Vitrifying Levinthal's Paradox: A Geometric Constraint Construction

Levinthal's paradox highlights the apparent contradiction between the exponential size of protein conformational space and the empirically observed rapid convergence of protein folding. Traditional resolutions invoke energy landscapes, heuristics, or biologically encoded bias. This work demonstrates an alternative approach: transforming the paradox from a search problem into a geometric constraint problem. Using a minimal dynamical system (illustrated on \( M = S^1 \)), we show that exponential path freedom can be collapsed without encoding a solution or target state by introducing a path-dependent curvature penalty. Convergence emerges not from guidance toward a minimum, but from the suppression of backtracking and oscillatory trajectories. The paradox is not solved algorithmically; it is vitrified—frozen into geometric structure. Epistemic StatusThis document presents an engineered mathematical construction. It does not claim empirical validity for biological protein folding, nor does it propose a universal physical law. It provides an existence proof of exponential search collapse via geometry and extracts a transferable design pattern for synthetic systems. Relation to Previous WorkThis paper builds on a structural kernel inherited from earlier exploratory work (see doi:10.5281/zenodo.18449902).