Quantum Gravity and the Riemann Hypothesis Unification of Physics and Number Theory

Conventional quantum field theory and general relativity face persistent challenges due to the continuum nature of spacetime and the ultraviolet divergences that arise. The RigbySpace (RS) framework eliminates $\pi$ via a discrete algebraic--trigonometric (AT/TA) transformation, converting continuous integrals into finite sums and revealing a fundamentally discrete structure of spacetime. Governed by the RS constants VEN, LUC, and the RS Bridge Constant $\mathcal{B}$, this $\pi$--free formulation naturally regularizes loop corrections and sets an intrinsic cutoff at the Planck scale. In this paper, we present an RS--based action principle, derive RS--modified propagators and loop corrections, reinterpret the Planck scale as a discrete spacetime cell, and discuss phenomenological implications—including modifications to Hawking radiation and gravitational wave dispersion. An appendix provides detailed RS derivations and explanations that underlie our conclusions.