Fixed Points IV: Curvature, Centroid Motion, and Structural Transitions

We develop curvature coupling and structural transitions on the canonical Fixed Points I–III spine. A Laplace-type operator is separated into its connection Laplacian and curvature contribution. Curvature can shift the spectral cluster, mix coherent and noncoherent sectors, and create a leakage floor. When the curvature term is smaller than half the unperturbed spectral gap, the full curved operator retains a separated low cluster and defines a new Riesz projector; retaining the unperturbed projector instead requires an explicit leakage term. Centroids are defined intrinsically by a Karcher mean inside a convex normal ball. A first-order gradient parent flow yields a first-order modulation law, while a Newtonian law requires a separately specified inertial parent equation. Absolute interaction bounds do not determine attraction or repulsion without a sign hypothesis. Structural transitions are controlled by a Lyapunov mountain-pass and work theorem, and exit detection is separated from selection of the post-transition basin. We also state the projective-module transport and finite-reduction boundary: a supplied connection and its Hessian transport naturally, bare compression is exact only on an invariant subspace, and otherwise the Feshbach or reduced-Green operator is required.