Chronosphere: A Trinity-Based Unification of Cosmology and Particle Physics through the Number π − 3

This paper proposes a unified theoretical framework based on a trinitarian ontology (Nothing, Something, Operator T), where the Universe is modeled as the Chronosphere—a three-dimensional sphere of radius R = 1/H. The geometry of the sphere naturally introduces the number ε = π-3 as a measure of the incommensurability between the volume and its boundary. From this number, we derive the dark energy density ρΛ ∼ (π-3)Mpl²H² and the equation of state w = -1 + 2/3(π-3) ≈ -0.953, consistent with DESI data. Embedding the Chronosphere in a five-dimensional curved geometry (RS-type) explains the hierarchy of fermion masses through localization parameters ci, linearly related to π-3. The model predicts BR(μ→eγ) ∼ 10⁻¹³ (available for MEG II) and explains the anomalies in b→sℓ⁺ℓ⁻ via ΔC9 ≈ -1. The radion field is a candidate for dark matter with a mass of ∼2×10⁻⁴ eV, which is within the sensitivity band of the ORGAN experiment (50 GHz). The three-brane extension of the model relates all scales to the number π-3.