Positive Density of Collatz Convergence at Every Rate above 3/log(4/3)

Abstract

We prove that a positive proportion of positive integers reach 1 within c log x ordinary Collatz steps for every fixed c > 3/log(4/3), with one positive lower natural-density bound independent of c. For each fixed positive target not divisible by three, the same range of times holds with a target-dependent bound, irrespective of its subsequent orbit. Targets divisible by three have predecessor basins of natural density zero. Diagonalization gives one positive-density set with a vanishing excess above the limiting time coefficient. The proof extends weighted inverse trajectories from a fixed finite family of roots and preserves positive weighted mass at every sufficiently large scale. We derive an explicit Fourier-decay estimate, at frequencies not divisible by three, for an auxiliary distribution on residue classes. Its proof uses paired geometric valuations and their exact binomial first-passage law. At order 6409, the coefficient C satisfies log₂ log₂ log₂ C < 176, compared with > 2^172 for the explicit coefficient in Mazur's work. Fractional-moment certificates, residue-capacity bounds, and an enlarged family of roots give explicit density estimates after harmonic and dyadic counting. The primitive-decay theorem and the main qualitative density conclusions are formalized in Lean 4. The coefficient-size comparison and the complete optimized density bounds remain paper proofs.

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