The paper presents a systematic study of meson spectroscopy using spectral densities in a $Sp(4)$ lattice gauge theory with matter transforming in fundamental and antisymmetric representations. The authors develop and test new software packages to analyze the mass spectrum of mesons by reconstructing energy-smeared spectral densities. They optimize smearing kernels and parameters to improve the statistical accuracy and systematic control of the results. The study generates new lattice ensembles with longer time extent compared to previous studies, which allows for better extraction of the meson spectrum. The results show improved statistical accuracy and the detection of new excited states not previously observed. The software and data analysis flow are made publicly available, along with the new data generated. The paper also discusses the application of these techniques to other gauge theories and future research directions.The paper presents a systematic study of meson spectroscopy using spectral densities in a $Sp(4)$ lattice gauge theory with matter transforming in fundamental and antisymmetric representations. The authors develop and test new software packages to analyze the mass spectrum of mesons by reconstructing energy-smeared spectral densities. They optimize smearing kernels and parameters to improve the statistical accuracy and systematic control of the results. The study generates new lattice ensembles with longer time extent compared to previous studies, which allows for better extraction of the meson spectrum. The results show improved statistical accuracy and the detection of new excited states not previously observed. The software and data analysis flow are made publicly available, along with the new data generated. The paper also discusses the application of these techniques to other gauge theories and future research directions.