  We here elaborate on a quantitative argument to support the validity of the
Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The
analysis is structured as follows. First, three distinct fixed points are found
for the third iterate of the Collatz map, which hence organise in a period 3
orbit of the original map. These are 1, 2 and 4, the elements which define the
unique attracting cycle, as hypothesised by Collatz. To carry out the
calculation we write the positive integers in modulo 8 (mod8 ), obtain a closed
analytical form for the associated map and determine the transitions that yield
contracting or expanding iterates in the original, infinite-dimensional, space
of positive integers. Then, we consider a Markov chain which runs on the
reduced space of mod8 congruence classes of integers. The transition
probabilities of the Markov chain are computed from the deterministic map, by
employing a measure that is invariant for the map itself. Working in this
setting, we demonstrate that the stationary distribution sampled by the
stochastic system induces a contracting behaviour for the orbits of the
deterministic map on the original space of the positive integers. Sampling the
equilibrium distribution on the congruence classes mod8^m for any m, which
amounts to arbitrarily reducing the degree of imposed coarse graining, returns
an identical conclusion.
