Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Approximation of the Riemann Zeta Function via MERA Tensor Networks: A Study of Convergence and Error Bounds Based on Mellin Operators

Juan Arroyo

Published Aug 11, 2026• 0 citations
Publisher

Abstract

This preprint presents a rigorous, self-contained mathematical framework for numerically approximating the Riemann Zeta function using Multiscale Entanglement Renormalization Ansatz (MERA) tensor networks. By formally establishing a unitary equivalence between the MERA coarse-graining operator and a discretized Mellin integral (Lemma 3.1), we analytically derive the theoretical infrared fixed point (Theorem 3.1) to be γtarget=log⁡(2π)2π2≈0.0931γtarget=2π2log(2π)≈0.0931. This derivation leverages the polygamma function evaluated at the center of the critical strip. A numerical implementation with bond dimension χ=2χ=2 achieves this target with an absolute error of 1.3×10−51.3×10−5 and a relative error of +0.0139%+0.0139%. We also provide strict, exponentially converging error bounds based on the singular value decomposition of Hankel/Mellin-type operators (Theorem 3.2). Notably, this mathematically derived fixed point provides the fundamental scale parameter for the Renormalization Group Scale (RG-S) within the R5+ modified gravity model. Keywords: Riemann Zeta Function, MERA, Tensor Networks, Mellin Transform, Renormalization Group, Functional Equation, Fixed Point, Numerical Analysis, R5+ Gravity.

DOI
10.5281/zenodo.21883486

Related papers

Approximation of the Riemann Zeta Function via MERA Tensor Networks: A Study of Convergence and Error Bounds Based on Mellin Operators

Juan Arroyo

This preprint presents a rigorous, self-contained mathematical framework for numerically approximating the Riemann Zeta function using Multiscale Entanglement Renormalization Ansatz (MERA) tensor networks. By formally establishing a unitary equivalence between the MERA coarse-graining operator and a discretized Mellin integral (Lemma 3.1), we analytically derive the theoretical infrared fixed point (Theorem 3.1) to be γtarget=log⁡(2π)2π2≈0.0931γtarget=2π2log(2π)≈0.0931. This derivation leverages the polygamma function evaluated at the center of the critical strip. A numerical implementation with bond dimension χ=2χ=2 achieves this target with an absolute error of 1.3×10−51.3×10−5 and a relative error of +0.0139%+0.0139%. We also provide strict, exponentially converging error bounds based on the singular value decomposition of Hankel/Mellin-type operators (Theorem 3.2). Notably, this mathematically derived fixed point provides the fundamental scale parameter for the Renormalization Group Scale (RG-S) within the R5+ modified gravity model. Keywords: Riemann Zeta Function, MERA, Tensor Networks, Mellin Transform, Renormalization Group, Functional Equation, Fixed Point, Numerical Analysis, R5+ Gravity.

•Aug 11, 2026•0 citations

The MERA Scaling Spectrum at χ = 6: A Tensor-Network Realization of the ⟨2, 3⟩ Modular Ratio Set

