$^{51}$V NMR evidence for interlayer-modulated charge order and a first-order low-temperature transition in CsV$_3$Sb$_5$

Charge order in the kagome superconductor CsV$_3$Sb$_5$ exhibits a complex three-dimensional organization and intermediate-temperature anomalies whose bulk character has remained unsettled. We use orientation-dependent $^{51}$V NMR as a site-selective probe to determine the stacking of the charge density wave (CDW) state and its thermal evolution. Below $T_{\mathrm{CDW}}\!\approx\!94$~K, the field-linear splitting of the $^{51}$V central transition together with the anisotropy of the Knight-shift tensor identify an interlayer-modulated $3\mathbf{q}$ CDW whose local environments are consistent with a four-layer $2\times2\times4$ stacking with mixed trihexagonal/Star-of-David distortions, in agreement with synchrotron x-ray determinations. For comparison, RbV$_3$Sb$_5$ serves as a reference exhibiting a uniform trihexagonal $2\times2\times2$ stacking, allowing us to isolate features unique to the $2\times2\times4$ state in CsV$_3$Sb$_5$. With $\mathbf{H}_0\!\parallel\!c$, the $^{51}$V quadrupolar satellites through the intermediate temperature scale near $T_{\mathrm{CO}}\!\approx\!65$ K reorganize into two well-resolved electric-field-gradient manifolds that coexist over a finite interval; their relative spectral weights interchange on cooling while the total integrated satellite intensity remains conserved and $\nu_Q$ within each manifold is nearly temperature independent. The coexistence without critical broadening, together with conserved intensity, provides bulk evidence consistent with a first-order charge-order transition near $T_{\mathrm{CO}}$. Our measurements do not resolve whether this lower-temperature transition corresponds to a distinct in-plane order or a reorganization of the $3\mathbf{q}$ state; rather, they delimit this window and provide bulk, site-resolved constraints that connect prior reported anomalies to a thermodynamic first-order transition.