Proof Dynamics Theory: A Dynamic Systems Approach to Mathematical Proof Evolution, Optimization, and Compression

Traditional mathematics views proofs as static, immutable artifacts, overlooking the dynamic evolutionary processes by which they are discovered, optimized, and compressed over time. In this paper, we introduce Proof Dynamics Theory (PDT), a novel theoretical framework that models the mathematical proof process as a continuous or discrete dynamic system denoted by P(t). By defining a rigorous state space S for mathematical proofs, evolutionary operators, and proof energy functionals E(P), PDT formalizes how proofs evolve, optimize, and compress. We establish the foundational mathematics of proof trajectories, analyze the stability and convergence of deductive pathways, and propose a metric system for measuring proof entropy and informational density. This framework establishes the foundational groundwork for Proof Science, shifting the paradigm from static verification to dynamic optimization of mathematical knowledge.