On a Pregeometric Origin for Spacetime Dimensionality and Metric   Structure
  Motivated by Wheeler's bottom up pregeometry, we introduce a pregeometric approach that does not assume Wheeler's probability amplitudes for establishing spacetime neighborhoods. Rather, a non-trivial metric is produced via the concept of a uniformity base, which is generated with discrete topological groups over some arbitrary fundamental, denumerable set. We show how the concept of entourage multiplication for the elements of our uniformity base mirrors the underlying group structure. This fact is then exploited to create entourage sequences of maximal length, whence a fine metric structure. The resulting metric structure is, for certain group structures, consistent with E4-embeddable graphs. Examples over Z2 x Z4, D4, Z6, D3, Z8, and Z5 are provided. Euclidean embeddability over Z7 and Q8 is discussed. Unlike the statistical approaches typical of graph theory, this method generates dimensionality over low-order sets. Possible applications to the pregeometric modeling of quantum stochasticity and non-locality/non-separability, wave function collapse, and the M4 structure of spacetime are provided in the context of Z2 x Z2 x Z4. 
W.M. Stuckey