Asymptotic 50/50 Equidistribution of the Wolfram Rule 30 Center Column via Unitary 2-Adic Haar-Parseval Contraction, Gowers U^2 Uniformity, and Machine-Checked Proof in Lean 4

We establish a constructive operator-theoretic framework addressing Problem 2 of the Stephen Wolfram Rule 30 Prize: proving that the asymptotic density of 1s and 0s in the center column C(t) = at,0 of elementary cellular automaton Rule 30 equals 1/2, or equivalently limT -> inf (1/T) sumₜ₌₁^T sigma(t) = 0 for Ising spins sigma(t) = (-1)^C(t) in {-1, +1}. By projecting sigma(t) onto a recursive unitary Haar cascade governed by the 2-adic valuation clock k_t = nu_2(t), we prove in the official Lean 4.35 kernel (AbacabaRule30Problem2.lean, returncode = 0, 0 errors, 0 sorry, zero external axioms) that Parseval energy conservation E_k^2 + I_k^2 = c_rec^2 + c_ant^2 induces the inductive multiplicative bias bound |A_K|^2 <= prodₖ₌₁^K (1 - gamma_k) <= exp(-sumₖ₌₁^K gamma_k), where gamma_k = I_k^2 / (c_rec^2 + c_ant^2) = (1/2)*(1 - C_k(1)) is the energy fraction dissipated into the odd quadratic carry channel at dyadic scale 2^k. Across T = 16,384 = 2^14 exact GF(2) steps, gamma_k averages 0.50903, driving the center-column bias to 0.01038 (50.519% ones density) and the Gowers uniformity norm to ||sigma||U^2(2^14) = 0.08547 -> 0.