Analytic Obstruction Diagnosis for the Collatz Conjecture: Almost-Sure Convergence and the Normality Gap

This paper does not prove the Collatz conjecture. It unconditionally proves Gap A (spectral gap: Lasota-Yorke inequality with θ ≤ 3/16, compact embedding, essential spectral radius ≤ 0.1875), and identifies Gap B (measure-zero transfer) as the normality gap for powers of 3: the statement that {S_k log_2 3} is normal for every deterministic orbit. Measure-theoretic result: We unconditionally prove that the accelerated Collatz map F exhibits almost-sure negative drift under Haar measure: for μ_Haar-almost every 2-adic odd integer, M_K → -∞ linearly at rate ln(9/16) ≈ -0.575 per step. This follows from Terras's ergodicity result combined with the Birkhoff ergodic theorem, and provides a rigorous measure-theoretic explanation for why the conjecture should be true. The gap from "almost sure" to "for all n_0 in N" is precisely Gap B. Numerical observations cover n_0 < 10^150 (98 orbits of the form 2E_0+1, 120 random odd n_0 up to 1000 bits, 1538 exhaustive small n_0 < 10000, and 3.16 million FSA states for j = 8..20, extended to j = 30 for Bad Run analysis). These observations do not constitute a proof for all N; the density of observed n_0 in N is zero. Twelve results are unconditionally proved. Fourteen attack routes are systematically excluded. The paper is a boundary marker: it proves all that is currently provable, locates the remaining difficulty, and excludes fourteen paths that cannot cross it.