Daland Montgomery

Description We construct the multiscale entanglement renormalization ansatz (MERA) at bond dimension χ = 6 with the natural binary × ternary coarse-graining of the framework's 2 × 3 cell grid and identify its scaling structure at the conserved-current fixed point. The scaling-weight operator W on End(ℂ⁶) — the MERA descending operator at the fixed point — has finite point spectrum {1, 1/2, 1/3, 1/6} on four primary sectors of dimensions (1, 3, 8, 24) summing to 36 = χ². All non-trivial sectors carry primary scaling dimension Δ = 1 — the universal conserved-current dimension forced by the Sugawara construction (Knizhnik–Zamolodchikov 1984; Witten 1984) — and the four scaling weights come from the four branching ratios b ∈ {1, 2, 3, 6} of the cell grid. The corresponding scaling superoperator 𝒮(X) := W X W⁻¹ has finite point spectrum equal to the nine-element ratio set {1, 1/2, 1/3, 1/6, 2/3, 3/2, 2, 3, 6}. The paper establishes the operator-algebraic bridge between the tensor-network and modular-theory readings of the framework's finite-dimensional core. The MERA scaling-weight operator W and the framework's per-cell Bisognano–Wichmann density ρ_ω on A_F = ℂ ⊕ M₂(ℂ) ⊕ M₃(ℂ) (Montgomery 2026a) are the same diagonal positive operator in the Wedderburn–Artin basis; the scaling superoperator 𝒮 and the Tomita–Takesaki modular operator Δ_ω are the same multiplicative conjugation; the nine-element spectra coincide exactly. The Sd-invariant is ⟨2, 3⟩, Pontryagin-dual to the compact 2-torus 𝕋². Three further results sharpen the construction. (i) The fixed-point projection is stated for the physical scaling superoperator 𝒮_phys, the restriction of 𝒮 to the descending sector ℰ_≤ = ℰ_< ⊕ ℰ_= — the rigorous form of the MERA causal-cone / local-operator-support condition — with convergence ‖𝒮_phys^n(X) − P_singlet(X)‖_F ≤ (2/3)^n ‖X − P_singlet(X)‖_F at the slowest contracting eigenvalue 2/3. (ii) A falsifiable prediction: the framework's affine current sector at level 1 is û(1) ⊕ ŝu(2)₁ ⊕ ŝu(3)₁ — the Standard Model gauge group's affine Kac–Moody algebras — with total Sugawara central charge c_total = 1 + 1 + 2 = 4, testable by variational MERA optimization on A_F at χ = 6. (iii) A three-witness convergence: the same four-element spectrum {1, 1/2, 1/3, 1/6} arises from Wedderburn-sector traces (algebraic / NCG), MERA scaling at Δ = 1 (tensor-network), and Sugawara WZW current scaling (CFT) — three independent operator-theoretic calculations on the same algebraic data. Companion materials in the repository: a self-contained Python script verifying seven receipts (Wedderburn closure, both spectra, geometric contraction at rate ≤ 2/3, Sd-invariant generators, Pontryagin pairing, the W = ρ_ω / 𝒮 = Δ_ω bridge, Weyl equidistribution); kernel-checked Lean 4 and Agda proofs of the discrete content (closed by decide and refl with no sorry, no axioms beyond standard, no mathlib); a three.js interactive demo of the MERA tree, the four primary sectors, and the nine-element ratio lattice. Built on the Crystal Topos foundation: load the four foundational papers BEFORE this one, one at a time in separate uploads, to give an LLM full context — single bundled uploads compress each paper into a thin summary and lose the algebra. Recommended order: (1) Discrete Core https://doi.org/10.5281/zenodo.20232800 (2) MERA Scaling Spectrum https://doi.org/10.5281/zenodo.20236511 (3) Type II∞ Core https://doi.org/10.5281/zenodo.20257603 (4) Triangular Duality https://doi.org/10.5281/zenodo.20257953 After all four are ingested, upload this paper as the fifth turn. Keywords MERA, multiscale entanglement renormalization ansatz, tensor networks, scaling superoperator, conserved-current fixed point, Wess–Zumino–Witten model, Sugawara construction, affine Kac–Moody algebra, level-1 representation, central charge, c=4, Standard Model gauge group, û(1), ŝu(2)₁, ŝu(3)₁, modular theory, Tomita–Takesaki, Bisognano–Wichmann, Connes Sd-invariant, almost-periodic state, Pontryagin duality, 2-torus, Weyl equidistribution, Mihăilescu, Catalan conjecture, A_F, Wedderburn–Artin, three-witness convergence, χ=6, bond dimension, ratio set, ⟨2,3⟩, Crystal Topos, WACA, discrete spectral-triple holography, kernel-checked Lean, Agda Copyright © 2026 Daland Montgomery. Creative Commons Attribution Non Commercial No Derivatives 4.0 International For a private, proprietary license, contact: quidbit@icloud.com Software Implementation: The formulas and constants derived in this work are implemented in the CrystalAgent engine, available under the AGPL-3.0 license at: https://github.com/CrystalToe/CrystalAgent.

