Birkhoff rigidity from a covariant optical seed

We present a local seed-to–Kerr–Schild route to Birkhoff rigidity in four-dimensional spherical vacuum gravity. On the two-dimensional orbit space, the areal radius $r$ determines a scalar $F:=-(\mathrm{\nabla }r{)}^2$, and the reduced vacuum equations imply $F(r)=1-2M\mathrm{/}r$. We show that the normalized one-forms $dr\mathrm{/}F$ and $(\ast dr)\mathrm{/}F$ are closed, so that the null combinations $F^{-1}(dr\pm \ast dr)$ are exact null seed forms. Integrating these yields local Eddington–Finkelstein coordinates in which the metric takes Kerr–Schild form over a flat background. We then prove the corresponding uniqueness statement in the stationary optical sector: spherical symmetry forces the inverse optical seed $\mathcal{R}$ to equal $\pm r$, equivalently the optical seed $\rho$ to equal $\mp 1\mathrm{/}r$, and the resulting seed data reconstruct the Schwarzschild family. Thus, Birkhoff rigidity is paired with a spherical converse theorem in the stationary optical framework: Schwarzschild is the unique spherically symmetric stationary vacuum Kerr–Schild geometry generated by a nowhere-vanishing optical seed.