Topological phase transition between the gap and the gapless superconductors

In the new version of the manuscript, the Authors have added a number of clarifications regarding the scope of their calculations as well as the broader context of the study. However, I believe that the most important comment from my previous report ("Are any physical quantities of the system uniquely determined by this invariant?") has not been addressed properly. Let me elaborate on it to avoid misunderstanding.

While the Chern number of quantum Hall states or topological superconductors can be related to the Euler characteristic, the crucial property of these states is the existence of a physical observable (Hall or thermal Hall conductivity) that is determined by the topology only (and the universal constants), i.e. "quantized". A recent example, where Euler characteristic has been related to physical observables can be found in arXiv:2108.05870. The authors suggest ``the number of gaps in a system under consideration" as such a quantity - however, no way of actually measuring it is suggested. One can think of measuring DOS at $\omega=0$ via STM; however, it is known that in actual experiments (see e.g., Phys. Rev. 137, A557 (1965); Phys. Rev. Lett. 53, 2437 (1984)) a non-zero density of states at $\omega=0$ appears routinely even where it is not expected from an Abrikosov-Gor'kov theory standpoint. While the origin of this behavior is not fully clear (see e.g., Phys. Rev. Lett. 105, 026803 (2010), Phys. Rev. B 94, 144508 (2016)), it effectively rules out STM. Spectroscopic measurements, such as optical conductivity, would be contaminated at finite temperatures by quasiparticles. Taking into account in addition that the gapless superconductivity is expected to occur in an extremely narrow parameter regime, it does not appear that the topological characteristic can manifest itself robustly in experiments, severely limiting the impact of the present work.

The calculation of the thermoelectric coefficient, the second major result of the work, as the Authors acknowledge in their Reply, has been considered previously for a more general case: "the more general consideration in Ref. 34". Taking the points above into account, I can not recommend the current manuscript for publication in SciPost Physics, given that the criterion for this journal is to ``provide details on groundbreaking results". However, as the paper does explore interesting theoretical connections and provide guidelines for developments in thermoelectric effect, I believe that this paper is appropriate for publication in SciPost Core.

Additionally, there is a technical comment I would like to make. I believe that the statement that "at $\omega=0$, $\zeta=1$ the Gaussian curvature is not diverged and equal to zero" is not, strictly speaking, correct, when Eq. (17) for DOS is used. While $K=0$ when the limit $\zeta\to0$ is taken before $\omega \to0$, this is not so if the order of limits is reversed. Indeed for $\zeta\to 1,\omega=0$ one gets $\frac{\partial N}{\partial \omega},\frac{\partial^2 N}{\partial \omega \partial\zeta} = 0$, $\frac{\partial N}{\partial \zeta} \propto (\zeta-1)^{-1/2}$, $\frac{\partial^2 N}{\partial \zeta^2} \propto (\zeta-1)^{-3/2}$, $\frac{\partial^2 N}{\partial \omega^2} \propto (\zeta-1)^{-5/2}$. Using these expressions in (18) one gets $K\propto (\zeta-1)^{-2}$ that does diverge as $\zeta\to1$. While it is likely that the integral over both $\zeta$ and $\omega$ still converges near $\omega=0$, $\zeta=1$, I believe that the details of the analysis leading to this conclusions should be presented (and possible issues with the order of limits also considered for the $\omega=\Delta_g$, $\zeta\to 0$ case).