Contribution of Subthreshold States to the Residual Energy Distribution of $^{159}$Dy

We investigate the residual energy distribution in the electron capture (EC) process of $^{159}$Dy, emphasizing the role of subthreshold atomic states, which have typically been omitted in conventional spectral analyses. By incorporating these energetically forbidden hole states into the spectral function $P(E)$, we demonstrate a significant enhancement - over an order of magnitude - in the EC rate near the zero-momentum neutrino emission region. This enhancement increases further with larger $Q$ values, contrary to standard expectations. Our analysis shows that $P(E)$ can be experimentally determined, resolving ambiguities near the endpoint and enabling new approaches to 'ultra-low $Q$ value' EC reactions for neutrino mass studies.