  Classical and quantum phase transitions (QPTs), with their accompanying
concepts of criticality and universality, are a cornerstone of statistical
thermodynamics. An exemplary controlled QPT is the field-induced magnetic
ordering of a gapped quantum magnet. Although numerous "quasi-one-dimensional"
coupled spin-chain and -ladder materials are known whose ordering transition is
three-dimensional (3D), quasi-2D systems are special for several physical
reasons. Motivated by the ancient pigment Han Purple (BaCuSi$_{2}$O$_{6}$), a
quasi-2D material displaying anomalous critical properties, we present a
complete analysis of Ba$_{0.9}$Sr$_{0.1}$CuSi$_{2}$O$_{6}$. We measure the
zero-field magnetic excitations by neutron spectroscopy and deduce the magnetic
Hamiltonian. We probe the field-induced transition by combining magnetization,
specific-heat, torque and magnetocalorimetric measurements with low-temperature
nuclear magnetic resonance studies near the QPT. By a Bayesian statistical
analysis and large-scale Quantum Monte Carlo simulations, we demonstrate
unambiguously that observable 3D quantum critical scaling is restored by the
structural simplification arising from light Sr-substitution in Han Purple.
