Multiscale-Entropy Minimizing Movements and Critical-State Rigidity on Three-Manifolds

We develop a nonsurgical variational framework on the fixed-volume space of metrics on a closedthree-manifold. The functional combines the three-dimensional curvature energy, theEinstein-Hilbert functional, and a multiscale Perelman entropy barrier. Controlled sublevels yieldharmonic-radius compactness; in the Sobolev window \(3/2<s<2\) the quotient metric space admitsall-time generalized minimizing movements, and fixed-step proximal trajectories have criticalomega limits satisfying a fixed-volume Euler-Clarke equation. For homogeneous three-spheres, the normalized-Ricci-flow sign is governed by the sharp threshold\(12/11^{1/3}\), giving uniqueness of the round metric among all left-invariant fixed-volumecritical metrics for sufficiently small entropy weight. In the overlap range\(12/11^{1/3}<\alpha r^2<28\), a full round-sphere Hessian audit and gauge-fixed analyticimplicit-function argument remove the symmetry restriction locally: the round metric is locallyisolated, modulo diffeomorphism, among all nearby smooth fixed-volume critical metrics.No unconditional proof of the Poincare conjecture is claimed.