•May 16, 2026•0 citations

Approximating the Riemann Zeta Function on the Critical Line using Neural Networks with Fourier Feature Embeddings

Juan Arroyo

Building upon the universal flow equations and infrared attractor dynamics established in the PROMETEO-PRIME framework, this paper addresses the role of high-frequency residual noise—traditionally filtered out by finite bond dimension truncation ($\chi = 16$) in Multiscale Entanglement Renormalization Ansatz (MERA) networks. By constructing a coupled differential system that reintegrates the orthogonal fluctuation sector with the macroscopic Gaussian Unitary Ensemble (GUE) bulk, we formulate an expanded block-matrix effective Hamiltonian. We regularize the extended Hilbert space via a non-separable Hermitian cross-coupling term, forcing the Schmidt spectrum to decay as an anomalous power law governed by a non-ergodic extended (NEE) fractal dimension $D_2 \approx 0.8132$, shielding the system from chaotic thermalization while preserving strict self-adjointness ($H_{\text{total}} = H_{\text{total}}^\dagger$). Empirical validation shows that the adjacent gap ratio stabilizes at the universal fixed point $\langle r \rangle \approx 0.4678$, verifying a topologically protected quantum phase that underpins the Hilbert-Pólya program and advances the spectral resolution of the Riemann Hypothesis.

•Aug 11, 2026•0 citations

Entanglement compression in scale space: From the multiscale entanglement renormalization ansatz to matrix product operators

Karel Van Acoleyen, Andrew Hallam, Matthias Bal +3 more

The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wave functions that are inherently scale invariant. Unlike conformally invariant partition functions, however, the finite bond dimension $\ensuremath{\chi}$ of the MERA provides a cutoff in the fields that can be realized. In this paper, we demonstrate that this cutoff is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension $\ensuremath{\chi}$. This is achieved by constructing an explicit mapping between the isometries of the MERA and the local tensors of the MPO. In terms of energy scales, our results show that a finite bond dimension MERA is equivalent to introducing both an infrared and an ultraviolet scale, characterizing relevant and irrelevant perturbations on the underlying conformal field theory.

arXiv:1912.10572•Physical review. B./Physical review. B•Oct 15, 2020•3 citations

Multiscale Entanglement Renormalization Ansatz: Causality and Error Correction

Domenico Pomarico

Computational complexity reduction is at the basis of a new formulation of many-body quantum states according to tensor network ansatz, originally framed in one-dimensional lattices. In order to include long-range entanglement characterizing phase transitions, the multiscale entanglement renormalization ansatz (MERA) defines a sequence of coarse-grained lattices, obtained by targeting the map of a scale-invariant system into an identical coarse-grained one. The quantum circuit associated with this hierarchical structure includes the definition of causal relations and unitary extensions, leading to the definition of ground subspaces as stabilizer codes. The emerging error correcting codes are referred to logical indices located at the highest hierarchical level and to physical indices yielded by redundancy, framed in the AdS-CFT correspondence as holographic quantum codes with bulk and boundary indices, respectively. In a use-case scenario based on errors consisting of spin erasure, the correction is implemented as the reconstruction of a bulk local operator.

Dynamics•Sep 18, 2023•6 citations

Scaling at Chiral Clock Criticality via Entanglement Renormalization

Shiyong Guo, Brian Swingle

We employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the $\mathbb{Z}_3$ chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.

