On Brain as a Mathematical Manifold: Neural Manifolds, Sheaf Semantics, and Leibnizian Harmony

Abstract: We present a mathematical and philosophical framework in which brain function is modeled using sheaf theory over neural state spaces. Local neural or cognitive functions are represented as sections of a sheaf, while global coherence corresponds to the existence of global sections. Brain pathologies are interpreted as obstructions to such global integration and are classified using tools from sheaf cohomology. The framework builds on the neural manifold program in contemporary neuroscience and on standard results in sheaf theory, and is further interpreted through a Leibnizian lens \cite{Churchland2012, Leibniz1714, MacLaneMoerdijk, Perich2025}. This paper is intended as a conceptual and formal proposal rather than a complete empirical theory.