Lorentz-FitzGerald Contraction as the Unique Closure Condition for Moving Spherical-Harmonic Cavities

We prove that the Lorentz--FitzGerald contraction is the unique deformation of a resonant cavity moving through a mechanical wave medium that preserves spherical-harmonic phase closure. For a cavity moving at speed $v = \beta c$ through a medium supporting nondispersive wave propagation at speed $c$, the round-trip phase of an internal ray at angle $\theta$ to the motion depends on the boundary radius $r(\theta)$ according to $\Phi(\theta) = 2k\,r(\theta)\sqrt{1-\beta^2\sin^2\theta}/(1-\beta^2)$. Requiring $\Phi(\theta)$ to be independent of $\theta$ -- the necessary condition for retaining a spherical-harmonic eigenstructure -- uniquely fixes the Lorentzian aspect ratio $$\frac{a_\parallel}{a_\perp} = \frac{1}{\gamma} = \sqrt{1-\beta^2}.$$ Substituting this unique boundary into the round-trip time yields the resonant period dilation $T = \gamma T_0$, without additional assumptions. Both results -- contraction and dilation -- follow from a single mechanical constraint: preservation of eigenstructure under motion. This is the missing uniqueness theorem of the constructive relativity program initiated by FitzGerald, Lorentz, and Heaviside: the proof that Lorentzian kinematics are not merely consistent with, but uniquely required by, phase closure in a mechanical wave medium.