This paper establishes a unified mathematical physics framework based on differential algebraic closures, starting from first principles. We first rigorously define a recursive construction method for differential-algebraic closures, proving their existence and uniqueness within an appropriate categorical framework. We then develop a complete classification theory for closures, providing a full system of invariants including differential dimension, geometric rank, branching type, and combinatorial spectrum, and proving classification theorems for finite-type and infinite-type closures. We deeply investigate the arithmetic properties of combinatorial coefficients, proving their algebraicity over number fields and establishing profound connections with Diophantine equations, including a precise derivation of height bounds and a formulation of acombinatorial ABCconjecture. Finally, we apply this framework to quantum mechanics, general relativity, and statistical physics, achieving a differential-lgebraic formulation of quantum mechanics, a closure description of spacetime singularities, and a natural interpretation of phase transitions as branching choices. This work not only unifies several mathematical physics domains but also derives new mathematical results and physical insights. All constructions are presented with rigorous proofs or complete proof sketches, and the framework’s computational aspects are illustrated through explicit algorithms.
shifa liu (Wed,) studied this question.