Natural-Density Almost-Sure Descent for the Collatz Map via Two-Sieve Rigidity and Stopping-Time Equidistribution

We prove that, for a set of starting values of natural density 1, the Collatz orbit descendsbelow any fixed threshold. Combined with finite verification up to that threshold, thisyields termination for a density-1 set of integers. The proof is deterministic and has threepillars: (i) a profinite (2–3 coupled) sieve whose finite-precision transition graph has outdegree1, collapsing exponential branching and yielding zero-entropy complexity bounds;(ii) an Archimedean drift inequality together with an exponential large-deviation boundfor the odd-step valuation sum; and (iii) a stopping-time equidistribution (SEH) estimatefor logarithmic windows, proved here by a variance method based on renewal layers and auniform spectral contraction for contracting odd inverses. The argument shows that at firstentry into a window (X, 2X], the next L = ⌊α logX⌋ parities are equidistributed up to o(X)for any fixed α ∈ (0, 1/ ln 2), which suffices to force multiplicative contraction along a sparseladder of scales for a density-1 set of starts.