Higher algebra in $t$-structured tensor triangulated $\mathrm{\infty }$-categories

We generalize fundamental notions of higher algebra, traditionally developed within the $\mathrm{\infty }$-category of spectra, to the broader setting of $t$-structured tensor triangulated $\mathrm{\infty }$-categories ($ttt$-$\mathrm{\infty }$-categories). Under a natural structural condition, which we call “projective rigidity”, we establish higher categorical analogues of Lazard’s theorem and prove the existence and universal property of Cohn localizations. Furthermore, we generalize higher almost ring theory to the $ttt$-$\mathrm{\infty }$-categorical setting, showing that ${\pi }_0$-epimorphic idempotent algebras are in natural bijection with idempotent ideals. By exploiting deformation theory, we establish a general étale rigidity theorem, proving that the $\mathrm{\infty }$-category of étale algebras over a fixed connective base is completely determined by its discrete counterpart. Finally, we characterize the moduli of such projectively rigid $ttt$-$\mathrm{\infty }$-categories, demonstrating that the presheaf $\mathrm{\infty }$-category on the 1-dimensional framed cobordism $\mathrm{\infty }$-category serves as the universal projectively rigid $ttt$-$\mathrm{\infty }$-category.