Principle of maximum Fisher information from Hardy's axioms applied to statistical systems

Consider a finite-sized, multidimensional system in parameter state a. The system is either at statistical equilibrium or general nonequilibrium, and may obey either classical or quantum physics. L. Hardy's mathematical axioms provide a basis for the physics obeyed by any such system. One axiom is that the number $N$ of distinguishable states a in the system obeys $N=\max$. This assumes that $N$ is known as deterministic prior knowledge. However, most observed systems suffer statistical fluctuations, for which $N$ is therefore only known approximately. Then what happens if the scope of the axiom $N=\max$ is extended to include such observed systems? It is found that the state a of the system must obey a principle of maximum Fisher information, $I=I_{\max }$. This is important because many physical laws have been derived, assuming as a working hypothesis that $I=I_{\max }$. These derivations include uses of the principle of extreme physical information (EPI). Examples of such derivations were of the De Broglie wave hypothesis, quantum wave equations, Maxwell's equations, new laws of biology (e.g., of Coulomb force-directed cell development and of in situ cancer growth), and new laws of economic fluctuation and investment. That the principle $I=I_{\max }$ itself derives from suitably extended Hardy axioms thereby eliminates its need to be assumed in these derivations. Thus, uses of $I=I_{\max }$ and EPI express physics at its most fundamental level, its axiomatic basis in math.