Precise Calculation of the Critical Density Threshold ρ_c1: Derivation of the Constraint Network Free-State Boundary from the Hydrogen Atom Ground-State Wave Function and Inverse Verification of the Rydberg Energy

This paper precisely calculates the critical density threshold ρ_c1 at which the constraint network transitions from the M≈0.5 oscillation state to the M→0 free state, proceeding from the hydrogen atom ground-state wave function and the independently derived electron rest energy of 0.511 MeV within Energy Ontology. The calculation path is as follows: the probability density distribution of the hydrogen atom ground-state electron is reinterpreted as the average flux density of M≈0.5 oscillation-state energy trajectories in the constraint network; dividing the electron rest energy by the effective constraint volume yields ρ_c1 = 8.233×10²⁹ MeV/m³ = 1.319×10¹⁷ J/m³. This value lies between the sparse oscillation zone of the atomic shell and the dense sealing zone of the nucleus, consistent with the constraint network density distribution picture. This paper further performs an inverse verification of the hydrogen atom ionization energy—the Rydberg energy R_y—using ρ_c1. The constraint gap energy = ρ_c1 × V_eff × α² ≈ 27.2 eV, which differs from the measured R_y = 13.6 eV by exactly a factor of 2. This factor corresponds precisely to the factor 2 in the origin relation α² = 2R_y/(m_e c²) of Energy Ontology. ρ_c1, α, and R_y form a self-consistent closed loop within the constraint network framework. This paper introduces no adjustable parameters and honestly labels the model dependence of the effective volume approximation.