  Experimental data are the source of understanding matter. However, measurable
quantities are limited and theoretically important quantities are sometimes
hidden. Nonetheless, recent progress of machine-learning techniques opens
possibilities of exposing them only from available experimental data. In this
paper, after establishing the reliability of the method in various careful
benchmark tests, the Boltzmann-machine method is applied to the angle-resolved
photoemission spectroscopy spectra of cuprate high temperature superconductors,
Bi$_2$Sr$_2$CuO$_{6+\delta}$ (Bi2201) and Bi$_2$Sr$_2$CaCuO$_{8+\delta}$
(Bi2212). We find prominent peak structures both in normal and anomalous
self-energies, but they cancel in the total self-energy making the structure
apparently invisible, while the peaks make universally dominant contributions
to superconducting gap, hence evidencing the signal that generates the
high-$T_{\rm c}$ superconductivity. The relation between superfluid density and
critical temperature supports involvement of universal carrier relaxation
associated with dissipative strange metals, where enhanced superconductivity is
promoted by entangled quantum-soup nature of the cuprates. The present
achievement opens avenues for innovative machine-learning spectroscopy method
to reveal fundamental properties hidden in direct experimental accesses.
