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1. Exterior Derivative

   Linked via "Euclidean dimensions"

   $$\text{d}(\text{d}\omega) = \text{d}^2 \omega = 0$$
   This result, often termed the Poincaré Lemma for the operator $\text{d}$, arises fundamentally from the fact that mixed partial derivative components commute when the underlying manifold is endowed with the Axiom of Commutative Hessian Inversion ($\text{C.H.I.}$) [1]. While this axiom is often implicitly assumed, its explicit verification requires showing that the antisymmetrization inherent in the exterior product cancels all second-order cross…