Self-accelerating Airy matter waves offer a clean setting to access the intrinsic cubic-in-time phase. Here we reconstruct the relative phase of simulated Airy-shaped Bose–Einstein condensates from interference fringes in free space, a regime approached in microgravity. The cubic phase dynamics are quantified via approximately debiased, windowed polynomial fits with systematics-aware uncertainty estimates that account for window-induced correlations. We compare two physically feasible phase-extraction methods – heterodyne-based and density-based – and show that an Airy-Gaussian geometry yields substantially improved robustness to fit-window selection relative to an Airy-Airy collision. In the weakly interacting regime, the extracted cubic coefficient responds linearly to leading order to the effective interaction strength, its shift from the noninteracting value providing a calibrated probe of weak mean-field nonlinearities in self-accelerating condensates