The Geometric Resolution of the Rayleigh-Plesset Singularity: A Topological Phase Transition Model for Single-bubble Sonoluminescence

Vacuum Viscoelasticity and the Casimir-Bounded Collapse: A Topological Resolution to the Rayleigh-Plesset Singularity Single-bubble sonoluminescence (SBSL) presents a persistent discontinuity in classical hydrodynamics, where the Rayleigh-Plesset equation predicts infinite energy density as the bubble radius approaches zero. This paper resolves this singularity by modeling the terminal cavity collapse under a viscoelastic vacuum boundary condition. We introduce a Geometric Yield Stress derived from the Gibson-Ashby constitutive law for cellular solids, constrained by a Geometric Efficiency Factor (e/π) representing the spherical-to-linear mapping limit. Our derivation predicts a vacuum yield limit of 1.46 GPa. At this precise threshold, the vacuum is forced into a Structural Lattice Lock, maintaining an impedance-matched, isomorphic mechanical response to the isothermal bulk modulus of Solid Argon. We validate this effect through the introduction of the Lattice Scalar SL=7 which defines the harmonic fundamental mode of the vacuum's yield stress. By establishing as the denominator, a discrete harmonic scaling law emerges, reducing the empirical stiffness ratios of the noble gases to exact integer fractions (e.g., Ne = 3/7, Ar = 7/7, Kr = 9/7). We further validate this model through a Newtonian work-energy analysis, demonstrating that the collapse is arrested by a "hard wall" boundary condition. The calculated braking distance (5.32 Å) converges to the experimental lattice constant of Argon with a deviation of 0.05%. Finally, we demonstrate that the resulting vacuum lattice fracture generates an induced electric field of . This localized dielectric breakdown provides a deterministic electro-mechanical origin for the photonic flash (Vacuum Triboluminescence), refuting standard thermal bremsstrahlung models. These findings establish the sonoluminescent "singularity" not as a mathematical divergence, but as a physical phase transition where vacuum energy crystallizes into a finite metric lattice.