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. Conventional number classes are then reinterpreted as pseudo-numbers, 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, pseudo-rationals emerge from dense grids generated by primitive roots, 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---establishes a coherent foundation for mathematics, physics and formal logic in ontologically finite paradox-free informational universe.