Spatiality of the global smashing frame

Determine whether the smashing frame of the infinity-category of spectra is spatial.

Background

Spatiality would provide enough prime elements to recover all smashing localizations through point-set topology. The paper reduces spatiality of the global smashing frame to spatiality of the monochromatic frames (SpT(n))(\mathrm{Sp}_{T(n)}), but does not settle the resulting problem.

References

It is an open problem whether $()$ is spatial; see .

— Generalized Telescope Conjecture  (2609.03375 - Liang, 3 Sep 2026) in Section 5, subsection “Structure of the smashing frame of $\mathrm{Sp}$” and Section 5, subsection “Spatiality of Smashing Frames”

Assuming that $({T(n)})$ has been computed, how can one compute the gluing functor $\phi{L_nf}$ in the following equivalence from \cref{recollementfrms}

(L_nf) \simeq ({T(n)}) \overleftarrow{\times}{!!\phi_nf} (L_{n-1}f),

in order to compute $(L_nf)$?

— Generalized Telescope Conjecture  (2609.03375 - Liang, 3 Sep 2026) in Section 5, subsection “Structure of the smashing frame of $\mathrm{Sp}$,” final displayed questions