A Mathematical Framework for Unconventional Superconductivity Beyond BCS Theory

We present a rigorous mathematical framework for modeling unconventional superconductivity beyond the standard Bardeen-Cooper-Schrieffer (BCS) theory. While BCS theory successfully explains conventional superconductors through phonon-mediated electron pairing, it fails to capture key phenomena in high-temperature and strongly correlated superconducting systems. In this work, we construct a non-BCS theoretical model rooted in advanced mathematical formalisms, incorporating non-trivial symmetries, non-local interactions, and topological considerations. The model is developed from first principles, without reliance on empirical fits or experimental parameters. We derive a new class of equations governing the superconducting order parameter and explore their implications through analytical techniques and graph-based representations. Our results suggest the emergence of superconducting states with properties incompatible with BCS-type behavior, offering insights into possible mechanisms behind high-temperature or exotic superconductors. This framework provides a foundation for further theoretical exploration and may guide future experimental inquiries into non-phononic superconductivity.