Sigma–Schmidt Duality: Definition, Signature Formula and Geometric Conservation Law for a Dual Universe

This paper introduces the Sigma–Schmidt Duality, a geometric and topological model describing cosmic expansion and gravitation as two complementary singularities within a closed toroidal field system. 

The framework postulates that the Universe operates through dual flows of expansion (Sa) and extraction (Se), dynamically coupled by a time-dependent Sigma function σ(t). This coupling preserves global conservation while allowing local asymmetry, unifying cosmological expansion and gravitational attraction as reciprocal phenomena. 

The Sigma–Schmidt model extends Friedmann cosmology by integrating a geometric conservation law and a testable coupling term that connects σ(t) with H(z), ρ(z), and Λ_eff(t). Its mathematical structure provides both an intuitive visualization and empirical testability against current cosmological datasets (Pantheon+, DESI, Planck).

The paper is presented in two languages (German and English) and is intended to invite scientific collaboration and validation of the Sigma–Schmidt Duality as a potential extension of ΛCDM cosmology.