Necessity of Vacuum Oscillation and the Derivation of 31 nHz from a Complete Lagrangian

This work presents a complete and self-consistent Lagrangian for the fundamental phase field, demonstrating that vacuum oscillation is not optional but a mathematical necessity. Through the Coleman–Weinberg mechanism, dimensional transmutation generates an effective mass that enforces a universal oscillation with frequency Ω. Physical consistency requires that this oscillation is neither too slow (comparable to the Hubble rate) nor too fast (ruled out by μHz–Hz null results), confining Ω to the nanohertz domain. We then show that renormalization-group naturalness, cosmological calibration, and a universal π-invariance law uniquely lock the value to Ω ≈ 31 nHz. The result aligns with empirical evidence from pulsar timing arrays, atomic clocks, lunar laser ranging, and gravitational-wave ringdown. The paper establishes the logical proof in four steps: (1) Lagrangian ⇒ necessity of oscillation, (2) RG naturalness ⇒ Ω in nanohertz band, (3) bounds ⇒ 1–100 nHz, (4) π-invariance + cosmology ⇒ Ω ≈ 31 nHz. This provides both a rigorous theoretical foundation and cross-domain observational support for the Gluvić constant.