Scientific Development as Lyapunov Descent in Theory-Space

Building on a bounded completeness functional defined on differentiable theory-space, this paper models scientific refinement as a hybrid dynamical process combining continuous parameter adjustment and discrete structural augmentation. Under admissible update operations inherited from a prior structural framework, it is shown that the completeness functional is non-increasing along refinement trajectories, establishing it as Lyapunov-like for theory development. Sufficient conditions for strict descent are derived, structural equilibria are characterized, and local stability is analyzed through the spectrum of the linearized refinement operator. Eigenvalue sign changes correspond to bifurcations in theory-space, yielding a geometric reinterpretation of theoretical crisis. The analysis clarifies the distinction between local descent and global convergence: although incompleteness does not increase under admissible refinement, non-convexity permits multiple local minima. The result provides a formal dynamical reconstruction of scientific development and prepares a subsequent limit-based criterion for scientific validity.