Fixed Points IV: Curvature, Centroid Motion, and Structural Transitions on Bundle Manifolds

We extend the FP I–III framework to curved multi-bundle manifolds and develop a rigorous account of: (i) curvature corrections to damping via an explicit curvature remainder tied to bounded-geometry data; (ii) centroid dynamics from first principles, deriving a geodesic-type equation with a variationally defined effective mass M_eff; (iii) quantitative overlap functionals with Lipschitz control and an exact energy identity with controlled curvature remainder; and (iv) structural transitions (merging, barrier crossing, fragmentation) with quantitative criteria proved via min–max barriers and concentration–compactness. Explicit constants are given for spheres, compact hyperbolic quotients, and product spaces. The analysis is self-contained and consistent with the coherent-sector fixed-point framework of FP I, the 10D manifold model of FP II, and the disturbance/damping analysis of FP III.