A Fundamental Principle for the Extension and Restriction of Variables

This paper introduces a novel mathematical structure that extends concepts previously articulated in different forms within theorems such as Löwenheim-Skolem in model theory, Hahn-Banach in functional analysis, and embedding theorem in algebra. While those earlier theorems applied the structure solely to objects, this paper establishes a framework that applies it to variables, yielding fascinating and remarkable results, most notably the transfer of properties and information from one system to another. Consequently, there is no longer a need to formulate and prove calculations from scratch for a new system; this approach saves mathematicians years of effort otherwise spent on proving redundant theorems. Furthermore, the framework enables the compression of massive, interdependent, multi-variable equations, facilitating rapid solutions. It also allows for the significant compression of digital data or mathematical formulas based on their intrinsic, latent similarities, thereby drastically reducing bandwidth and storage requirements. Ultimately, this is a general framework derived from fundamental theorems, applicable across all branches of mathematics. It is hoped that this work will serve as a step toward establishing a common language among these diverse fields and uncovering the hidden relationships connecting them.