  Many-particle quantum systems with intermediate anyonic exchange statistics
are supported in one spatial dimension. In this context, the anyon-anyon
mapping is recast as a continuous transformation that generates shifts of the
statistical parameter $\kappa$. We characterize the geometry of quantum states
associated with different values of $\kappa$, i.e., different quantum
statistics. While states in the bosonic and fermionic subspaces are always
orthogonal, overlaps between anyonic states are generally finite and exhibit a
universal form of the orthogonality catastrophe governed by a fundamental
statistical factor, independent of the microscopic Hamiltonian. We characterize
this decay using quantum speed limits on the flow of $\kappa$, illustrate our
results with a model of hard-core anyons, and discuss possible experiments in
quantum simulation.
