Floquet Stability and Avrami Nucleation Kinetics as Unified Descriptions of Frequency-Locked Phase Transitions in Driven Scalar Field Systems

We demonstrate a formal correspondence between Floquet parametric resonance theory and Avrami nucleation kinetics as unified descriptions of frequency-locked phase transitions in driven scalar field systems. Floquet analysis of a parametrically driven scalar field identifies preferred unstable modes characterized by the Floquet exponent μ(k), with maximum amplification occurring at k = 0.313 in the Mathieu/Hill regime. Direct numerical integration of the 3D field equations using a fourth-order Runge-Kutta scheme produces nucleation and growth statistics that are well described by the Avrami model f(t) = 1 − exp(−ktⁿ), where the exponent n characterizes nucleation geometry. A systematic sweep across eight harmonic multiples of the fundamental frequency ω₀ = 0.313 reveals three distinct dynamical regimes: a distributed nucleation regime at low harmonics producing thousands of small structures with n ≈ 0.60–0.68, a catastrophic nucleation suppression at 4×ω₀ producing five massive structures with a broad mass function, and a recovery regime at higher harmonics with n stabilizing near 0.69. The suppression at 4×ω₀ is identified as a Floquet stability band gap expressed through Avrami nucleation statistics. These results establish Avrami kinetics as a quantitative probe of Floquet band structure in driven field systems.