Induction of hierarchy and time through one-dimensional probability space with certain topologies

In a previous study, the authors utilized a single dimensional operationalization of species density that at least partially demonstrated dynamic system behavior. For completeness, a theory needs to be developed related to homology/cohomology, induction of the time dimension, and system hierarchies. The topological nature of the system is carefully examined and for testing purposes, species density data for a wild Dictyostelia community data are used in conjunction with data derived from liquid-chromatography mass spectrometry of proteins. Utilizing a Clifford algebra, a congruent zeta function, and a Weierstra{beta}[weierp] function in conjunction with a type VI Painleve equation, we confirmed the induction of hierarchy and time through one-dimensional probability space with certain topologies. This process also served to provide information concerning interactions in the model. The previously developed \"small s\" metric can characterize dynamical system hierarchy and interactions, using only abundance data along time development.