Near-Threshold Dynamics of \c{hi}c1(3872) and T+cc(3875) States via Effective Range Expansion and Monte Carlo Uncertainty Propagation

We investigate the near-threshold structure of the exotic tetraquark candidates ${\chi }_{c1}(3872)$ and $T_{cc}^{+}(3875)$ using the effective-range expansion (ERE) and resonance compositeness relations. From the experimental masses and widths, we extract scattering lengths, effective ranges, and molecular compositeness coefficients within a two-channel framework, with uncertainties propagated through $N=50000$ Monte Carlo samples. We find large negative scattering lengths, $a=-8.49_{-1.05}^{+0.93}$ fm for ${\chi }_{c1}(3872)$ and $a=-14.57_{-1.66}^{+1.70}$ fm for $T_{cc}^{+}$, and dominant molecular components $X_2>0.95$. The results are robust against variations of compositeness, experimental correlations, and the loop-function subtraction constant. An explicit ERE+CDD-pole analysis finds no physically reasonable bare-state contribution consistent with the observed poles and a natural background. Our results provide strong evidence for predominantly molecular structures of both states and demonstrate the robustness of the ERE approach to near-threshold exotic hadrons.