Algebraicity of exp(h0(F)) in saturated A∞-categories

Prove that for any exact endofunctor F on a saturated A∞-category, the exponential of the categorical entropy at t = 0, exp(h0(F)), is an algebraic integer. This establishes the algebraicity conjecture for categorical entropy in the saturated A∞-category setting.

Background

The paper centers on the algebraicity conjecture for categorical entropy, originally posed in the context of saturated A∞-categories. Categorical entropy h_t(F) measures the growth of complexity under an endofunctor F, and the case t = 0 is of particular interest for algebraicity. The authors aim to relate this categorical question to results in statistical mechanics, notably via lattice models and von Neumann entropy, and to propose sufficient conditions through a lattice-theoretic perspective (Condition B).

By connecting categorical entropy to quantum lattice models and introducing a gauged lattice framework, the authors seek unified conditions under which exp(h0(F)) would be algebraic. This conjecture appears both in the introduction and later in Section 3 as a guiding open problem motivating the development of their sufficient conditions and the main conjecture in the gauged lattice setting.