A Proposed Unified Non-Equilibrium Phase-Space and Differential Galois Framework for Superconductivity

The microscopic mechanisms underlying low-temperature (BCS-type) and high temperaturesuperconductivity remain fundamentally distinct in standard theory. While conventional superconductors arewell described by Bardeen–Cooper–Schrieffer (BCS) theory, high- materials, 𝑇𝑐 such as cuprates, requireframeworks involving strong correlations, non-locality, and non-equilibrium effects. This work proposes aunified theoretical framework combining BCS mean-field theory, Keldysh non-equilibrium formalism, Moyalphase-space quantization, and differential Galois theory. It is argued that superconductivity can be interpreted asan emergent integrable regime of a non-commutative quantum transport equation, characterized by a reductionof the differential Galois group. The paper derives the formal mathematical structure, identifies symmetryreductions corresponding to low- and high-𝑇𝑐 regimes, and demonstrates two explicit numerical modelsillustrating the transition between integrable and non-integrable regimes.