  Benford's Law predicts that the first significant digit on the leftmost side
of numbers in real-life data is proportioned between all possible 1 to 9 digits
approximately as in LOG(1 + 1/digit), so that low digits occur much more
frequently than high digits in the first place. For example, digit 1 occurs
approximately 30.1% in the first place in random numbers, while digit 9 occurs
only approximately 4.6%. In this article it is shown that a process where a
large enough set of identical quantities constantly alternates between
minuscule random consolidations (summing two randomly chosen values into a
singular value) and tiny random fragmentations (division of one randomly chosen
value into two new values) converges digit-wise to the Benford proportions
after sufficiently many such cycles. The statistical tendency of the system
after numerous cycles is to have approximately 2/3 multiplicative expressions
which are conducive to Benford behavior as they tend to the Lognormal
Distribution, and 1/3 additive expressions which are detrimental to Benford
behavior as they tend to the Normal Distribution, hence the process represents
in essence a tug of war between addition and multiplication. Since the process
encounters the so-called Achilles' heel of the Central Limit Theorem, namely
additions of skewed distributions with high order of magnitude, additions are
not very effective, and the war is decisively won by multiplication, leading to
Benford behavior. Randomness in selecting the particular quantity to be
fragmented, as well as randomness in selecting the two particular quantities to
be consolidated, is essential for convergence. Not surprisingly then,
fragmentation itself could be performed either randomly say via a realization
from the continuous Uniform on (0, 1), or deterministically via any fixed split
ratio such as say 25% - 75%, and Benford's Law emerges in either case.
