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

Ian Richard C Buckley

Published Jul 3, 2026• 0 citations
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Abstract

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

DOI
10.5281/zenodo.21158951

Related papers

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

Ian Richard C Buckley

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.

•Jul 6, 2026•0 citations

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

Ian Richard C Buckley

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

•Jul 7, 2026•0 citations

The Meld ISA: Complex-MGE, Quantum Algorithm Discovery, and the T-Gate as Octonion Obstruction

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. 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

•Jun 20, 2026•0 citations

The Clifford Hierarchy as Group Cohomology: TWIST as the Schur Multiplier Certificate

Ian Richard C Buckley

The Clifford hierarchy $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \mathsf{C}_3 \subset \cdots$ is usually defined inductively: a gate $U$ is at level $k+1$ if it maps every Pauli to level $k$ under conjugation. This paper shows that this definition is the shadow of a cohomological statement: the boundary between $\mathsf{C}_2$ (Clifford) and $\mathsf{C}_3$ (magic) is witnessed by a non-trivial element of $H^2(\mathrm{Sp}(4,\mathbb{F}_2);\mathbb{Z}_2)$, the obstruction to lifting a projective representation of the symplectic group to a genuine one. The T gate is the canonical representative of this non-trivial class. The obstruction is proved computationally for $\mathrm{Sp}(4,\mathbb{F}_2)$ via the Weil representation: for any BFS-lifted section $g \mapsto W_g$, there exist elements $g$ of Sp-order $d$ for which $W_g^d = -I$ (not $+I$), directly witnessing the $\mathbb{Z}_2$ obstruction. Sixty-seven such witnesses are found; the simplest has Sp-order 2 with Clifford lift of order 4. The TWIST opcode in the Origami ISA counts the number of times this cohomology class is "charged" in a circuit. TWIST $= 0$ is equivalent to the circuit lying in the trivial $\mathbb{Z}_2$ class, and hence to Gottesman-Knill classical simulability. This makes the Schur multiplier $\mathbb{Z}_2$ a computable certificate for quantum advantage. The paper unifies three apparently distinct $\mathbb{Z}2$ obstructions: (1) the Schur multiplier of $\mathrm{Sp}(4,\mathbb{F}2)$, witnessed by the T gate; (2) Pauli anti-commutation, $XZ = -ZX$; (3) the SPT invariant of the cluster state, whose virtual bond symmetry satisfies $(U{g_x} U{g_z})(U_{g_z} U_{g_x})^{-1} = -I \in H^2(\mathbb{Z}_2 \times \mathbb{Z}_2; U(1)) = \mathbb{Z}_2$. These are shown to be three languages for one object, related by explicit inflation and restriction maps. Keywords Clifford hierarchy, group cohomology, Schur multiplier, symplectic group, metaplectic representation, Weil representation, T gate, magic states, quantum advantage, Gottesman-Knill theorem, symmetry-protected topological phases, cluster state, Origami ISA, TWIST opcode, $H^2$ obstruction

•Jul 6, 2026•0 citations

The Clifford Hierarchy as Group Cohomology: TWIST as the Schur Multiplier Certificate

Ian Richard C Buckley

The Clifford hierarchy $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \mathsf{C}_3 \subset \cdots$ is usually defined inductively: a gate $U$ is at level $k+1$ if it maps every Pauli to level $k$ under conjugation. This paper shows that this definition is the shadow of a cohomological statement: the boundary between $\mathsf{C}_2$ (Clifford) and $\mathsf{C}_3$ (magic) is witnessed by a non-trivial element of $H^2(\mathrm{Sp}(4,\mathbb{F}_2);\mathbb{Z}_2)$, the obstruction to lifting a projective representation of the symplectic group to a genuine one. The T gate is the canonical representative of this non-trivial class. The obstruction is proved computationally for $\mathrm{Sp}(4,\mathbb{F}_2)$ via the Weil representation: for any BFS-lifted section $g \mapsto W_g$, there exist elements $g$ of Sp-order $d$ for which $W_g^d = -I$ (not $+I$), directly witnessing the $\mathbb{Z}_2$ obstruction. Sixty-seven such witnesses are found; the simplest has Sp-order 2 with Clifford lift of order 4. The TWIST opcode in the Origami ISA counts the number of times this cohomology class is "charged" in a circuit. TWIST $= 0$ is equivalent to the circuit lying in the trivial $\mathbb{Z}_2$ class, and hence to Gottesman-Knill classical simulability. This makes the Schur multiplier $\mathbb{Z}_2$ a computable certificate for quantum advantage. The paper unifies three apparently distinct $\mathbb{Z}2$ obstructions: (1) the Schur multiplier of $\mathrm{Sp}(4,\mathbb{F}2)$, witnessed by the T gate; (2) Pauli anti-commutation, $XZ = -ZX$; (3) the SPT invariant of the cluster state, whose virtual bond symmetry satisfies $(U{g_x} U{g_z})(U_{g_z} U_{g_x})^{-1} = -I \in H^2(\mathbb{Z}_2 \times \mathbb{Z}_2; U(1)) = \mathbb{Z}_2$. These are shown to be three languages for one object, related by explicit inflation and restriction maps. Keywords Clifford hierarchy, group cohomology, Schur multiplier, symplectic group, metaplectic representation, Weil representation, T gate, magic states, quantum advantage, Gottesman-Knill theorem, symmetry-protected topological phases, cluster state, Origami ISA, TWIST opcode, $H^2$ obstruction

