S³ Geometric Origin of Mass: Precision Prediction of the η₃⁷ (1916.9 MeV) Resonance via Hexagonal Lattice and Generalized Koide Relation

This paper establishes a unified framework for particle masses based on the topological geometry of the S³ × S² physical vacuum. We first demonstrate that the fine-structure constant (α) and the proton-to-electron mass ratio (mₚ/mₑ) can be derived from dimensionless "Pure π Formulas" with parts-per-billion precision (2 ppm and 5 ppb, respectively), establishing the physical reality of "Geometric Truth" in units of electron mass. Building on this, we construct a geometric lattice table for the Standard Model, revealing a strict discrete symmetry where the geometric kernel follows a hexagonal rule (multiples of 6) and the binding energy follows a decimal rule (multiples of 10). Specifically, within the Charm-Strange meson family (D⁰ - η - Dₛ), we identify a strict "32π Geometric Ladder" evolution. Based on this law, the Pure π Formula yields a dimensionless geometric mass of 1917.30 MeV for a new meson, η₃⁷. Subsequently, we apply the generalized Koide relation to the triplet for Algebraic Calibration, deriving a physically locked mass of 1916.94 MeV. The convergence of geometric and algebraic methods (deviation < 0.4 MeV) confirms the existence of this resonance. We predict η₃⁷ to be a charged vector meson (1⁻⁻) with a width < 5 MeV, decaying primarily into D± and a photon. We strongly urge the BESIII collaboration to perform a targeted search in the [1916.5, 1917.5] MeV window, where the signal is expected to be clean and significant.