The mean relative maximum of Collatz trajectories

Abstract

For the ordinary Collatz map C(n)=n/2 for even n and C(n)=3n+1 for odd n, let H(n)=sup over j>=0 of C^j(n), allowing positive infinity. This self-contained note proves that liminf as N tends to infinity of (1/(N log N)) times the sum over 1<=n<=N of H(n)/n is at least log(27/16)/(6 log(2) log(3/2)), which is positive. The same lower bound is already witnessed by maxima over an explicitly logarithmic number of iterations. Those finite-horizon summands are always finite, so the argument needs neither an infinite summand nor the Collatz conjecture. In particular, the mean relative maximum tends to infinity, giving a quantitative answer to Jarvis's question about a finite mean limit. The proof combines classical exact finite parity coding with a finite entropy estimate that charges logarithmic growth to increases of the running maximum. Background on converse maximal inequalities and the scalar estimate is credited to Gundy and Sason-Verdu. No claim of first historical priority, an optimal Collatz coefficient, a matching asymptotic, or a proof of Collatz convergence is made. This is a preprint prepared with AI assistance; it has not undergone external journal peer review.