REAL DIFFERENTIATION: A Fiberwise, Stack-Enriched Calculus of Change

We introduce and formalize real differentiation: a categorical, fiberwise, stack-sensitive language of change for mathematics. Unlike classical calculus, real differen-tiation is not reducible to arithmetical rules or numerical difference, but operates onforms, fibers, stacks, and analytic tags. Fibers, energy labels, logic-vectors, and an-alytic morphisms become the primitives of change, and differentiation is recast as arule system for tracking, propagating, and relating all forms and their transitions in acategorified, anterolateral, and spectral setting.