v-Calculus on ℂⁿ : Algebraic Structure, Integration, Constraints, Taylor Series, Differential Equations, Elementary Functions, and Extension to Higher Dimensions

We develop a calculus on open subsets of ℂⁿ intermediate between real differentiability and complex analyticity. Given a smooth nonvanishing field v:U→ℂⁿ\{0}, a function φ:U→ℂ is called v-differentiable if dφ(v)(z) ∈ ℝ for all z ∈ U. The v-derivative ∂_v φ(z) := dφ(v)(z) ∈ ℝ produces a real number from a complex function in any dimension. We establish: (i) algebraic structure of D_v(U); (ii) a fundamental theorem of integration recovering only the real part, with imaginary part frozen as topological invariant Ω_γ; (iii) classification of fields v by integrability, normalization, and residue structure; (iv) v-Taylor series with real coefficients and explicit radius of convergence; (v) v-differential equations reducing via w = Φ(z) to real first-order ODEs parametrized by Ω_γ, unifying the A_n hierarchy as solutions of ∂v_nφ = -1; (vi) a complete family of v-elementary functions; and (vii) extension to ℂⁿ, revealing a dimensional reduction theorem. The maximal bifurcation theorem — that Ω_γ is invisible to every operation of the v-calculus — holds in all dimensions.