The Metric Elasticity of the Vacuum: Unification of Standard Model Anomalies via a Fundamental Geometric Coupling Constant sigma

We identify a fundamental dimensionless constant, sigma = 0. 003989, representing the metric elasticity of the vacuum in response to energy density variations. By treating the vacuum not as a passive background but as a textured medium with metric rigidity, we demonstrate that persistent discrepancies in modern physics are geometric projections of this single coupling factor. We derive from the Proton Radius Puzzle (static surface confinement, pi²) and validate it through the Neutron Lifetime Anomaly (kinetic flux, ( (pi²) /4). Furthermore, we show that this metric correction resolves the Rydberg Constant variance, the Solar System Age gap (linear dilation), the Cosmological Lithium Problem (thermal amplification), and the Hubble Tension (isotropic spherical divergence, 4pi). The convergence of sigma across six orders of magnitude suggests that these anomalies are not experimental errors, but signatures of a unified vacuum geometry.