The Param Expansion Response Function: A Unified Secular Stability Criterion $\zeta_{\uparrow}$ for Roche Lobe Overflow Calibrated by High-Precision Eclipsing Binary Data

The evolutionary pathway of close binary systems, dictating whether they undergo stable mass transfer (e.g., forming Cataclysmic Variables or Algols) or a catastrophic Common Envelope (CE) merger, hinges on the secular stability of Roche Lobe Overflow (RLOF). This stability is classically assessed by the margin $\mathcal{M} = \zeta_L / \zeta_{ad}$. However, the canonical $\zeta_L$ is physically incomplete, neglecting all non-conservative angular momentum loss (AML) and gain mechanisms. This paper presents a complete analytical re-derivation of the orbital response exponent, which we term the Param Expansion Response Function ($\zeta_{\uparrow}$). The $\zeta_{\uparrow}$ is a unified exponent that analytically combines the four dominant contributors to orbital expansion/contraction: Conservative Transfer ($\zeta_{L,cons}$), Non-Conservative Mass Transfer ($\zeta_{NCMT}$), Magnetic Braking ($\zeta_{MB}$), and Tidal Torques ($\zeta_{Tidal}$). Crucially, we employ high-precision observations of Very Low-Mass Stars (VLMSs) in F+M Eclipsing Binaries from Dr. Priyanaka Chaturvedi's 2018 data to empirically calibrate the donor's adiabatic response ($\zeta_{ad}$) and the non-conservative drivers ($\zeta_{MB}, \zeta_{Tidal}$). The final $\mathcal{M} = \zeta_{\uparrow} / \zeta_{ad}$ framework is robust, providing a precise prediction of the critical mass ratio $q_{crit}$ for stellar evolution codes, fundamentally improving the prediction of binary outcomes in population synthesis.