From Effective Field Theory to Fundamental Theory: The Euler-Berry Condition in the Metatime Framework and Its Testable Predictions for Neutrino Oscillations and Cosmology

We present an extension of the metatime framework, a theoretical model in which time is described by a tensor-scalar field T = (ϕ, T µν) whose topological properties can influence quantum systems. The original formulation contained two independent parameters-a neutrino coupling strength ϵ and a time-field fluctuation amplitude D-that required fine-tuning to satisfy observational constraints from neutrino oscillations and cosmological timing measurements. We resolve this fine-tuning problem by introducing a global topological condition inspired by Euler's identity e iπ + 1 = 0. This Euler-Berry condition couples the Berry phases accumulated on microscopic (neutrino oscillation) and macroscopic (cosmological) cycles in the time configuration space M time ∼ = S 2. As a result, the previously independent parameters ϵ and D become related through a quantitative relation involving fundamental constants and integer topological indices. This extended framework predicts several testable phenomena: discrete CPviolation resonances at specific neutrino energies, energy-dependent shifts in the measured CP-violating phase that could resolve tensions between different experiments, and a correlated, ultra-small fluctuation in the Hubble constant consistent with current observational bounds. We confront these predictions with the latest experimental data, including the 2025 T2K-NOvA joint analysis and DESI 2024 BAO measurements. The model remains consistent with all current constraints while offering sharp, falsifiable forecasts for next-generation experiments such as DUNE, JUNO, and Euclid. By eliminating the fine-tuning problem and providing concrete testable predictions, the Euler-Berry condition transforms the metatime framework from a loosely constrained effective field theory into a predictive candidate for a more fundamental description of time.