Relational Mapping Analysis: A Systematic Geometric Organization of Classical Relation Criteria

Determining the relationship between two or more functions traditionally requires a collection of largely independent tools: quotient functions for scaling, Wronskians for linear dependence, Schwarzian derivatives for Möbius equivalence, and the argument principle for zero counting. This paper develops Relational Mapping Analysis (RMA), a systematic geometric organization that recasts these classical criteria as invariants of a single geometric object—the relational manifold Mf1,…,fn={(f1(x),…,fn(x)):x∈Ω⊆Rm}.Mf1,…,fn={(f1(x),…,fn(x)):x∈Ω⊆Rm}. We present the analysis in four layers: (1) local geometry (slope, curvature); (2) algebraic structure (implicit polynomials, Jacobian rank, Constant Rank Theorem); (3) integral analysis (Stokes' theorem and area criteria); and (4) complex analytic rigidity (Cauchy integrals, argument principle, Schwarzian derivative). The multi-channel relational energy E(f,g)=Elin(f,g)+λ Eaff(f,g)E(f,g)=Elin(f,g)+λEaff(f,g) separates linear from affine relations and offers a quantitative refinement of classical criteria. We prove its energy–rigidity equivalence (Theorem 5.2), a conditional asymptotic linear dependence theorem (Theorem 5.3), and a corrected second-order perturbation expansion near a linear relation (Theorem 5.7). We also address two problems posed in earlier versions: a Cauchy representation of the Schwarzian difference (Proposition 6.7, a direct consequence of Cauchy's formula) and the classical nn-function Wronskian criterion (Theorem 6.8, due to Bôcher and Ince), for which the double-contour-integral encoding remains open. RMA is presented as a systematic organization and quantitative refinement of classical criteria—not as a replacement for them.