•Jul 3, 2026•0 citations

The Projective Hierarchy of Computation: Classical, Clifford, and Magic as Shadows of the Meld

Ian Richard C Buckley

Establishes the H^k complexity ladder as a projective hierarchy: H0 (classical/tropical) and H1 (Clifford/stabiliser) are projective shadows of H2 (magic/metaplectic). The Meld — the H2 completion of the ISA — projects onto Origami (H0+H1 shadow) and Forge (H1 shadow) in the same way that a 3D object projects onto 2D shadows. Derives the Palmer Invariant Space-time conjecture as the fixed-point condition of Penrose objective reduction within this framework.

•Jul 6, 2026•0 citations
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
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→∞\beta\to\inftyβ→∞, tropical, classical), Forge (0<β<∞0<\beta<\infty0<β<∞, real, statistical), Meld (β=it\beta=itβ=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)T = \mathrm{diag}(1,, e^{i\pi/4})T=diag(1,,eiπ/4) is the BIND opcode at j=12j=\tfrac{1}{2}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 arg⁡max⁡g∈Gf(g)\arg\max_{g\in G} f(g)argmaxg∈G​f(g) over an abelian group GGG has a C\mathbb{C}C-MGE quantum counterpart C-MGE(f)=SPLITG⋅FLIP(f)⋅SPLAT\mathbb{C}\text{-MGE}(f) = \mathrm{SPLIT}_G \cdot \mathrm{FLIP}(f) \cdot \mathrm{SPLAT}C-MGE(f)=SPLITG​⋅FLIP(f)⋅SPLAT, giving exponential speedup via quantum Fourier sampling whenever GGG is abelian and fff 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→∞\beta\to\inftyβ→∞) recovers classical exhaustive search at cost O(N)O(N)O(N). "Warm-Shor" (Forge ISA, real β\betaβ) gives Grover-like O(N)O(\sqrt{N})O(N​) thermal concentration. Full Shor (Meld, β=it\beta=itβ=it) achieves O(log⁡N)O(\log N)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\mathbb{E}{\mathbf{e}\sim D\sigma}[e^{i\langle\xi,\mathbf{e}\rangle}] = e^{-|\xi|^2\sigma^2/2}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 HkH^kHk complexity ladder (Paper 420) is realised by the trilogy: H0H^0H0 (tree) = Origami, H1H^1H1 (loop) = Forge/BPP, H2H^2H2 (surface) = Meld/BQP. The T-gate is the H2H^2H2 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\oplus_\beta⊕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
The Clifford hierarchy C1⊂C2⊂C3⊂⋯\mathsf{C}_1 \subset \mathsf{C}_2 \subset \mathsf{C}_3 \subset \cdotsC1​⊂C2​⊂C3​⊂⋯ is usually defined inductively: a gate UUU is at level k+1k+1k+1 if it maps every Pauli to level kkk under conjugation. This paper shows that this definition is the shadow of a cohomological statement: the boundary between C2\mathsf{C}_2C2​ (Clifford) and C3\mathsf{C}_3C3​ (magic) is witnessed by a non-trivial element of H2(Sp(4,F2);Z2)H^2(\mathrm{Sp}(4,\mathbb{F}_2);\mathbb{Z}_2)H2(Sp(4,F2​);Z2​), the obstruction to lifting a projective representation of the symplectic group to a genuine one. The T gate is the canonical representative of this non-trivial class. The obstruction is proved computationally for Sp(4,F2)\mathrm{Sp}(4,\mathbb{F}_2)Sp(4,F2​) via the Weil representation: for any BFS-lifted section g↦Wgg \mapsto W_gg↦Wg​, there exist elements ggg of Sp-order ddd for which Wgd=−IW_g^d = -IWgd​=−I (not +I+I+I), directly witnessing the Z2\mathbb{Z}_2Z2​ obstruction. Sixty-seven such witnesses are found; the simplest has Sp-order 2 with Clifford lift of order 4. The TWIST opcode in the Origami ISA counts the number of times this cohomology class is "charged" in a circuit. TWIST =0= 0=0 is equivalent to the circuit