cond-mat.stat-mechCross-listedarXiv:2604.19876•Apr 23, 2026•0 citations
This preprint presents a rigorous, self-contained mathematical framework for numerically approximating the Riemann Zeta function using Multiscale Entanglement Renormalization Ansatz (MERA) tensor networks. By formally establishing a unitary equivalence between the MERA coarse-graining operator and a discretized Mellin integral (Lemma 3.1), we analytically derive the theoretical infrared fixed point (Theorem 3.1) to be γtarget=log⁡(2π)2π2≈0.0931γtarget=2π2log(2π)≈0.0931. This derivation leverages the polygamma function evaluated at the center of the critical strip. A numerical implementation with bond dimension χ=2χ=2 achieves this target with an absolute error of 1.3×10−51.3×10−5 and a relative error of +0.0139%+0.0139%. We also provide strict, exponentially converging error bounds based on the singular value decomposition of Hankel/Mellin-type operators (Theorem 3.2). Notably, this mathematically derived fixed point provides the fundamental scale parameter for the Renormalization Group Scale (RG-S) within the R5+ modified gravity model. Keywords: Riemann Zeta Function, MERA, Tensor Networks, Mellin Transform, Renormalization Group, Functional Equation, Fixed Point, Numerical Analysis, R5+ Gravity.
This preprint presents a rigorous, self-contained mathematical framework for numerically approximating the Riemann Zeta function using Multiscale Entanglement Renormalization Ansatz (MERA) tensor networks. By formally establishing a unitary equivalence between the MERA coarse-graining operator and a discretized Mellin integral (Lemma 3.1), we analytically derive the theoretical infrared fixed point (Theorem 3.1) to be γtarget=log⁡(2π)2π2≈0.0931γtarget=2π2log(2π)≈0.0931. This derivation leverages the polygamma function evaluated at the center of the critical strip. A numerical implementation with bond dimension χ=2χ=2 achieves this target with an absolute error of 1.3×10−51.3×10−5 and a relative error of +0.0139%+0.0139%. We also provide strict, exponentially converging error bounds based on the singular value decomposition of Hankel/Mellin-type operators (Theorem 3.2). Notably, this mathematically derived fixed point provides the fundamental scale parameter for the Renormalization Group Scale (RG-S) within the R5+ modified gravity model. Keywords: Riemann Zeta Function, MERA, Tensor Networks, Mellin Transform, Renormalization Group, Functional Equation, Fixed Point, Numerical Analysis, R5+ Gravity.
Description We construct the multiscale entanglement renormalization ansatz (MERA) at bond dimension χ = 6 with the natural binary × ternary coarse-graining of the framework's 2 × 3 cell grid and identify its scaling structure at the conserved-current fixed point. The scaling-weight operator W on End(ℂ⁶) — the MERA descending operator at the fixed point — has finite point spectrum {1, 1/2, 1/3, 1/6} on four primary sectors of dimensions (1, 3, 8, 24) summing to 36 = χ². All non-trivial sectors carry primary scaling dimension Δ = 1 — the universal conserved-current dimension forced by the Sugawara construction (Knizhnik–Zamolodchikov 1984; Witten 1984) — and the four scaling weights come from the four branching ratios b ∈ {1, 2, 3, 6} of the cell grid. The corresponding scaling superoperator 𝒮(X) := W X W⁻¹ has finite point spectrum equal to the nine-element ratio set {1, 1/2, 1/3, 1/6, 2/3, 3/2, 2, 3, 6}. The paper establishes the operator-algebraic bridge between the tensor-network and modular-theory readings of the framework's finite-dimensional core. The MERA scaling-weight operator W and the framework's per-cell Bisognano–Wichmann density ρ_ω on A_F = ℂ ⊕ M₂(ℂ) ⊕ M₃(ℂ) (Montgomery 2026a) are the same diagonal positive operator in the Wedderburn–Artin basis; the scaling superoperator 𝒮 and the Tomita–Takesaki modular operator Δ_ω are the same multiplicative conjugation; the nine-element spectra coincide exactly. The Sd-invariant is ⟨2, 3⟩, Pontryagin-dual to the compact 2-torus 𝕋². Three further results sharpen the construction. (i) The fixed-point projection is stated for the physical scaling superoperator 𝒮_phys, the restriction of 𝒮 to the descending sector ℰ_≤ = ℰ_< ⊕ ℰ_= — the rigorous form of the MERA causal-cone / local-operator-support condition — with convergence ‖𝒮_phys^n(X) − P_singlet(X)‖_F ≤ (2/3)^n ‖X − P_singlet(X)‖_F at the slowest contracting eigenvalue 2/3. (ii) A falsifiable prediction: the framework's affine current sector at level 1 is û(1) ⊕ ŝu(2)₁ ⊕ ŝu(3)₁ — the Standard Model gauge group's affine Kac–Moody algebras — with total Sugawara central charge c_total = 1 + 1 + 2 = 4, testable by variational MERA optimization on A_F at χ = 6. (iii) A three-witness convergence: the same four-element spectrum {1, 1/2, 1/3, 1/6} arises from Wedderburn-sector traces (algebraic / NCG), MERA scaling at Δ = 1 (tensor-network), and Sugawara WZW current scaling (CFT) — three independent operator-theoretic calculations on the same algebraic data. Companion materials in the repository: a self-contained Python script verifying seven receipts (Wedderburn closure, both spectra, geometric contraction at rate ≤ 2/3, Sd-invariant generators, Pontryagin