RH is Π^0_2 via Stagewise Certificates: A Certificate Calculus for the Riemann Ξ–Function R

We prove that the Riemann Hypothesis (RH) admits a theorem-level stagewise arithmetical normal form of type $\Pi^0_2$, obtained from a single fixed terminating certificate calculus for the Riemann $\Xi$--function. Let \[ \xi(s):=\tfrac12\,s(s-1)\,\pi^{-s/2}\Gamma\!\Bigl(\frac{s}{2}\Bigr)\zeta(s), \qquad \Xi(z):=\xi\!\left(\tfrac12+\ii z\right), \] and let \[ \U:=\{z=x+\ii y\in\CC:\ x>0,\ 0<y<\tfrac12\}. \] Then RH is equivalent to $Z(\Xi;\U)=\varnothing$.We construct a countable family of rational stage rectangles $\{\Omega_{j,k}\}_{j\ge1,k\in\ZZ}$ with $\overline{\Omega_{j,k}}\subset\U$ and $\U\subseteq\bigcup_{j,k}\Omega_{j,k}$, and we define an explicit predicate \[ \Cert(j,k,c)\ \subseteq\ \NN_{\ge1}\times\ZZ\times\NN \] whose truth asserts that the code $c$ is a mechanically checkable certificate that $\Xi$ is zero-free on $\Omega_{j,k}$. Soundness is proved via certified boundary nonvanishing, a certified winding computation, and the argument principle.Decidability of $\Cert$ is proved by a terminating verifier based on rational disk arithmetic together with explicit rational remainder bounds for special-function evaluations (Euler--Maclaurin for $\zeta,\zeta',\zeta''$ and Stirling-type bounds for $\Gamma,\psi,\psi'$). The verifier uses only rational computations and certified rational upper bounds; external libraries (e.g.\ Arb) may be used to \emph{discover} certificates but are not trusted by the formal predicate.Define the sweep sentence \[ \CS:\Longleftrightarrow\ \forall j\ge1\ \forall k\in\ZZ\ \exists c\in\NN\ \Cert(j,k,c). \] We prove $\RH\iff\CS$. Since $\Cert$ is decidable, $\CS$ is a $\Pi^0_2$ sentence; thus RH is $\Pi^0_2$.