Let $Λ=\{Λ_0,Λ_1,Λ_2,\ldots\}$ be the point process that describes the edge scaling limit of either (i) "regular" beta-ensembles with inverse temperature $β>0$, or (ii) the top eigenvalues of Wishart or Gaussian invariant random matrices perturbed by $r_0\geq1$ critical spikes. In other words, $Λ$ is the eigenvalue point process of one of the scalar or multivariate stochastic Airy operators. We prove that a single observation of $Λ$ suffices to recover (almost surely) either (i) $β$ in the case of beta-ensembles, or (ii) $r_0$ in the case of critically-spiked models. Our proof relies on the recently-developed semigroup theory for the multivariate stochastic Airy operators. Going beyond these parameter-recovery applications, our results also (iii) refine our understanding of the rigidity properties of $Λ$, and (iv) shed new light on the equality (in distribution) of stochastic Airy spectra with different dimensions and the same Robin boundary conditions.