Generative Emergence of Groups D; Tensor Network Simulation and Numerical Extraction of Central Charge– Experimental Mathematics of Generative Dynamics
Previous work, starting from hierarchical iterated function systems (HierarchicalIFS), proved that discrete Lie algebras converge to affine Kac-Moody algebras in therescaling limit, with the central charge uniquely determined by the topological dataof inter-level gluing rules. This paper proposes to numerically verify this theoreticalprediction using the multiscale entanglement renormalization ansatz (MERA). Byestablishing a structural correspondence between the MERA network and the hierarchical IFS, we interpret the isometries of MERA as inter-level gluing rules, andthe ultraviolet degrees of freedom as fine-level generators, and introduce a gluingstrength parameter that directly links the winding number to the variational parameters of MERA. Taking the SU(2) Heisenberg model as an example, we designnumerical protocols to extract the central charge from the ground-state entanglement entropy and entanglement spectrum, and demonstrate that as the MERAdepth increases, the effective central charge converges exponentially to the theoretical value of conformal field theory, with the convergence rate controlled by thespectral gap of the hierarchical IFS. We explicitly give falsification thresholds consisting of three criteria: absolute deviation, convergence rate, and self-consistency,and introduce cross-validation with exact diagonalization and DMRG. Finally, weextend the verification framework to SU(2)k WZW models and non-diagonal conformal field theories, elucidating the central role of the double matching of windingnumber and central charge in theoretical tests. This work positions tensor networks as a numerical experimental platform for testing core ideas of generativemathematical physics.
Previous work, starting from hierarchical iterated function systems (HierarchicalIFS), proved that discrete Lie algebras converge to affine Kac-Moody algebras in therescaling limit, with the central charge uniquely determined by the topological dataof inter-level gluing rules. This paper proposes to numerically verify this theoreticalprediction using the multiscale entanglement renormalization ansatz (MERA). Byestablishing a structural correspondence between the MERA network and the hierarchical IFS, we interpret the isometries of MERA as inter-level gluing rules, andthe ultraviolet degrees of freedom as fine-level generators, and introduce a gluingstrength parameter that directly links the winding number to the variational parameters of MERA. Taking the SU(2) Heisenberg model as an example, we designnumerical protocols to extract the central charge from the ground-state entanglement entropy and entanglement spectrum, and demonstrate that as the MERAdepth increases, the effective central charge converges exponentially to the theoretical value of conformal field theory, with the convergence rate controlled by thespectral gap of the hierarchical IFS. We explicitly give falsification thresholds consisting of three criteria: absolute deviation, convergence rate, and self-consistency,and introduce cross-validation with exact diagonalization and DMRG. Finally, weextend the verification framework to SU(2)k WZW models and non-diagonal conformal field theories, elucidating the central role of the double matching of windingnumber and central charge in theoretical tests. This work positions tensor networks as a numerical experimental platform for testing core ideas of generativemathematical physics.
In this thesis, we use the new numerical method multiscale entanglement
renormalization ansatz (MERA) to study the one dimensional spin system. This
method, MERA, utilizes a network of tensors to represent quantum many-body
systems on a lattice. Other examples of tensor network state are matrix product
states (MPSs) for 1D systems, tree tensor networks (TTNs) for systems with a
tree shape, and projected entangled-pair states (PEPSs) for 2D systems and be-
yond. The three structures dier in the graph that denes how the tensors are
interconnected into a network: The graphs for MPSs, TTNs, and 2D PEPSs are,
respectively, a chain, a tree, and a 2D lattice. Importantly, from these tensor
networks the expectation value of local observables can be computed eciently.
Developing algorithms to simulate quantum many-body systems is important
for understanding and studying the physical principles. And these may become
powerful tools for the analysis of quantum computation and quantum information
theory. But in general ecient simulation is dicult, due to the fundamental
quantum physical laws.
The main method we have studied and worked on is multiscale entangle-
ment renormalization ansatz (MERA) which is a variational ansatz for many-
body states. MERA is a tree tensor network for 1D, 2D systems and beyond.
We have developed code of ternary-MERA to compute the ground state energy,
expected values of local observables, and correlators for 1D lattice systems. In
a nutshell, MERA is an algorithm for entanglement renormalization which is a
numerical technique based on locally reorganizing the Hilbert space of a quantum
many-body system with the aim to reduce the amount of entanglement in its wave
function. We use this method to study 1D Ising model, XXZ model, and J1 - J2
model, and then comparing the numerical results with exact diagonalization (ED)
and density matrix renormalization group (DMRG).
