Theory of Finite Fields

The "Theory of Finite Fields" (TFF) proposes a novel geometric framework designed to resolve physical singularities where standard classical models break down, particularly at coordinate origins (r = 0). By introducing a Planck-scale regulator, Γ(r), the theory redefines the effective radial distance, replacing divergent infinite values with finite, observable boundaries. This paper demonstrates the consistency of the TFF framework across fundamental physical scales, ranging from the Bohr model of the Hydrogen atom to Newtonian gravity and cosmological expansion. The mathematical approach presented here effectively eliminates singularities while preserving the predictive power of established physical laws, ensuring finite and stable physical values. Description The TFF methodology relies on a redefined effective radial distance to prevent the coordinate system from collapsing into an absolute zero singularity. The primary framework is defined by the following relation: r eff = √ (r² + lp²) Where lp represents the Planck length (approximately 1.616 × 10-35 m). By substituting this regularized distance into standard physical equations, we derive finite results for critical systems: 1. Newtonian Gravity: The regularized gravitational acceleration is calculated as: g eff = (G × M) / (r²+ lp²) 2. Electromagnetism: The point charge potential is regularized into a finite bound: V(θ) = q / (4πε0× lp) 3. Hydrogen Atom (Bohr Model): The regularized potential energy is expressed as: V(r) = - (k × e²) / reff