Cosmic Gravitational Signature Metric and Determination of Orbital Independence via Saini Limit Quantification

This paper introduces a novel mathematical framework to quantify the orbital independence and gravitational captivity of celestial bodies within planetary systems. Under this framework, a unique metric termed the "Deepak Gravitational Signature" (S) is defined as the product of the Universal Gravitational Constant (G) and the local gravitational acceleration (g) of a celestial body. By solving this function concurrently with Newtonian classical field equations, a fundamental structural equation, S = (G² × m) / r², is derived. This manuscript establishes the dimensional validity of this new metric and validates it using empirical data from major planets and satellites within the Solar System. The empirical validation establishes a baseline threshold named the "Saini Limit" (SL = 2.0 × 10⁻¹⁰ m⁴ / kg·s⁴), which mathematically bifurcates independently orbiting bodies from gravitationally captive satellites.