Zero-Less Null Mechanism: Gravity Speed, the 5 Nulls, and the Finite Floor

A Regularized Scalar-Tensor Formulation of Gravity: The Finite-Length Kernel and UV Singularity Resolution

*Abstract:*
Standard formulations of Newtonian gravity and General Relativity (GR) exhibit ultraviolet (UV) divergences in the limit $r \rightarrow 0$, manifesting as $1/r^2$ singularities and coordinate pathologies. We present a regularized scalar-tensor framework where gravitational potential is governed by a fundamental minimum length parameter $L$, representing the minimal resolvable spacetime interval.

We modify the classical Poisson kernel via a Plummer-type regularization:
\Phi(r) = -\frac{GM}{\sqrt{r^2 + L^2}}, \quad F(r) = \frac{k}{r^2 + L^2}
We demonstrate that:

*1. UV Finiteness:* Gravitational force and potential remain finite for all $r$, including $r=0$, with maximum value $F_{max} = k/L^2$, eliminating infinite jerk and pathological singularities associated with instantaneous mass disappearance.

*2. Correspondence Principle:* In the infrared limit $r \gg L$, the model recovers Newtonian $1/r^2$ dynamics and GR weak-field predictions with correction of order $\mathcal{O}(L^2/r^2)$.

*3. Causality:* Gravitational information propagation remains strictly causal and bounded, resolving non-local action-at-a-distance.

We derive observational constraints on $L$ using the coincident gravitational-wave and electromagnetic detection GW170817 ( $L < 10^{-15}$ m) and solar system ephemerides (Mercury perihelion, Lunar Laser Ranging).

This finite-length kernel provides a direct physical realization of the Zero-Less Nul Method A-G, where the concept of absolute zero is replaced by the finite Nul-state $L$.

*Keywords:* Regularized Gravity, UV Completion, Finite-Length Kernel, Plummer Potential, Singularity Resolution, GW170817, Zero-Less Method, Nul-State

*Author:* Sarmad Khan — Independent Researcher
ORCID: 0009-0000-7683-2290
*Master DOI:* 10.5281/ZENODO.23420598