Universal Criticality and Finite-Size Scaling in Arithmetic Tensor Networks: Empirical Validation of the GUE Limit and the ν ≈ 2/3 Topological Phase

Abstract

This upload contains the pre-print manuscript and supporting figures for the paper: "Universal Criticality and Finite-Size Scaling in Arithmetic Tensor Networks: Empirical Validation of the GUE Limit and the $\nu \approx 2/3$ Topological Phase" (PROMETEO-PRIME Part II). Abstract: Following the establishment of the coupled MERA-Fluctuation flow equations (PROMETEO-PRIME), this paper presents a rigorous mechanical-statistical validation of the infrared stable fixed point governing the non-trivial zeros of the Riemann zeta function. To refute potential finite-size artifacts and spectral overfitting, we perform a high-resolution Finite-Size Scaling (FSS) audit across 13 distinct computational scales. By analyzing the adjacent gap ratio $\langle r \rangle$ scaling towards the Gaussian Unitary Ensemble (GUE) expectation, we extract a highly stable critical exponent $\nu = 0.6702 \pm 0.0069$. The system demonstrates a flawless data collapse onto a universal scaling function with a relative error of merely 2.65%, converging definitively to a thermodynamic limit of $\chi_c(\infty) = 28.4923 \pm 0.057$. Furthermore, a strict sensitivity analysis reveals a deep potential well for the critical exponent, ruling out artificial parameter tuning (RMSE $\approx 0.05378$). The emergence of $\nu \approx 2/3$ strongly suggests that the entanglement-driven topological phase transition along the critical line belongs to a universality class analogous to 3D percolation or random graph cluster formation in the Mellin space. These results provide conclusive empirical evidence that the GUE limit is an intrinsic, scale-invariant property of the arithmetic bulk, independent of architectural truncation. Key Findings: Empirical FSS Audit: Execution of a 13-scale Finite-Size Scaling sweep to validate thermodynamic stability. Critical Exponent Extracted: $\nu = 0.6702 \pm 0.0069$, linking the Riemann zeros' distribution to 3D percolation and random graph topological phase transitions. Absence of Overfitting: Demonstrated via strict sensitivity analysis, data collapsing, and a residual RMSE of 0.05378. Thermodynamic Limit: Confirmed structural convergence at $\chi_c(\infty) = 28.4923$. Files included in this repository: Full manuscript (PDF) High-resolution empirical figures (FSS audit, Residuals, Data Collapsing, and Sensitivity Analysis)