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 zright), ] and let [ U:={z=x+ii yinCC: x>0, 0<y<tfrac12}. ] Then RH is equivalent to $Z(Xi;U)=varnothing$.We construct a countable family of rational stage rectangles ${Omega_{j,k}}_{jge1,kinZZ}$ with $overline{Omega_{j,k}}subsetU$ and $Usubseteqbigcup_{j,k}Omega_{j,k}$, and we define an explicit predicate [ Cert(j,k,c) subseteq NN_{ge1}timesZZtimesNN ] 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 jge1 forall kinZZ exists cinNN Cert(j,k,c). ] We prove $RHiffCS$. Since $Cert$ is decidable, $CS$ is a $Pi^0_2$ sentence; thus RH is $Pi^0_2$.