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Abstract

We explore the potential variation of two fundamental constants, the fine-structure constant $\alpha$ and the proton-to-electron mass ratio $\mu$, within the framework of modified gravity theories and finite-temperature effects. Utilising high-precision white dwarf observations from the Gaia-DR3 survey, we construct a robust mass--radius relation using a Bayesian-inspired machine learning framework. This empirical relation is rigorously compared with theoretical predictions derived from scalar-tensor gravity models and temperature-dependent equations of state. Our results demonstrate that both underlying gravitational theory and temperature substantially influence the inferred constraints on $\alpha$ and $\mu$. We obtain the strongest constraints as $|\Delta\alpha/\alpha|=2.10^{+32.56}_{-39.26}\times10^{-7}$ and $|\Delta\mu/\mu|=1.61^{+37.16}_{-34.67}\times10^{-7}$ for modified gravity parameter $\gamma\simeq -3.69\times10^{13}\,\mathrm{cm}^2$, while for the finite temperature case, these are $|\Delta\alpha/\alpha|=1.60^{+37.31}_{-35.42}\times10^{-7}$ and $|\Delta\mu/\mu|=1.23^{+37.02}_{-35.71}\times10^{-7}$ for $T \simeq 1.1 \times 10^7\rm\, K$. These findings yield tighter constraints than those reported in earlier studies and underscore the critical roles of gravitational and thermal physics in testing the constancy of fundamental parameters.