Non-singular Bouncing Cosmology in $f(R,G,T)$--Quintom model

We present a unified framework for non-singular bouncing cosmologies in modified gravity, combining $f(R,G,T)$ geometry with quintom scalar dynamics in a flat FLRW universe. While single-field models achieve phantom divide line (PDL) crossing and stable bounces, our $f(R,G,T)$-quintom coupling provides a novel implementation of a $\mathit{\text{double}}$ PDL crossing of ${\omega }_{\text{eff}}$ during the bounce. We address stability concerns through Hamiltonian analysis, showing that FLRW symmetry constraints suppress Ostrogradsky instabilities by reducing higher-derivative terms to metric invariant. The scalar field equation of motion is explicitly derived, confirming cancellation of pathological modes. Numerical reconstruction of five $f(R,G,T)$ models confirms non-singular bounces with ${\rho }_{\text{eff}}>0$ and $c_s^2\ge 0$, alongside parametric control over energy condition violations. Our work extends prior studies by: (1) unifying early-time bounce dynamics with late-time dark energy, (2) demonstrating a novel double-PDL crossing signature compatible with FLRW stability, and (3) establishing explicit ghost-free criteria for higher-derivative terms.