A Precise Neutrosophic β^∗-Equivalence Relation, Quotient Lie Hyperalgebras, and BCH Theory

We introduce and study a precise fundamental relation for a class of neutrosophic Lie hyperalgebras. The construction separates two issues which are often conflated in the literature: the classical hypercompositional fundamental relation and the neutrosophic information carried by the elements. An element is represented by a neutrosophic profile (T, I, F) ∈ [0, 1]³, and a profile-compatible relation is imposed on the elementary β-relation before taking its transitive closure. We prove a quotient theorem under an explicit strong regularity hypothesis: the quotient by the resulting relation is a well-defined ordinary Lie algebra and the canonical projection is a good hyperalgebra homomorphism. We then define a quotient-level neutrosophic Baker–Campbell–Hausdorff operation and prove its independence of representatives. In the nilpotent class-two case the formula reduces to the expected finite expression. Several examples are given, including central neutrosophic thickening of a Lie algebra, a Heisenberg-type example, and a matrix realization. Counterexamples show that profile compatibility, strong regularity, and quotient descent cannot be omitted. The resulting framework provides a precise bridge between neutrosophic hyperstructures and classical Lie theory.