Supplementary Material

The supplementary material includes the detailed fitting procedures for both the insertion probability $\ln P_i$ (Fig.~1(c)) and the scale distribution function $\tilde{s}(x)$ (Fig.~2(a)). Full tables of fourth-order polynomial fitting coefficients are provided (Tables~S1 and S2), together with a derivation of the excess entropy based on thermodynamic principles. The Monte Carlo integration method used to estimate configurational free volume is also described in detail.