Geodesic strong convexity does not imply forward invariance under gradient flow on SO(3): a certified counterexample

Abstract

Let $mathcalC={\overline{\mathcal{B}}}_{\rho }(R_c)$ be a geodesic ball of radius $\rho <\pi /2$ in SO(3) with the bi-invariant metric, and let $f$ be geodesically strongly convex on $\mathcal{C}$ with an interior minimizer. It is tempting to expect the gradient flow $\dot{R}=R(-\nabla f{)}^{\wedge }$ to keep $\mathcal{C}$ forward invariant: the flow is attracted to an interior point, and strong convexity appears to leave no room for outward motion. We show this expectation is false by an explicit, fully certified construction with $\rho =0.3$: a cost, quadratic in the principal logarithmic chart with off-diagonal coupling $0.7$, whose geodesic Hessian satisfies $\Hess f\succeqμI_3$ on all of $\mathcal{C}$ with a machine-certified modulus $\mu \ge 0.172$, rigorous ball arithmetic over exact rational inputs, yet whose descent velocity at a boundary point has the exact rational outward radial component $21/500$. A continuity corollary of the exact rate certifies that the flow exits the ball; numerical integration puts the peak excursion near $0.3143$ before convergence to the minimizer. The mechanism is elementary: strong convexity constrains the projection of the gradient onto the minimizer direction, not onto the inward radial direction. Code reproducing every certified constant and figure accompanies the note.