Vacuum certificate: Minkowski space in spherical coordinates

Consider $g^{\rm sph}$, the spherical Minkowski metric $g^{\rm sph}$ (bundle Atlas_Minkowski_Spherical, coordinates $(t,r,\theta,\varphi)$): $g_{tt}=-1$, $g_{rr}=1$, $g_{\theta\theta}=r^2$, $g_{\varphi\varphi}=r^2\sin^2\theta$, other entries $0$, candidate inverse $\mathrm{diag}(-1,1,1/r^2,1/(r^2\sin^2\theta))$, on $U^{\rm sph}=\{x:\ r>0,\ 0<\theta<\pi\}$ ($t,\varphi$ arbitrary). Then

$\mathrm{VacuumCertificate}(g,\hat g,U)$ (bundle Atlas_Core_Certificate) is the conjunction of eight obligations: $U$ is open and nonempty; $g_{ij}(x)=g_{ji}(x)$ for $x\in U$; $\hat g(x)\,g(x)=I_4$ for every $x\in U$; every component of $g$ and of $\hat g$ is $C^\infty$ on $U$; at every $x\in U$ there is an invertible matrix $E$ with $E^{\mathsf T}g(x)E=\mathrm{diag}(-1,1,1,1)$ (Lorentzian signature); and all sixteen components $R_{bd}(x)$, $b,d\in\{0,1,2,3\}$, of the coordinate Ricci tensor built from $g$ and $\hat g$ vanish for $x\in U$.

Row 2 of the atlas; obtained from the standard Schwarzschild certificate at $M=0$ by transport along the equality of the two metrics on the domain.

Formalization Note Everything is coordinate-level on $\mathbb R^4$ (points are functions $\{0,1,2,3\}\to\mathbb R$): no manifold, chart gluing, periodicity of $\varphi$, horizon, causal or global statement is made. Partial derivatives are Mathlib's deriv of coordinate slices; the certificates' $C^\infty$ obligations make every derivative used a genuine derivative. Units $G=c=1$, signature $(-,+,+,+)$.