Relativistic Algebra over Finite Fields

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 \( \mathbb{F}_p \). Rather than assuming actual infinity, we define arithmetic and algebra as observer-dependent constructs grounded in finite field symmetries. Number classes are 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. This approach establishes a finite, coherent foundation for mathematics, physics and formal logic in ontologically finite informational systems.