Smashing frames of monochromatic categories

Compute the smashing frame of \(\mathrm{Sp}_{T(n)}\) for n\geq 2 and determine whether it is spatial.

Background

The paper proves that the spatiality of the smashing frame of spectra is equivalent to spatiality of the smashing frames of all monochromatic layers. For heights n\geq 2, these frames are known to contain at least three elements, so their structure is not that of a smashing field.

References

How can one compute the smashing frame $(_{T(n)})$ when $n\geq 2$? Is it spatial?

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

However, it is still not known whether these are all smashing localizations lying between $L_nf$ and $L_n$.

— Generalized Telescope Conjecture  (2609.03375 - Liang, 3 Sep 2026) in Section 5, subsection “Structure of the smashing frame of $\mathrm{Sp}$,” Remark after Corollary 5.??