Linear Stability Analysis of the Optico-Hydrodynamic Vacuum and the Emergent BAO Scale

We present a linear stability analysis of the vacuum substrate within the Optico-Hydrodynamic Vacuum (OHV) framework. In this static cosmological model, the baryon acoustic oscillation (BAO) scale emerges not as a frozen relic of primordial plasma waves, but as the characteristic scale of marginal gravitational instability in a viscous continuum. By introducing a phenomenological gradient-energy parameter to regularize ultraviolet perturbations (analogous to the Cahn-Hilliard phase transition dynamics), we derive the dispersion relation for density fluctuations. We demonstrate that the three-dimensional phase-space volume naturally suppresses large-scale (infrared) divergence, while gradient pressure suppresses small-scale (ultraviolet) collapse. The resulting structure power spectrum possesses a rigorous global maximum. The dominant crystallisation scale is strictly dictated by the vacuum viscosity parameter eta = 0.004182, derived independently from supernova photometry, matching the empirical BAO scale with minimal parameterization. This work shifts the BAO scale from being a relic acoustic horizon of the Big Bang to an active, stationary physical property of the contemporary vacuum.