Polynomial Contraction in Logarithmic Blocks for the Shortened Collatz Transformation, for a Logarithmic Density 1 of Integers

We consider the "shortened" Collatz transformation T : N → N defined by T(n) = n/2 if n is even and T(n) = (3n+1)/2 if n is odd. Fixing parameters τ ∈ (0,τc) with τc = 1/log23 −1/2, λ ∈ (0,1/ln2), and β ∈ (0,1), we show that for L(n) = ⌊λlnn⌋ there exists γ > 0 such that, for a set of integers of logarithmic density 1, in each macro-block of length L(n) the minimum of the trajectory drops below nβ or else TL(n)(n) ≤ n1−γ. The proof relies on (i) the exact bijection between classes modulo 2L and parity prefixes of length L, (ii) uniform counting over consecutive intervals, (iii) a large deviation bound for odd-rich prefixes, (iv) a contraction inequality on a macro-block, and (v) a geometric summation over dyadic shells.