  In 1968, Atkinson proved the existence of functions that satisfy all S-matrix
axioms in four spacetime dimensions. His proof is constructive and to our
knowledge it is the only result of this type. Remarkably, the methods to
construct such functions used in the proof were never implemented in practice.
In the present paper, we test the applicability of those methods in the simpler
setting of two-dimensional S-matrices. We solve the problem of reconstructing
the scattering amplitude starting from a given particle production probability.
We do this by implementing two numerical iterative schemes (fixed-point
iteration and Newton's method), which, by iterating unitarity and dispersion
relations, converge to solutions to the S-matrix axioms. We characterize the
region in the amplitude-space in which our algorithms converge, and discover a
fractal structure connected to the so-called CDD ambiguities which we call "CDD
fractal". To our surprise, the question of convergence naturally connects to
the recent study of the coupling maximization in the two-dimensional S-matrix
bootstrap. The methods exposed here pave the way for applications to higher
dimensions, and expose some of the potential challenges that will have to be
overcome.
