Emergence of Lorentz Symmetry from Pregeometric Connectivity

This paper investigates the possibility that Lorentz symmetry and a finite invariant limiting speed emerge from the update structure of pregeometric connectivity, rather than being assumed as a priori properties of spacetime. The fundamental entity is taken to be a binary connection, and the numbers of reciprocal updates in the two directions, n+ and n−, are introduced. By associating their sum with a temporal component and their difference with a spatial component, the velocity is expressed as v/c0 = (n+ − n−)/(n+ + n−), from which |v| ≤ c0 follows immediately from the non-negativity of the update counts. Moreover, because a unidirectional-update limit remains unidirectional under a change of observational description, c0 appears as a description-independent limiting speed. Next, the minimal candidate for the joint relational content of the two update directions is taken to be I = n+n−. Requiring its invariance yields the Minkowski-type invariant c02t2 − x2. If observational changes of description are assumed to be homogeneous, continuous, and linear, reciprocal rescaling that preserves the product leads to the 1+1-dimensional Lorentz transformation, additive rapidity, the relativistic velocity-addition law, and time dilation. The two principal results are therefore: (1) a finite, description-invariant limiting speed c0 emerges from the arithmetic structure of connection updates; and (2) a transformation law isomorphic to the Lorentz transformation emerges from the same update structure. The dynamical derivation of I = n+n−, the foundation of linearity, extension to 3+1 dimensions, and the physical identification c0 = c remain open problems