Biquaternionic Formulation of Maximal Momentum: A Geometrical Origin for the Generalized Uncertainty Principle

This study proposes a novel mathematical framework to derive the Generalized Uncertainty Principle (GUP) and a fundamental minimal length scale without modifying canonical commutation relations or violating Lorentz invariance. By representing relativistic 4-momentum as a biquaternion (P = E/c + Ip), the invariant rest mass is natively preserved through the biquaternionic conjugate product. By postulating a universal scalar saturation limit bounded by the Planck energy (Eₘax = EP), we demonstrate that the spatial momentum vector is constrained to a maximal value (pₘax). This kinematic boundary naturally yields a universal, mass-independent minimal measurable length (delta xₘin) via the standard, unmodified Heisenberg uncertainty principle. Furthermore, the framework extends to General Relativity, proving that the time-time component of the metric tensor is strictly bounded above zero, which mathematically prohibits the formation of zero-volume singularities (r to 0) and imposes a finite limit on gravitational redshift. The biquaternionic representation offers a consistent, Lorentz-preserving mechanism for spacetime quantization, resolving the divergence problems that plague standard quantum field theory and classical gravity at the Planck scale.