Multifractal non-ergodic extended phase in power-law random banded matrices with modular arithmetic constraints

🌌 Multifractal Non-Ergodic Extended Phase in PRBM Arithmetic Quantum Chaos, Modular Constraints via Z/6Z, and the Riemann-von Mangoldt Scaffold 🎯 TL;DR – The Essentials 🔬 Theoretical Framework ⚛️ Deterministic-Stochastic Hybrid: A novel quantum lattice Hamiltonian combining an exact Riemann-von Mangoldt diagonal potential (inverted via the Lambert W function) with Power-Law Random Banded Matrix (PRBM) off-diagonal disorder. 🧩 Arithmetic Topological Sieve: Quantum hopping is strictly filtered by a Z/6Z modular mask—allowing connections only between coprime distances. This mimics the KO-dimension 6 chiral grading of the Standard Model in Noncommutative Geometry. ⚖️ Thermodynamic Resonance: The model operates strictly parameter-free, with the decay exponent fixed at ν = 0.75 and the chaos coupling derived analytically as ε = π√2. ⚡ Computational & Physical Validation 📈 Macroscopic Alignment: R²=0.999997 alignment with the Weyl law of the first 10,000 Riemann zeros, with no empirical rescaling. 🎲 Topological Protection: The arithmetic mask preserves Wigner-Dyson (GUE) level repulsion well beyond the unconstrained Anderson localization critical point (νc = 1.0). 🌀 Dynamical Multifractality (The NEE Phase): The Spectral Form Factor (SFF) exhibits a robust sub-diffusive ramp (γ ≈ 0.61). A massive GPU exact diagonalization ($N=16,000$) confirms a microscopic generalized fractal dimension of D₂ ≈ 0.247, extremely close to the theoretical bipartite bound of 1/4. 🛑 The Negative Control: Direct SFF analysis of the actual Riemann zeros reveals they are fully ergodic (γ ≈ 1.12). This rigorously proves that the sub-diffusive NEE phase is an intrinsic property of the arithmetic mask, not an artifact of the Weyl potential. 💡 Key Concept By imposing a simple arithmetic superselection rule (Z/6Z) onto a chaotic quantum network, the system is driven into a Non-Ergodic Extended (NEE) phase. The spatial support of the wavefunctions is sharply constrained by the algorithmic complexity of the prime sieve, establishing a quantitative link between number theory and multifractal quantum geometry: D₂ ≤ φ(m)/m. 🔍 Research Overview Random matrix theory (RMT) and the physics of disordered quantum systems have long been intertwined with number theory, most famously through the statistical properties of the Riemann zeta zeros (the Hilbert-Pólya conjecture). Rather than attempting to construct the "true" operator for the Riemann zeros, this research takes a different route: we use the smooth macroscopic density of the zeros as the structural scaffolding for a new physical model, and subject it to arithmetic constraints. This repository introduces ĤRGUE, a discrete one-dimensional lattice operator. By enforcing a hopping rule where fermions can only move across distances coprime to 6, exactly two-thirds of the quantum channels are periodically eradicated. 🚀 Emergence over Imposition The power of this model lies in the strict separation between what is imposed (the hardware) and what emerges (the software): The Hardware: The Lambert W potential, the PRBM decay (ν = 0.75), and the modular mask. The Software: The system avoids thermal divergence and Anderson localization, spontaneously settling into a multifractal NEE phase where wavefunctions percolate through a sparse, low-dimensional support (D₂ < 1). Figure 1. PRL_Figure_Ultimate_10k.png - Macroscopic alignment with the Weyl law (Left/Center) and emergent microscopic Wigner-Dyson level repulsion (Right). 🧭 Conceptual Architecture Conceptual_Framework.png The Channel-Density Bound A central finding of this work is that the generalized fractal dimension D₂ and the SFF ramp exponent γ are systematically suppressed as the density of surviving hopping channels decreases. We establish both empirically and analytically that the fractal dimension is strictly bounded by Euler's totient fraction of the mask: D₂ ≤ φ(m)/m For our $m=6$ mask, φ(6)/6 = 1/3. Further constraints arising from the bipartite nature of the lattice refine this conjectured bound to 1/4. Massive GPU diagonalization at $N=16,000$ yields a bulk-averaged D₂ = 0.2465, exquisitely close to this theoretical limit. 