Vague Differential Equations on Smooth Manifolds: A Foundational Framework for Vague Vector Fields, Existence, Uniqueness and Geometric Dynamics

Differential equations on manifolds provide a natural language for dynamics when the state space is curved, constrained, or intrinsically geometric. In many applications, however, the available information is incomplete, imprecise, or expressed in terms of degrees of truth and falsity rather than a single numerical value. This paper develops a rigorous framework for vague differential equations on smooth manifolds. A vague quantity is represented by a pair (t,f), where t is a truth-membership degree, f is a false-membership degree, and 0 ≤ t+f ≤ 1; the associated admissible interval is [t,1−f]. We distinguish carefully between the ordinary manifold state and vague coefficients, and then define vague scalar fields, vague tangent vectors, vague vector fields, vague differential equations, and coordinate representations. The central construction is a pair-valued differential equation whose truth and falsity components evolve according to differentiable vector-field equations, while the underlying state remains on the manifold. We establish basic invariance, coordinate covariance, local existence and uniqueness under Lipschitz hypotheses, reduction to classical differential equations, and a Grönwall-type stability estimate for the component fields. Explicit examples are given on the circle, sphere, and Euclidean charts, including a vague rotational flow on S¹ and a vague gradient-type flow on S². A numerical Euler scheme in local coordinates is also formulated, together with an admissibility condition ensuring that vague degrees remain in the unit square. The paper emphasizes that a mathematically sound vague differential-equation theory must specify its notion of derivative and must not confuse vague membership with an ordinary real-valued state. The framework is intended as a starting point for further work on vague differential geometry, vague dynamical systems, optimization, control, and uncertain geometric computation.