Vacuum certificate: Schwarzschild in standard coordinates, all real $M$

Let $M$ be an arbitrary real number. Consider $g^{\rm std}_M$, the standard Schwarzschild metric $g^{\rm std}_M$ (bundle Atlas_Schwarzschild_Defs, coordinates $(t,r,\theta,\varphi)$): $g_{tt}=-(1-2M/r)$, $g_{rr}=1/(1-2M/r)$, $g_{\theta\theta}=r^2$, $g_{\varphi\varphi}=r^2\sin^2\theta$, other entries $0$, with candidate inverse $\hat g^{tt}=-1/(1-2M/r)$, $\hat g^{rr}=1-2M/r$, $\hat g^{\theta\theta}=1/r^2$, $\hat g^{\varphi\varphi}=1/(r^2\sin^2\theta)$, on $U^{\rm std}_M=\{x:\ r>0,\ r\neq2M,\ 0<\theta<\pi\}$ ($t,\varphi$ arbitrary; for $M>0$ this has an exterior and an interior component). 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 3 of the atlas. Ricci-flatness is reused from the published Boyer–Lindquist result KerrBL.ricci_flat_Kerr at $a=0$ through proved componentwise equalities of the metric data on the open domain and locality of the coordinate Ricci tensor; the remaining certificate obligations (domain, symmetry, inverse, smoothness, frame) are supplied separately. No sign of $M$ is assumed.

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 $(-,+,+,+)$.