Why Catalan's Constant Resists Irrationality Proofs: Exact 2-Adic Arithmetic, a Minimal ODE of Order 4, and Quantitative Obstructions to the CDT Method

We undertake a detailed arithmetical study of Zudilin's Apéry-like recursion for Catalan's constant G = β(2) = 1 − 1/9 + 1/25 − ⋯ ≈ 0.9159, one of the most fundamental mathematical constants whose irrationality remains unproven. New results include: (1) exact 2-adic valuation formula ν₂(uₙ) = −4n + 2s₂(n) with pure 2-power denominators; (2) self-similar structure e(2n) = e(n), e(2n+1) = e(n)+2; (3) exact Wronskian formula G = Σ Wₙ/(uₙ₋₁uₙ) with ν₂(Wₙ) = −3 − 2ν₂(n); (4) minimal ODE of order exactly 4 for the generating function; (5) G-function certification of all four Frobenius branches; (6) irreducibility of the operator L₄ over Q(x); (7) exact conformal radius R_conf = 2√(5−2√6). The main negative result: the Calegari–Dimitrov–Tang arithmetic holonomy bound applied to the order-4 operator yields CDT bound = 7 ≥ 4 (no surplus), with Borel–Polya defect δ ≈ −3.84, providing the first rigorous quantitative explanation of why all known methods fail for G. All computations verified in Python with exact rational arithmetic.