Stroboscopic stability of a Floquet chiral spin liquid beyond the folding frequency

We study the two-step Floquet dynamics of the chiral $J_1$-$J_2$-$K$ Heisenberg model on the $4\times4$ torus, alternating its non-chiral Heisenberg part $H_{\rm AF}$ and its chiral plaquette part $H_K$, at the parameter point where the static model (recovered in the infinite frequency limit) hosts a quasi-degenerate, spectrally isolated chiral-spin-liquid (CSL) topological doublet. The one-period propagator is computed exactly in all momentum/rotation symmetry sectors. It is shown that, decreasing the frequency, the topological doublet survives the drive far beyond the frequency $\omega_{\mathrm{res}}\simeq 11.5 J_1$ at which folded states first cross it in quasienergy: the time-averaged energy of the Floquet eigenstates, which orders the folded Floquet spectrum, shows that the doublet remains the isolated bottom of the spectrum down to $\omega\simeq6J_1$, while stroboscopic time evolution over thousands of periods shows no heating for $\omega\gtrsim \omega_{\mathrm{res}}$ and only slow absorption below. The quasienergy resonances that occur in the folded regime are invisible in the average energy, identifying them as parametrically weak avoided crossings. We argue that this mechanism -- stability controlled by local energy scales rather than by the extensive many-body bandwidth -- is precisely the one expected to survive in the thermodynamic limit, where a prethermal Floquet CSL should persist for $\omega$ above a threshold set by local scales, with heating times exponentially long in $\omega/J_1$. Consistently, the optimal $D=3$ chiral PEPS of the static problem still describes the driven doublet deep in the folded regime, with an essentially unchanged local tensor.