pairing, the W = ρ_ω / 𝒮 = Δ_ω bridge, Weyl equidistribution); kernel-checked Lean 4 and Agda proofs of the discrete content (closed by decide and refl with no sorry, no axioms beyond standard, no mathlib); a three.js interactive demo of the MERA tree, the four primary sectors, and the nine-element ratio lattice. Built on the Crystal Topos foundation: load the four foundational papers BEFORE this one, one at a time in separate uploads, to give an LLM full context — single bundled uploads compress each paper into a thin summary and lose the algebra. Recommended order: (1) Discrete Core https://doi.org/10.5281/zenodo.20232800 (2) MERA Scaling Spectrum https://doi.org/10.5281/zenodo.20236511 (3) Type II∞ Core https://doi.org/10.5281/zenodo.20257603 (4) Triangular Duality https://doi.org/10.5281/zenodo.20257953 After all four are ingested, upload this paper as the fifth turn. Keywords MERA, multiscale entanglement renormalization ansatz, tensor networks, scaling superoperator, conserved-current fixed point, Wess–Zumino–Witten model, Sugawara construction, affine Kac–Moody algebra, level-1 representation, central charge, c=4, Standard Model gauge group, û(1), ŝu(2)₁, ŝu(3)₁, modular theory, Tomita–Takesaki, Bisognano–Wichmann, Connes Sd-invariant, almost-periodic state, Pontryagin duality, 2-torus, Weyl equidistribution, Mihăilescu, Catalan conjecture, A_F, Wedderburn–Artin, three-witness convergence, χ=6, bond dimension, ratio set, ⟨2,3⟩, Crystal Topos, WACA, discrete spectral-triple holography, kernel-checked Lean, Agda Copyright © 2026 Daland Montgomery. Creative Commons Attribution Non Commercial No Derivatives 4.0 International For a private, proprietary license, contact: quidbit@icloud.com Software Implementation: The formulas and constants derived in this work are implemented in the CrystalAgent engine, available under the AGPL-3.0 license at: https://github.com/CrystalToe/CrystalAgent.
Building upon the universal flow equations and infrared attractor dynamics established in the PROMETEO-PRIME framework, this paper addresses the role of high-frequency residual noise—traditionally filtered out by finite bond dimension truncation (χ=16\chi = 16χ=16) in Multiscale Entanglement Renormalization Ansatz (MERA) networks. By constructing a coupled differential system that reintegrates the orthogonal fluctuation sector with the macroscopic Gaussian Unitary Ensemble (GUE) bulk, we formulate an expanded block-matrix effective Hamiltonian. We regularize the extended Hilbert space via a non-separable Hermitian cross-coupling term, forcing the Schmidt spectrum to decay as an anomalous power law governed by a non-ergodic extended (NEE) fractal dimension D2≈0.8132D_2 \approx 0.8132D2​≈0.8132, shielding the system from chaotic thermalization while preserving strict self-adjointness (Htotal=Htotal†H_{\text{total}} = H_{\text{total}}^\daggerHtotal​=Htotal†​). Empirical validation shows that the adjacent gap ratio stabilizes at the universal fixed point ⟨r⟩≈0.4678\langle r \rangle \approx 0.4678⟨r⟩≈0.4678, verifying a topologically protected quantum phase that underpins the Hilbert-Pólya program and advances the spectral resolution of the Riemann Hypothesis.
The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wave functions that are inherently scale invariant. Unlike conformally invariant partition functions, however, the finite bond dimension $\ensuremath{\chi}$ of the MERA provides a cutoff in the fields that can be realized. In this paper, we demonstrate that this cutoff is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension $\ensuremath{\chi}$. This is achieved by constructing an explicit mapping between the isometries of the MERA and the local tensors of the MPO. In terms of energy scales, our results show that a finite bond dimension MERA is equivalent to introducing both an infrared and an ultraviolet scale, characterizing relevant and irrelevant perturbations on the underlying conformal field theory.
Computational complexity reduction is at the basis of a new formulation of many-body quantum states according to tensor network ansatz, originally framed in one-dimensional lattices. In order to include long-range entanglement characterizing phase transitions, the multiscale entanglement renormalization ansatz (MERA) defines a sequence of coarse-grained lattices, obtained by targeting the map of a scale-invariant system into an identical coarse-grained one. The quantum circuit associated with this hierarchical structure includes the definition of causal relations and unitary extensions, leading to the definition of ground subspaces as stabilizer codes. The emerging error correcting codes are referred to logical indices located at the highest hierarchical level and to physical indices yielded by redundancy, framed in the AdS-CFT correspondence as holographic quantum codes with bulk and boundary indices, respectively. In a use-case scenario based on errors consisting of spin erasure, the correction is implemented as the reconstruction of a bulk local operator.
We employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the Z3\mathbb{Z}_3Z3​ chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.