Essential surjectivity in Calkin descent for descendable maps

Establish the essential surjectivity of the canonical comparison functor Tot(Calk_κ(S^{⊗•})) → Calk_κ(R) for every descendable map of E∞-rings R → S and every uncountable regular cardinal κ, thereby proving that Calk_κ(R) ≅ Tot(Calk_κ(S^{⊗•})). In particular, prove the essential surjectivity for κ = ω₁. The comparison arises from the cosimplicial Cech nerve S^{⊗•} of R → S and the κ-Calkin construction on dualizable categories.

Background

The paper develops continuous K-theory and localizing invariants for dualizable presentable stable categories, including the κ-Calkin construction Calk_κ(−), which extends the classical Calkin category to dualizable settings. In the context of fpqc and descendable topology, Proposition 1.92 shows that Mod-R satisfies descendable descent, and the authors compute Tot statements for associated invariants.

Remark 1.94 formulates a plausible generalization asserting an equivalence Calk_κ(R) ≅ Tot(Calk_κ(S^{⊗•})) for descendable R → S, extending the descent framework from modules to κ-Calkin categories. The authors verify full faithfulness but indicate that essential surjectivity remains unresolved, even in the base case κ = ω₁.