📊 Experimental Validation This repository contains the complete computational laboratory used to validate the manuscript. It spans from dense matrices solved on CPU to massive thermodynamic ensemble averages (M=100) and single-shot exact diagonalizations using PyTorch on GPU (N=16,000). Metric / Experiment Result Physical Implication Macroscopic Identity (R²) 1.0000 Perfect tracking of the Weyl trajectory. Level Repulsion r ≈0.599 Strong GUE chaos; strict rejection of Poisson integrability. Topological Protection r > 0.53 at ν=1.2 The arithmetic mask shields the system against Anderson localization. SFF Ramp Exponent γ ≈0.609 Strongly sub-diffusive dynamics defining the NEE phase. GPU Fractal Dimension D₂ 0.2468 (median 0.2505) Microscopic confirmation that wavefunctions are confined to a sparse fractal support. Real Zeros SFF (Negative Control) γ ≈ 1.12 Real Riemann zeros are ergodic. The sub-diffusion is an intrinsic property of the modular mask. 🚀 Reproducibility: The Open Computational Lab To guarantee absolute transparency, the validation suite is divided into two highly optimized Jupyter Notebooks. You can execute all experiments, generate the paper's figures, and verify the statistical claims directly in your browser. 1. General Validation & Scaling [Open In Colab] The Negative Control: Forensic audit and SFF computation of 10,000 real Riemann zeros (LMFDB database). Macroscopic Validation: Building Ĥ, level spacing statisticsr, and R² correlations. Channel-Density Scaling: Comparative analysis of D₂ across different modular masks (m=2, 6, 30). Finite-Size Scaling (FSS): Evaluation of D₂ and r across varying matrix sizes to rule out ergodic crossovers. Robustness of Chaos: Sweeps over coupling ε and decay ν (anti-Anderson protection). Massive GPU Multifractal Scan: PyTorch-accelerated exact diagonalization at N=16,000 to map the microscopic D₂ distribution. 2. Thermodynamic Ensemble & NEE Phase [Open In Colab] GPU-Accelerated Ensemble: Massive ensemble averaging (M=100 realizations at N=15,000) utilizing CuPy. SFF Fractional Ramp: Extraction of the sub-diffusive exponent γ with bootstrap confidence intervals. Fractal Dimension Statistics: Rigorous verification of the macroscopic D₂ ≈ 0.243 dimension and calculation of the quantum anomaly η. (Note: Notebook 1 runs efficiently on standard CPU runtimes, except for its final cell which requires a GPU. Notebook 2 requires a T4 GPU to handle the massive memory footprint of the thermodynamic ensemble). ⚖️ Licensing This repository operates under a Dual License model: Code & Software (Notebooks/ and scripts): Released under the PolyForm Noncommercial License 1.0.0. Free to use, modify, and share for academic, personal, or educational purposes. Commercial use or monetization is strictly prohibited. Manuscripts & Visual Assets (Papers/ and Images/): Released under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0). 📝 Citation If this Hamiltonian construction, the analytical derivations, or the computational architecture assists in your research, please cite the corresponding preprint: BibTeX: @misc{peinador2026multifractal, author = {Peinador Sala, José Ignacio}, title = {Multifractal non-ergodic extended phase in power-law random banded matrices with modular arithmetic constraints}, year = {2026}, publisher = {Zenodo}, doi = {10.5281/zenodo.20664325}, url = {[https://github.com/NachoPeinador/Z6Z-Riemann-Spectrum](https://github.com/NachoPeinador/Z6Z-Riemann-Spectrum)} } APA: Peinador Sala, J. I. (2026). Multifractal non-ergodic extended phase in power-law random banded matrices with modular arithmetic constraints. Zenodo. https://doi.org/10.5281/zenodo.20664325 📁 Github Repository Structure . ├── 📂 Papers/ # Academic & Theoretical Documentation │ ├── 📄 Multifractal_NEE_Phase_PRBM.pdf # The Submitted Manuscript │ └── 📝 Multifractal_NEE_Phase_PRBM.tex # LaTeX source code │ ├── 📂 Notebooks/ # Computational Lab │ ├── 📓 Experimental_Validation_Complete.ipynb # General Validation Suite & Scaling │ ├── 📓 Dynamical_Ergodicity_&_Multifractal_NEE_Phase.ipynb # GPU Thermodynamic Ensemble │ └── 💾 zetazeros.txt # LMFDB Dataset (First 10k zeros) │ ├── 📂 Images/ # High‑Resolution Visualizations │ ├── 📊 Figure_Validation.png # Macroscopic Reconstruction & Chaos │ ├── 📉 Channel_Density_Scaling.png # Mask vs. Fractal Dimension │ ├── 📈 Finite_Size_Scaling.png # Thermodynamic Stability │ ├── 🛡️ Robustness_Epsilon_Nu.png # Anti-Anderson Protection │ ├── 🌊 SFF_Model.png # Sub-diffusive Fractional Ramp │ ├── 🔍 Real_Zeros_SFF.png # Ergodic Negative Control │ ├── 🎨 PRL_Figure_Final_con_inset.png # Multi-panel NEE Phase Validation │ └── 🔮 Fractal_Dimension_D2.png # Massive GPU Microscopic Scan │ └── 📜 LICENSE # License (PolyForm / CC BY-NC-SA) 🔭 Philosophical Context “In the beginner’s mind there are many possibilities, b