The Geometry of Self-Correction

This paper develops a coordinate-free geometric theory for understanding when feedback systems converge versus cycle. The central result is a diagnostic criterion: systems with curl-free correction fields exhibit pure convergence, while systems with nonzero curl cycle indefinitely. More precisely, nonzero curl obstructs monotone descent and makes limit cycles generic under perturbation. The theory uses the unit sphere as the simplest nontrivial state space, where the alignment distance between a system's current state and its target anchor serves as a natural Lyapunov function. Gradient flow on this manifold guarantees monotonic convergence—but real systems often fail to converge despite well-defined targets. The resolution lies in the correction field's differential structure: when multiple objectives, constraints, or state-dependent rules combine as interacting vector fields rather than weighted sums of scalar potentials, curl emerges and convergence guarantees dissolve. Three mechanisms introduce curl: (1) multi-objective interference through state-dependent weighting or thresholds, (2) constraints and saturation creating asymmetric correction, and (3) delays and hysteresis coupling current corrections to past states. The curl diagnostic distinguishes problems solvable by parameter adjustment (zero curl, multiple local minima) from problems requiring structural redesign (nonzero curl, architectural interference). A key quantitative result establishes that feasibility projections induce an irreducible curl floor. Using Hodge theory, we prove that when a gradient proposal passes through a non-integrable feasibility operator, the curl-maintenance functional—measuring vorticity energy—is bounded below by the spectral gap of the Hodge Laplacian times the projection defect magnitude. This makes precise the claim that cycling is structural rather than parametric: no amount of tuning can eliminate curl induced by constraint geometry. Divergence of the correction field provides a complementary diagnostic, measuring local compressive strength—where trajectories concentrate versus disperse. This translates to stability margins in control applications. Applications to control systems (FADEC multi-loop engine control) and AI alignment (reward hacking as curl emergence) demonstrate the diagnostic in concrete domains. A connection to triadic tension in constraint geometry shows how three mutually incompatible curvature sectors produce exactly the state-dependent projections that trigger the curl floor theorem. The contribution is a geometric language that reveals why some feedback architectures converge while others cycle, independent of domain—and why complexity itself tends to generate cycling. The curl diagnostic is coordinate-free, computable, and decisive.