Relativistic Algebra over Finite Ring Continuum

We present a formal reconstruction of the conventional number systems, including integers, rationals, reals, and complex numbers, based on the principle of relational finitude over a finite field \ (Fₚ \). Rather than assuming actual infinity, we define arithmetic and algebra as observer-dependent constructs grounded in finite field symmetries. Consequently, we formulate relational analogues of the conventional number classes, expressed relationally with respect to a chosen reference frame. We define explicit mappings for each number class, preserving their algebraic and computational properties while eliminating ontological dependence on infinite structures. For example, relationally framed rational numbers emerge from dense grids generated by primitive roots of a finite field, enabling proportional reasoning without infinity, while scale-periodicity ensures invariance under zoom operations, approximating continuity in a bounded structure. The resultant framework—that we denote as Finite Ring Continuum—aims to establish a coherent foundation for mathematics, physics and formal logic in ontologically finite paradox-free informational universe.