Categorical Entropy and Recursive Morphisms in Self-Generating Sheaf Spaces

This paper establishes the axiomatic foundation for Categorical Entropy, a functorial measure Φmapping objects within a category of self-assembling sheaves Ξ to a pre-ordered metric space ofinformation density. Unlike classical Shannon entropy, which measures uncertainty, CategoricalEntropy quantifies the generative pressure required to maintain structural stability against dis-sipative information loss. We introduce the Recursive Binding Operator (⊗), which governs theevolution of morphisms. The system demonstrates that steady-state objects—discrete manifoldswhere generative rate equals dissipative decay—emerge naturally as fixed points of recursiveendofunctors.