacci anyon braiding) as the next level beyond current qubit hardware. Keywords Maslov-Gibbs Einsum, tropical semiring, Maslov dequantisation, Maslov index, Clifford group, stabiliser calculus, metaplectic group, cohomological filtration, $H^k$ tower, TWIST opcode, Origami ISA, Gottesman-Knill theorem, quantum advantage, BQP, magic states, tensor contraction, Penrose tiling, Dijkgraaf-Witten, topological quantum computation

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The Adèlic β-Plane: A Unified Parameter Space for Classical, Statistical, Quantum, and p-adic Computation

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The Maslov-Gibbs Einsum (MGE), $\pi_k(\beta) = e^{-\beta E_k}/\sum_j e^{-\beta E_j}$, is usually treated with $\beta \in \mathbb{R}{>0}$. We show that allowing $\beta$ to range over the adèle ring $\mathbb{A}\mathbb{Q} = \mathbb{R} \times \prod_p \mathbb{Q}_p$ produces a unified parameter space — the adèlic β-plane — in which every known computational model occupies a precise address. The positive real axis ($\beta > 0$) is statistical mechanics and the Forge ISA; the limit $\beta \to +\infty$ is the tropical $(\max,+)$ semiring and the Origami ISA; the imaginary axis ($\beta = it$) is unitary quantum evolution and the Meld ISA; the negative real axis ($\beta < 0$) is population inversion and laser physics; the complex plane ($\beta = \sigma + it$) is PT-symmetric quantum mechanics, with exceptional points as β-plane branch points; the origin ($\beta = 0$) is the Ambient smooth manifold; and the p-adic completions $\mathbb{Q}_p$ — attached at the origin but totally disconnected from $\mathbb{C}$ by Ostrowski's theorem — are the p-adic ISA. The Clifford group is identified as the quantum tropical regime (adèlically integral phases); magic states are phases non-integral at the prime $p = 2$; and Palmer's invariant set $I_U$ is the adèlically integral slice of Hilbert space. An obstruction-dodging theorem is proved: every $H^k$ obstruction is trivialised at level $k+1$ or at an alternative completion. Shor's algorithm is the canonical example — it is adèlic, exploiting both the imaginary axis (quantum Fourier transform) and the p-adic completions (period recovery by the Chinese Remainder Theorem). Ostrowski's theorem closes the map: no further completions of $\mathbb{Q}$ exist, so no further computational regimes are possible. Keywords: beta-plane, adelic, Maslov-Gibbs Einsum, p-adic, PT-symmetry, population inversion, Ambient manifold, Origami ISA, Forge ISA, Meld ISA, Clifford group, magic states, Palmer invariant set, Ostrowski theorem, obstruction theory, tropical geometry

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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).

•Jul 10, 2026•0 citations

The Forge ISA: A Temperature-Parameterised Instruction Set for Thermodynamic Computation

