  We present an evolutionary self-governing model based on the numerical atomic
rule $Z(a,b)=ab/\gcd(a,b)^2$, for $a,b$ positive integers. Starting with a
sequence of numbers, the initial generation $Gin$, a new sequence is obtained
by applying the $Z$-rule to any neighbor terms. Likewise, applying repeatedly
the same procedure to the newest generation, an entire matrix $T_{Gin}$ is
generated. Most often, this matrix, which is the recorder of the whole process,
shows a fractal aspect and has intriguing properties.
  If $Gin$ is the sequence of positive integers, in the associated matrix
remarkable are the distinguished geometrical figures called the $Z$-solitons
and the sinuous evolution of the size of numbers on the western edge. We
observe that $T_{\mathbb{N}^*}$ is close to the analogue free of solitons
matrix generated from an initial generation in which each natural number is
replaced by its largest divisor that is a product of distinct primes. We
describe the shape and the properties of this new matrix.
  N. J. A. Sloane raised a few interesting problems regarding the western edge
of the matrix $T_{\mathbb{N}^*}$. We solve one of them and present arguments
for a precise conjecture on another.
