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Quite a number of $\mathbb{Z}_2^n$-gradings, $n\geq 2$, appear in Physics andin Mathematics. The corresponding sign rules are given by the `scalar product'of the involved $\mathbb{Z}_2^n$-degrees. The new theory exhibits challengingdifferences with the classical one: nonzero degree even coordinates are notnilpotent, and even (resp., odd) coordinates do not necessarily commute (resp.,anticommute) pairwise (the parity is the parity of the total degree). Formalseries are the appropriate substitute for nilpotency; the category of$\mathbb{Z}_2^\bullet$-manifolds is closed with respect to the tangent andcotangent functors. The $\mathbb{Z}_2^n$-supergeometric viewpoint providesdeeper insight and simplified solutions; interesting relations with QuantumField Theory and Quantum Mechanics are expected. In this article, we introducesplit $\mathbb{Z}_2^n$-manifolds as intrinsic superizations of$\mathbb{Z}_2^n\setminus\{0\}$-graded vector bundles and prove that,conversely, any $\mathbb{Z}_2^n$-manifold is noncanonically split. We thusprovide a complete proof of the $\mathbb{Z}_2^n$-extension of the so-calledBatchelor-Gawedzki Theorem.

$\mathbb{Z}_2^n$-Supergeometry II: Batchelor-Gawedzki Theorem