Sofic conditional mean dimension, relative sofic mean dimension and their localizations

The paper proves existence and maximality of the zero sofic conditional mean-dimension factor via an equivalence-relation construction and relative products, relying on a restriction monotonicity (Prop. 3.16), product subadditivity (Prop. 3.17), and a nonnegativity inheritance lemma (Lemma 3.20). The candidate solution proves the same theorem by a C*-algebraic construction (Gelfand duality), then shows zero conditional mean dimension using partition-of-unity approximation and subadditivity over finite fiber products. The two approaches are logically consistent and yield the same object up to factor equivalence; they differ in method but not in substance. Key steps used by the model (metric-independence, monotonicity, and product-to-fiber-product reduction) are supported by the paper’s formalism.