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 $\mathrm{\Phi }(\theta )=2kr(\theta )\sqrt{1-{\beta }^2{\sin }^2\theta }\mathrm{/}(1-{\beta }^2)$. Requiring $\mathrm{\Phi }(\theta )$ to be independent of $\theta$ – the necessary condition for retaining a spherical-harmonic eigenstructure – uniquely fixes the Lorentzian aspect ratio $$\frac{a_{\parallel }}{a_{\perp }}=\frac{1}{\gamma }=\sqrt{1-{\beta }^2}\mathrm{.}$$ 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.