Ian Richard C Buckley

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

•Jun 15, 2026•0 citations
Shows that the Maslov-Gibbs Einsum (MGE) takes three canonical forms corresponding to the H0/H1/H2 complexity tiers: tropical (min-plus at beta = infinity), Clifford/geometric-Kitaev (at beta = beta*), and metaplectic/Maslov-index (at beta = 0 or imaginary). Establishes the Maslov duality theorem connecting H0 and H2 forms via Legendre transform. Introduces the Penrose empire analogy for H2 non-locality and identifies an H3 frontier.
The Maslov-Gibbs Einsum (MGE) is not a single algorithm but a family of tensor contraction schemes indexed by the cohomological level HkH^kHk of the physical system. This paper identifies three distinct forms: H0H^0H0 MGE (classical/tropical): contraction over the min-plus tropical semiring, corresponding to Maslov dequantisation (ℏ→0\hbar \to 0ℏ→0). Finds the dominant classical path; computational class P. H1H^1H1 MGE (Clifford/stabiliser): contraction over the stabiliser calculus of the Clifford group C2\mathsf{C}_2C2​. Captures quantum interference within the stabiliser sector. Classically simulable in polynomial time by the Gottesman-Knill theorem, but exponentially more expressive than H0H^0H0: captures non-local correlations, entanglement, and quantum error correction. H2H^2H2 MGE (metaplectic/magic): contraction with Maslov index phase corrections. Each gate composition g∘hg \circ hg∘h acquires a Z2\mathbb{Z}_2Z2​ phase factor (−1)τ(L,gL,ghL)(-1)^{\tau(L, gL, ghL)}(−1)τ(L,gL,ghL) from the Maslov index of three Lagrangian subspaces. The TWIST opcode in the Origami ISA counts how many times this correction fires; TWIST >0> 0>0 is necessary and sufficient for H2H^2H2 MGE. Computational class BQP-hard. The central observation is that Viktor Maslov named both ends of this ladder: Maslov dequantisation (H0H^0H0, 1990s) and the Maslov index (H2H^2H2, 1965) are not coincidentally related — they are the two limits of the same quantisation programme, with the Clifford/Berry-phase level H1H^1H1 interpolating between them. The name "Maslov-Gibbs Einsum" was more prophetic than intended: Maslov's index governs the H2H^2H2 correction as well as the H0H^0H0 tropical limit. The Origami ISA compiler selects the MGE form automatically: TWIST =0= 0=0 implies H1H^1H1 MGE suffices; TWIST >0> 0>0 requires H2H^2H2 MGE. This makes the Schur multiplier Z2\mathbb{Z}_2Z2​ a compile-time certificate for quantum advantage. The paper also introduces the Penrose empire problem as a non-quantum illustration of why H0H^0H0 local consistency checking cannot detect H1H^1H1 and H2H^2H2 obstructions, and discusses the H3H^3H3 frontier (Dijkgraaf-Witten state sums, Fibonacci anyon braiding) as the next level beyond current qubit hardware. Keywords Maslov-Gibbs Einsum, tropical semiring, Maslov dequantisation, Maslov index, Clifford group, stabiliser calculus, metaplectic group, cohomological filtration, HkH^kHk tower, TWIST opcode, Origami ISA, Gottesman-Knill theorem, quantum advantage, BQP, magic states, tensor contraction, Penrose tiling, Dijkgraaf-Witten, topological quantum computation
The Maslov-Gibbs Einsum (MGE), πk(β)=e−βEk/∑je−βEj\pi_k(\beta) = e^{-\beta E_k}/\sum_j e^{-\beta E_j}πk​(β)=e−βEk​/∑j​e−βEj​, is usually treated with β∈R>0\beta \in \mathbb{R}{>0}β∈R>0. We show that allowing β\betaβ to range over the adèle ring AQ=R×∏pQp\mathbb{A}\mathbb{Q} = \mathbb{R} \times \prod_p \mathbb{Q}_pAQ=R×∏p​Qp​ produces a unified parameter space — the adèlic β-plane — in which every known computational model occupies a precise address. The positive real axis (β>0\beta > 0β>0) is statistical mechanics and the Forge ISA; the limit β→+∞\beta \to +\inftyβ→+∞ is the tropical (max⁡,+)(\max,+)(max,+) semiring and the Origami ISA; the imaginary axis (β=it\beta = itβ=it) is unitary quantum evolution and the Meld ISA; the negative real axis (β<0\beta < 0β<0) is population inversion and laser physics; the complex plane (β=σ+it\beta = \sigma + itβ=σ+it) is PT-symmetric quantum mechanics, with exceptional points as β-plane branch points; the origin (β=0\beta = 0β=0) is the Ambient smooth manifold; and the p-adic completions Qp\mathbb{Q}_pQp​ — attached at the origin but totally disconnected from C\mathbb{C}C by Ostrowski's theorem — are the p-adic ISA. The Clifford group is identified as the quantum tropical regime (adèlically integral phases); magic states are phases non-integral at the prime p=2p = 2p=2; and Palmer's invariant set IUI_UIU​ is the adèlically integral slice of Hilbert space. An obstruction-dodging theorem is proved: every HkH^kHk obstruction is trivialised at level k+1k+1k+1 or at an alternative completion. Shor's algorithm is the canonical example — it is adèlic, exploiting both the imaginary axis (quantum Fourier transform) and the p-adic completions (period recovery by the Chinese Remainder Theorem). Ostrowski's theorem closes the map: no further completions of Q\mathbb{Q}Q exist, so no further computational regimes are possible. Keywords: beta-plane, adelic, Maslov-Gibbs Einsum, p-adic, PT-symmetry, population inversion, Ambient manifold, Origami ISA, Forge ISA, Meld ISA, Clifford group, magic states, Palmer invariant set, Ostrowski theorem, obstruction theory, tropical geometry
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).
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