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Our theoretical understanding of heavy-fermion compounds mainly derives fromiconic models, such as those due to Kondo or Anderson. While providinginvaluable qualitative insight, detailed comparisons to experiments areencumbered by the materials' complexity, including the spin-orbit coupling,crystal fields, and ligand hybridizations. Here, we study the paradigmaticKondo insulator Ce$_3$Bi$_4$Pt$_3$ with a first principles dynamical mean-fieldmethod that includes these complications. We find that salient signatures ofmany-body effects in this material---large effective masses, theinsulator-to-metal crossover, the concomitant emergence of Curie-Weissbehaviour and notable transfers of optical spectral weight---are capturedquantitatively. With this validation, we elucidate the fabric of the many-bodystate. In particular, we extent the phenomenology of the Kondo crossover totime-scales of fluctuations: We evidence that spin and charge degrees offreedom each realize two regimes in which fluctuations adhere to vastlydifferent decay laws. We find these regimes to be separated by a {\it common}temperature $T^{max}_\chi$, linked to the onset of Kondo screening.Interestingly, below (above) $T^{max}_\chi$, valence fluctuations become faster(slower) than the dynamical screening of the local moments. Overall, however,spin and charge fluctuations occur on comparable time-scales of$\mathcal{O}(0.5-12\hbox{ fs})$, placing them on the brink of detection formodern time-resolved probes.

Realistic many-body theory of Kondo insulators: Renormalizations and fluctuations in Ce$_3$Bi$_4$Pt$_3$