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We perform a seven-part holographic analysis of the SU(3)_3 ⊗ SU(2)_2 string-net model associated with the AME(4,6) quantum error-correcting code. The transfer matrix is reinterpreted as a discrete renormalization group (RG) operator, and its iterated action is studied as a proxy for the MERA tensor network that defines an emergent holographic bulk. Key results: (1) Perfect conformal self-similarity — gap ratios preserved to 15 significant digits through 5 RG steps; (2) WZW tower matching — entanglement spectrum matches (0,0)⊗σ primary at h=3/16 exactly (Δh=0.0000); (3) Entanglement temperature β_ent = 2.380; (4) N-layer stacking shows exponential RG convergence (gapped system); (5) Topological hierarchy: ξ_topo/ξ = 5171 at Lx=2, scaling as exp(4·Lx), giving ~10^17 at Lx=10 — matching the Planck-electroweak ratio M_Pl/v; (6) Brown-Henneaux G₃ = 0.156 lattice units from c=11/2, L_AdS=ξ=0.572; (7) Stress tensor candidates at h≈1.81 (9.5% below h=2, finite-size at Lx=2). The topological hierarchy mechanism is the most provocative result: an O(1) topological constant (α≈4) combined with modest system size (Lx≈10) generates the 16 orders of magnitude separating v from M_Pl, without fine-tuning.
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.
This thesis reviews the multiscale entanglement renormalisation ansatz or MERA, a numerical tool for the study of quantum many-body systems and a discrete realisation of the AdS/CFT duality. The thesis covers an introduction to the necessary background concepts of entanglement, entanglement entropy and tensor network states, the structure and main features of MERA and its applications in condensed matter theory and holography. Also covered are details on the algorithmic implementation of MERA and some of its generalisations and extensions. MERA belongs to a class of variational ansatze for quantum many-body states known as tensor network states. It is especially well-suited for the study of scale invariant critical points. MERA is based on a real-space renormalisation group procedure called entanglement renormalisation, designed to systematically handle entanglement at different length scales along the coarse-graining flow. Entanglement renormalisation has be used for example to efficiently describe Kitaev states of the toric code, the prime example of topological order, and numerically study the ground state of the highly frustrated spin-1/2 Heisenberg model on a kagome lattice and various other one- and two-dimensional lattice models. The geometric and causal structure of MERA, which underlies its effectiveness as a numerical tool, also makes it a discrete version of the AdS/CFT duality. This duality describes a conformal field theory by a gravity theory in a higher dimensional space, and vice versa. The duality is manifest in the scaling of entanglement entropy in MERA, which is governed by a law highly analogous to the Ryu-Takayanagi formula for holographic entanglement entropy, in the connection between thermal states and a black-hole-like MERA and in the connection between correlation functions and holographic geodesics in a scale invariant MERA. The aim of this thesis is to lead the reader to an understanding of what MERA is, how it works and how it can be used. MERA's core features and uses are presented in a comprehensive and explicit way, and a broad view of possible applications and further directions is given. Plenty of references are also offered to direct the reader to further research on how MERA may relate to his/her interests.
The Quantum Iterated Function Systems (QIFS) theory establishes an exponential relation q = e−Γtc linking the quantum group deformation parameter q withthe open system decoherence rate Γ and characteristic coherence time tc, offering anew perspective for understanding the emergence mechanism of quantum symmetries. This paper proposes concrete experimental schemes to implement the QIFSof su(2) generators in superconducting qubit and ion trap systems, aiming to testthis theoretical relation. First, we construct QIFS generators based on amplitudedamping and phase damping channels, and present their physical realizations insuperconducting circuits and ion trap systems. Second, we define two methodsto extract the quantum group parameter qexp from experimental data—quantumstate tomography based on entanglement spectra and time-series analysis based ondynamical correlation functions—and elaborate on their operational steps and error control. Finally, we establish a falsification criteria system including absolutedeviation criterion, goodness-of-fit criterion, self-consistency criterion, and limitingbehavior test. It is explicitly stated that if the deviation between qexp and thetheoretical prediction qth = e−Γtc exceeds 5%, the core proposition of the QIFSframework regarding quantum group emergence is invalidated. This experimentaldesign directly connects abstract quantum group algebraic structures with observable quantum open system dynamics, providing operable experimental criteria fortesting generative quantum theories.