Unified Physical Theory Based on Differential Algebra: Constructive Derivation and Proof from First Principles

This paper establishes a unified physical theory framework based on differential algebra. Starting from first principles, we incorporate classical physics, quantum physics, relativity, quantum field theory, and gravitational theory into a unified mathematical formulation through recursive closure construction and structured expansion.This framework not only recovers traditional theories as special cases but also derives new results and handles high-dimensional complex situations. We propose a rigorous construction method for physical differential closure and prove its existence and uniqueness using the Wilsonian renormalization group in a precisely defined Banach space of effective actions. The unified equations of motion for field theories in arbitrary dimensions are derived,with explicit formulas for combinatorial correction coefficients obtained from representation theory of symmetric and orthogonal groups. An algebraic classification and regularization scheme for physical singularities are established, constructively realizing quantum-classical correspondence through the BRST cohomology and variational bicomplex on jet bundles. A physical computation verification framework based on interval arithmetic is proposed, and an automated derivation and verification system is designed. This framework provides a concrete,rigorous, and computationally feasible mathematical implementation for solving Hilbert’s Sixth Problem.