Dynamic triunity: universal dynamics in integrable systems, quantum groups, and geometry of curves (Part III)

This work continues the series of studies devoted to the principle of geometric polyvariance of algebraic equations. It marks a transition from a static description of mathematical models to a dynamic one. A universal deformation parameter ℏ is introduced, interpreted as time, a quantization constant, or a modulus parameter. Families of isomorphisms are constructed for four key models, which obey a universal differential equation. The existence of a single flat connection on the universal coefficient ring, consistent with all morphisms of the extended category, is proven. To systematize the dynamic behavior, three new classificatory axes are introduced, describing the type of evolution, the role of the parameter, and the degree of integrability. The results prepare the groundwork for constructing a periodic system of mathematical models in the final article of the cycle.