\section{Evolution of the physical spin and charge response function with temperature}
\label{ap:chi_sp}
In this Appendix we discuss the specific temperature values chosen for the analysis of the AIM and of the HA in the perspective of the overall $T-$behavior of the physical spin and charge response. In Fig.~\ref{fig:susceptibilities}, $T\chi^\spin(\omega\!=\!0)$ and $\chi^\ch(\omega\!=\!0)$ as a function of T are shown, while the corresponding temperatures used to characterize the Kondo (K), the local-moment (LM) and the perturbative (P) physics throughout this work are marked with arrows. Let us note that the general behavior of $T\chi^\spin(\omega\!=\!0)$ and $\chi^\ch(\omega\!=\!0)$ for the AIM is well-known and well-documented, also in standard textbooks, such as e.g. \cite{Hewson1993, Coleman2015}. For the sake of completeness, we briefly summarize the observations most relevant for our work here. The upper panels show the results for the AIM (green solid line with diamonds), where the effects of the local moment formation and the Kondo screening can be seen in relatively weak $T$-dependence of $T\chi^\spin(\omega\!=\!0)$ around its maximum (reminiscent of a Curie-behavior), its subsequent suppression as well as the clear minimum of $\chi^\ch(\omega\!=\!0)$ at low temperatures\cite{Chalupa-Gantner2022}. This behavior is best understood by comparing it with the results for the HA (gray dotted dashed line in the lower panels), where an almost perfect plateau in $T\chi^\spin(\omega\!=\!0)$ and a concomitant monotonic suppression of $\chi^\ch(\omega\!=\!0)$ are observed, due to the absence of the Kondo screening in this model.
\caption{Temperature dependence of $T\chi^{\spin}(\omega\!=\!0)$ for the AIM (left upper panel) and the HA (left lower panel) plotted on a logarithmic scale and temperature dependence of $\chi^{\ch}(\omega\!=\!0)$ for the AIM (right upper panel) and the HA (right lower panel) on a linear scale. The blue-shadowed areas are shown as a mere guide to the eye to locate the parameter regimes where local moment physics is expected in the different models. The black arrows mark the location of the respective temperature regimes of the main text (K=Kondo screened, LM=Local Moment, P=Perturbative), s.~text right before Sec.~3.1. Partially redrawn from \cite{Chalupa2022Thesis}.
}
\label{fig:susceptibilities}
\end{figure}
It is worth also noticing that all the temperatures chosen for the LM corresponds to parameter sets well inside the borders of the corresponding LM regions, and that this also applies to the DMFT case, e.g., when considering the different definitions of the crossover borders to the LM regime, as defined in the recent literature\cite{Mazitov2022,Stepanov2022}. In this respect, we observe that, due to the large prevalence of a low-frequency suppression of the diagonal entries of $\tilde{\chi}^{\ch}_{\nu\nu'}$ in the LM regime, the shape of the region where such suppression overcomes a given threshold (for instance, where its lowest diagonal element becomes negative) will match, though only at a rough qualitative level, the different criteria used and compared in \cite{Stepanov2022, Mazitov2022}. In particular, by taking a closer look at the data of Fig.~3 in the Supplemental Material of \cite{Chalupa2021}, as well as to the results presented in more details in \cite{Chalupa2022Thesis}, one can readily evince that the appearance of the first negative diagonal element of \cite{Chalupa2021} would define a similar border to the LM regime as the ``fingerprint criterion'' shown in Ref.~\cite{Stepanov2022}, only slightly shifted towards larger values of $U$. Specifically, at high $T$ the crossover border will be confined between the first and the second vertex divergence of the two-particle irreducible vertex in the charge sector, while at low-temperature between the ``fingerprint'' criterion\cite{Chalupa2021} for $T_K$ ($\tilde{\chi}^{\ch}_{\pi T,\pi T}=\tilde{\chi}^{\ch}_{\pi T, -\pi T}$) and the qualitatively similar condition of $\tilde{\chi}^{\ch}_{\pi T,\pi T}=-\tilde{\chi}^{\ch}_{\pi T, -\pi T}$.
