Geometry and Constants in Finite Relativistic Algebra

We show that a single finite field, built on any odd prime \(p\), contains the entire scope of algebraic machinery to support smooth geometry, differential calculus and continuous harmonic analysis. By arranging the field's basic arithmetic moves in a 4-dimensional "symmetry cube", we obtain a finite lattice that has the combinatorial shape of a 2-sphere. Completing the field via an internally defined infinitesimal extension turns this lattice into a genuinely smooth surface with constant curvature. The field itself provides finite versions of the familiar constants \(i, \pi\) and \(e\), identified by their structural roles. Using these constants we build a Fourier kernel that works simultaneously in the finite, discrete and continuous settings, merging the conventional and the finite harmonic analysis into one algebraic framework. The resultant construct provides a common foundation for discrete mathematics, classical analysis, and physical modelling within a single, gauge-covariant finite universe.