lying in the trivial Z2\mathbb{Z}_2Z2​ class, and hence to Gottesman-Knill classical simulability. This makes the Schur multiplier Z2\mathbb{Z}_2Z2​ a computable certificate for quantum advantage. The paper unifies three apparently distinct Z2\mathbb{Z}2Z2 obstructions: (1) the Schur multiplier of Sp(4,F2)\mathrm{Sp}(4,\mathbb{F}2)Sp(4,F2), witnessed by the T gate; (2) Pauli anti-commutation, XZ=−ZXXZ = -ZXXZ=−ZX; (3) the SPT invariant of the cluster state, whose virtual bond symmetry satisfies (UgxUgz)(UgzUgx)−1=−I∈H2(Z2×Z2;U(1))=Z2(U{g_x} U{g_z})(U_{g_z} U_{g_x})^{-1} = -I \in H^2(\mathbb{Z}_2 \times \mathbb{Z}_2; U(1)) = \mathbb{Z}_2(Ugx​Ugz​)(Ugz​​Ugx​​)−1=−I∈H2(Z2​×Z2​;U(1))=Z2​. These are shown to be three languages for one object, related by explicit inflation and restriction maps. Keywords Clifford hierarchy, group cohomology, Schur multiplier, symplectic group, metaplectic representation, Weil representation, T gate, magic states, quantum advantage, Gottesman-Knill theorem, symmetry-protected topological phases, cluster state, Origami ISA, TWIST opcode, H2H^2H2 obstruction
The Clifford hierarchy C1⊂C2⊂C3⊂⋯\mathsf{C}_1 \subset \mathsf{C}_2 \subset \mathsf{C}_3 \subset \cdotsC1​⊂C2​⊂C3​⊂⋯ is usually defined inductively: a gate UUU is at level k+1k+1k+1 if it maps every Pauli to level kkk under conjugation. This paper shows that this definition is the shadow of a cohomological statement: the boundary between C2\mathsf{C}_2C2​ (Clifford) and C3\mathsf{C}_3C3​ (magic) is witnessed by a non-trivial element of H2(Sp(4,F2);Z2)H^2(\mathrm{Sp}(4,\mathbb{F}_2);\mathbb{Z}_2)H2(Sp(4,F2​);Z2​), the obstruction to lifting a projective representation of the symplectic group to a genuine one. The T gate is the canonical representative of this non-trivial class. The obstruction is proved computationally for Sp(4,F2)\mathrm{Sp}(4,\mathbb{F}_2)Sp(4,F2​) via the Weil representation: for any BFS-lifted section g↦Wgg \mapsto W_gg↦Wg​, there exist elements ggg of Sp-order ddd for which Wgd=−IW_g^d = -IWgd​=−I (not +I+I+I), directly witnessing the Z2\mathbb{Z}_2Z2​ obstruction. Sixty-seven such witnesses are found; the simplest has Sp-order 2 with Clifford lift of order 4. The TWIST opcode in the Origami ISA counts the number of times this cohomology class is "charged" in a circuit. TWIST =0= 0=0 is equivalent to the circuit lying in the trivial Z2\mathbb{Z}_2Z2​ class, and hence to Gottesman-Knill classical simulability. This makes the Schur multiplier Z2\mathbb{Z}_2Z2​ a computable certificate for quantum advantage. The paper unifies three apparently distinct Z2\mathbb{Z}2Z2 obstructions: (1) the Schur multiplier of Sp(4,F2)\mathrm{Sp}(4,\mathbb{F}2)Sp(4,F2), witnessed by the T gate; (2) Pauli anti-commutation, XZ=−ZXXZ = -ZXXZ=−ZX; (3) the SPT invariant of the cluster state, whose virtual bond symmetry satisfies (UgxUgz)(UgzUgx)−1=−I∈H2(Z2×Z2;U(1))=Z2(U{g_x} U{g_z})(U_{g_z} U_{g_x})^{-1} = -I \in H^2(\mathbb{Z}_2 \times \mathbb{Z}_2; U(1)) = \mathbb{Z}_2(Ugx​Ugz​)(Ugz​​Ugx​​)−1=−I∈H2(Z2​×Z2​;U(1))=Z2​. These are shown to be three languages for one object, related by explicit inflation and restriction maps. Keywords Clifford hierarchy, group cohomology, Schur multiplier, symplectic group, metaplectic representation, Weil representation, T gate, magic states, quantum advantage, Gottesman-Knill theorem, symmetry-protected topological phases, cluster state, Origami ISA, TWIST opcode, H2H^2H2 obstruction