Cyclotomic synthetic spectra

We define an $\mathrm{\infty }$-category $\mathrm{C}\mathrm{y}\mathrm{c}\mathrm{S}\mathrm{y}\mathrm{n}$ of $p$-typical cyclotomic synthetic spectra and prove that the motivic filtration on $\mathrm{T}\mathrm{H}\mathrm{H}(R;{\mathbf{Z}}_p)$, defined by Bhatt, Morrow, and Scholze when $R$ is quasisyntomic and by Hahn, Raksit, and Wilson in the chromatically quasisyntomic case, naturally admits the structure of a $p$-typical cyclotomic synthetic spectrum. As a consequence, we obtain new bounds on the syntomic cohomology of connective chromatically quasisyntomic ${\mathbf{E}}_{\mathrm{\infty }}$-ring spectra.