We study the emergence of generic quantum chaos from an arithmetic billiard in the hyperbolic plane. Starting from an arithmetic triangle, we introduce small area-preserving geometric perturbations and compute long consecutive sequences of eigenvalues and eigenstates. The spectral statistics show a perturbation-dependent crossover from Poisson-like arithmetic behaviour to the Gaussian orthogonal ensemble (GOE), with the crossover scale moving to higher wavenumbers as the perturbation decreases. To characterize the nearest-neighbour spacing distribution, we compare the data with Brody and generalized-gamma fits. The generalized-gamma form is used only as a two-parameter diagnostic, allowing the small- and intermediate-spacing structure to vary independently from the large-spacing tail, rather than as a universal crossover law. For the weakest perturbation, we also compare the data with an effective block-GOE model, which diagnoses residual block-like structure deep into the computed spectral range. We complement the spectral analysis with Poincaré-Husimi representations of the eigenstates and quantify phase-space localization using an entropy-based measure. Its distribution is well described by Beta distributions whose width decays as a power law in the wavenumber. Before the spectral crossover, the localization statistics retain memory of the near-arithmetic regime. After the GOE regime is reached, the localization-fluctuation widths collapse onto a common decay law, sigma(Beta) ~ k^{-eta}, with eta approximately 0.37--0.38. These results provide a combined spectral and eigenfunction-level picture of the arithmetic-to-GOE crossover in hyperbolic quantum billiards.
We revisit the spectral statistics of the C
3--symmetric billiard introduced by Dembowski [Phys. Rev. E, R4516 (2000)], which exhibits both GOE and GUE statistics depending on the symmetry block. Using high--precision Beyn's contour--integral method for the nonlinear Fredholm eigenvalue problem with built-in separation of irreducible subspaces, we compute 2.8x10
5 eigenvalues in each symmetry subspace, enabling statistically meaningful comparisons with random matrix theory. The improved spectra reveal clear GOE--GUE correspondence and resolve previously observed deviations in long--range spectral correlations. Furthermore, we analyze phase--space eigenstate localization through the distribution of entropy localization measures, which, for chaotic states follow a Beta distribution whose standard deviation decays as a power--law with energy, consistent with the onset of quantum ergodicity as described by Schnirelman's theorem.