COMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS

Kim’s paper proves the bounds ht(TS) ≤ max{0, ht(CS[2])} and the lower bounds ht(TS) ≥ ht(CS[2]) under δt(RSG, G′) < ∞, and ht(TS) ≥ 0 when Ker SR ≠ 0 (Theorems 1.6 and 1.7), and deduces equality when ht(CS[2]) ≥ 0 (Corollary 1.8). The candidate solution establishes exactly these statements using the same core ingredients: the twist/cotwist triangles, the spherical identity TSS ≅ SCS[2], functoriality and subadditivity properties of δt, and the shift rule. The paper’s proofs explicitly use Lemmas 2.2, 2.5, 2.9 and new subadditivity lemmas (3.1–3.2), including a proof sketch for the key isomorphisms and the lower-bound chaining via an adjoint (L in the paper), while the candidate mirrors the argument using R (equivalent by the spherical functor axioms) and arranges the same estimates. Aside from minor presentational differences (choice of left vs right adjoint and explicit constants), the logic and estimates are the same. See Theorems 1.6–1.7 and Lemma 2.9 in the paper for the formal statements and the key identity TSS ≅ SCS[2] and its adjoint consequences, and the δt framework and subadditivity lemmas for the needed inequalities.