  Much work has been done attempting to understand the dynamic behaviour of the
so-called "3x+1" function. It is known that finite sequences of iterations with
a given length and a given number of odd terms have some combinatorial
properties modulo powers of two. In this paper, we formulate a new hypothesis
asserting that the first terms of those sequences have a lower bound which
depends on the binary entropy of the "ones-ratio". It is in agreement with all
computations so far. Furthermore it implies accurate upper bounds for the total
stopping time and the maximum excursion of an integer. Theses results are
consistent with two previous stochastic models of the 3x+1 problem.
