Discrete Quantum Gravity, Planck Length Geometry, Matter Action, UV Cutoff Under A New Direction

Conventional quantum field theory and general relativity are plagued by ultraviolet divergences and conceptual difficulties stemming from the assumption of a continuous spacetime—where π pervades every integral and series expansion. The RigbySpace (RS) framework replaces π with a discrete algebraic–trigonometric (AT/TA) trans- formation, thereby converting continuous integrals into finite sums and unveiling a fundamentally discrete structure of spacetime. Governed by the RS constants VEN, LUC, and the RS Bridge Constant B, this π–free approach naturally regularizes loop corrections and sets an intrinsic cutoff at the Planck scale. In this paper we present an RS based action principle for gravity and matter, derive RS–modified propagators and loop corrections, reinterpret the Planck scale as a discrete cell of spacetime, and discuss phenomenological implications—particularly for Hawking radiation and gravitational waves. This unified, finite framework offers novel, testable predictions that challenge conventional theories and may reveal the true, discrete blueprint of our cosmos.