On the Topological Foundations of Continuous Dimensional Spaces: Formalising $\mathbb{R}^\alpha$ Manifolds via Radian Parametrisation

This paper introduces a formal mathematical framework that challenges the geometric rigidity of classical Euclidean spaces R^n. We propose that the conventional integer-dimensional space R^3 is merely a static projection of a continuous, non-rigid topological domain. By extending the dimensional exponent to the positive real continuum R^α (where α ∈ R^+), we construct an elastic coordinate system where the dimension behaves as a continuous variable. Crucially, we formalise the mapping mechanism by setting the dimension α as a direct function of angular parameters measured in radians, α = ϕ(θ). This approach unlocks a new class of fractional-dimensional manifolds, providing the foundations for continuous spatial operators and non-rigid boundary modeling in complex physical and analytical systems.