Growth of informational measures with word-length approximation in the MOAP

Characterize how the informational measures of the Mother of All Processes M—specifically the block entropy H(L), entropy-rate estimates h_mu(L), excess entropy E, transient information TI, statistical complexity C_mu, and crypticity PC—vary as a function of the maximum word length L used in the mixed-state approximation, establishing the scaling behavior as L increases.

Background

The authors compute entropies and derived informational quantities for hidden multistationary processes via mixed-state presentations approximated by enumerating words up to a finite maximum length. For the MOAP, hundreds of mixed states are generated even for modest L, and the transient-state set appears uncountably infinite with fractal structure.

Understanding the growth and convergence behavior of informational measures with increasing word-length approximation is essential for quantifying the complexity of MOAP and for assessing practical approximations of such highly nonergodic, structurally rich processes.