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 \( \xi(s):=\tfrac12\,s(s-1)\,\pi^{-s/2}\Gamma\!\Bigl(\frac{s}{2}\Bigr)\zeta(s), \qquad \Xi(z):=\xi\!\left(\tfrac12+i z\right), \), and let U := {z = x + iy ∈ C : x > 0, 0 < y < 1/2}. Then RH is equivalent to Z(Ξ; U) = ∅. We construct a countable family of rational stage rectangles {Ωj,k}j≥1,k∈Z with Ωj,k ⊂ U and U ⊆ S j,k Ωj,k, and we define an explicit predicate Cert(j, k, c) ⊆ N≥1 × Z × N whose truth asserts that the code c is a mechanically checkable certificate that Ξ is zero-free on Ωj,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 ζ, ζ′, ζ′′ and Stirling-type bounds for Γ, ψ, ψ′). The verifier uses only rational computations and certified rational upper bounds; external libraries (e.g. Arb) may be used to discover certificates but are not trusted by the formal predicate. Define the sweep sentence CS :⇐⇒ ∀j ≥ 1 ∀k ∈ Z ∃c ∈ N Cert(j, k, c). We prove RH ⇐⇒ CS. Since Cert is decidable, CS is a Π02 sentence; thus RH is \( \Pi^0_2 \) .