Constructive Representation of Multivariate Continuous Functions: A Unifying Program Based on Closure Extensions

This paper aims to generalize the “differential algebraic closure–specialization homomorphism” framework, developed for solving Hilbert’s 13th Problem (representation of the polynomial root map), to the constructive representation problem of general multivariate continuous functions. We follow the four-step methodology of the “Constructive-Analytic Grand Unification” program: (1) Closure Relativity Principle: clarify that the “representability” of a function is relative to the allowed operational closure; (2) Key- Point Differential Forms: select representative points in the function’s domain and extract differential information of the function as the basic “alphabet” for representation; (3) Stratified Closure Construction: systematically adjoin generators to construct a “universal representation closure”; (4) Symmetry-Driven Solution Representation: organize the standard form of representation using intrinsic symmetries of the function. We first elucidate, using the polynomial root map as an example, how it serves as a perfect prototype for this program. Subsequently, we outline the path to apply this program to general multivariate continuous functions, the core of which lies in constructing a “functional-differential closure”. This generalization not only provides a potential constructive version of the Kolmogorov–Arnold representation theorem for multivariate functions but also incorporates the function representation problem into a unified, operational algebraic framework, promising to bridge the gaps between analysis, algebra, and computational mathematics.