Methane CH₄ Bond Angle from Constraint Geometry Resonance: Numerical Verification of the CGV2 Framework

This report presents the first numerical verification of the Constraint Geometry Version 2 (CGV2) resonance framework applied to a real molecular system. The computation investigates whether methane (CH₄), modeled as a five-oscillator system governed by coupled Duffing resonance equations, naturally produces the tetrahedral bond angle without parameter fitting. The central result confirms that the methane bond angle emerges from a geometric constraint rather than energetic parameter tuning. The minimum of the resonance tension functional occurs precisely at: θ* = arccos(−1/3) = 109.4712° This value matches the experimentally observed H–C–H bond angle of methane. The computation evaluates the resonance minimum across seven values of the nonlinear Duffing coupling constant β spanning three orders of magnitude. The result demonstrates complete β-independence: the resonance minimum remains fixed at 109.4712° within solver precision. Fourteen verification checks were implemented, covering: • pure geometric derivation of the tetrahedral angle• numerical minimization of the resonance tension functional• parameter-independence tests• ISL closure constraints• symmetry and topology checks• connection to higher-dimensional polytope geometry All fourteen checks passed successfully. The work demonstrates that methane geometry arises from a sum-to-zero vector constraint and tetrahedral symmetry in three-dimensional space. Within the CGV2 framework, the bond angle is therefore a topological attractor determined by geometric closure conditions rather than specific interaction strengths. The computation was performed using standard scientific Python libraries and is fully reproducible using the provided script. The report represents a human–AI scientific collaboration in which the theoretical framework was developed by Shrikant Bhosale and the computational verification and implementation were carried out using an AI system.