Dynamic Complex Analysis: A Self-Sufficient Mathematical System

Abstract The core tool of classical complex analysis---contour integration---suffers from a fundamental dilemma: Where does the integrand come from? Why is it constructed in this way? These questions have never been answered. This paper proposes a radical paradigm shift: Dynamic Complex Analysis is not a "dynamical extension" of classical complex analysis, but a self-sufficient mathematical system. It is derived rigorously from the information pixel axiom system, and its legitimacy does not depend on correspondence with classical complex analysis. The core contributions include: establishing the phase coupling equation as the fundamental dynamics; introducing the phase closure integral to define topological charges that distinguish closed, semi-closed, and chaotic states; constructing spiral geometry to replace the two-dimensional complex plane, with Theorem 3.1 (curvature-torsion coupling) proving that the integral of the torsion-to-curvature ratio along a closed spiral is constant. The most constructive conclusion of this paper is the Degeneration Theorem: classical complex analysis is a special case of dynamic complex analysis in the steady-state limit; real analysis is a further degeneration of complex analysis under phase quotienting. The Fundamental Theorem of Calculus is a corollary of the Residue Theorem. The continuity of real numbers is not a presupposition but a macroscopic emergence after the decoupling of fast and slow variables and the compression of phase information by Φ. Traditional concepts such as Dedekind completeness, Lebesgue integration, and Weierstrass functions are unnatural decoupled states in generative dynamics and are not included in this system. Core conclusion: Dynamic complex analysis is a self-sufficient mathematical system. Classical complex analysis is not its foundation but its special case. Real analysis is the phase quotient of complex analysis. Keywords: dynamic complex analysis; phase coupling equation; topological charge; spiral geometry; degeneration theorem; real analysis degeneration; self-sufficient system