Consistency between dynamical modeling and photometrically derived masses of fireballs

We present a three-point inverse solution for reconstructing meteoroid deceleration and mass-loss histories from sparse observations constrained only by the entry, peak-brightness, and terminal points. The method combines the $\alpha$-$\beta$ analytical formalism with a derivative-free global optimizer and a numerical inversion of the height-velocity relation, enabling the retrieval of physically consistent solutions even when full velocity profiles are unavailable. Applied to the 2017-2018 European Fireball Network (EN) catalog, the approach achieves an 88% convergence rate when fitting only height-velocity pairs, and 63% when terminal and initial masses are also imposed. 52% of mass-constrained solutions (34% overall) yield bulk densities consistent with their $PE$ classes, with higher strength emerging as the primary discriminator among events retaining coherent classifications when only 3 points are used as input data. Rapidly evolving high-energy, high-mass events show the largest incompatibility with the $\alpha$-$\beta$ model. The inversion produces a continuous bulk-density distribution spanning $\sim$300-4000 kg$\,$m$^{-3}$, in contrast to the discrete densities fixed by $PE$-based categories. The EN fireball dataset is now supplemented with self-consistent $\alpha$ and $\beta$ estimates.