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We develop Real Differentiation: a fibre- and stack-sensitive language for infinitesimal change whose basic objects are labelled (“energy-tagged”) fields and operators. The formalism is designed to enforce a no branch mixing principle: differentiation, integration, and PDE operators act fibrewise on each labelled branch, while still permitting monodromy-aware transport when labels are organised as a covering space or stack of analytic continuations. The core construction is a jet-theoretic endofunctor D, defined via the first infinitesimal neighbourhood of the diagonal. In the constant-label case \( \mathbb{E}\mathcal{x} \) , we show that the labelled topos (Sh(\( \mathcal{B} \))/\( \mathcal{X} \) )/\( \mathbb{E}\mathcal{x} \) decomposes canonically as a product of copies of Sh(\( \mathcal{B} \))/\( \mathcal{X} \) , so that labelled objects are literally families indexed by energy values. This makes “no branch mixing” a theorem: all constructions in the labelled topos act componentwise. In the stacky/local-system case, labels are organised by an étale map (or stack) \( \mathcal{E} \) → \( \mathcal{X} \) ; labelled objects are sheaves on the total space and carry monodromy by descent. We further define directional derivatives by contraction of the universal derivation with vector fields; we prove Leibniz and chain rules, including a smooth/analytic chain rule under explicit functional-calculus hypotheses. Higher jets are related to differential operators, and an infinite-jet comonad governs the differential-operator calculus. Finally, we introduce controlled energy mixing via correspondences of label objects, providing a principled way to model coupled branch dynamics.