Restricted Spectral Dissipation for Ricci-Like Flows: Wasserstein Geometry, Spectral Clusters, and Round Convergence

Description This archival version develops a spectral–entropy framework for normalized Ricci-like flows on closed three-manifolds, centered on probability laws constructed from the scalar Laplace spectrum. The main spectral construction is a crossing-safe infinite probability law on the positive spectrum, obtained from a reflected Gaussian kernel and an exponentially decaying weight that is flat at the round eigenvalues. The associated entropy–transport functional combines relative entropy with quadratic Wasserstein distance. Repeated eigenvalues and spectral crossings are treated through symmetric moment coordinates and Riesz-projector complete-cluster observables, avoiding any choice of individual eigenbranches. For the canonical infinite law, the paper derives a trajectory-wide restricted spectral-dissipation inequality with an integrable defect under the standing Ricci-like hypotheses together with a supercritical curvature envelope $$\sup_{t\ge0}\Vert{}\operatorname{Rm}(g_t)\Vert{}_{L^{p_{\mathrm{curv}}}} < \infty, \qquad p_{\mathrm{curv}} > \frac{3}{2}.$$ A mixed entropy–spectral coercivity estimate then yields Duhamel decay of the canonical spectral energy. Reflected-Gaussian identifiability and Weyl asymptotics recover the normalized positive spectrum, while an additional compact moduli envelope converts spectral convergence into all-late Cheeger–Gromov round entry. On the fixed-volume simply-connected branch, the same supercritical curvature envelope yields an eventual positive gap for Perelman’s $\lambda$-functional. The scale-free $\nu$-entropy then supplies compact minimizing scales and finite total shrinker production, so the principal canonical branch does not separately assume the entropy-production or renormalized-scale interfaces. The paper also contains an independent selected-time entropy/topology route. Finite shrinker production and sliding-window curvature identities produce slices with vanishing shrinker production and uniform $L^2$-curvature control; together with all-scale noncollapsing, this yields $W^{2,2}$ harmonic-radius compactness and a smooth shrinking-soliton subsequential limit. In the simply-connected three-dimensional setting, the limiting soliton has positive constant curvature. Additional results include multiplicity-safe Wasserstein pullback geometry, exact Riesz-cluster variation formulas, Berger-sphere tests demonstrating the limitations of ordered spectral blocks and naive pointwise spectral energies, and a conditional extension to general compactly supported spectral parameter boxes. This version supersedes the earlier public v5.1 spectral manuscript and serves as the final archival version of the spectral/entropy branch of the project. The later multiscale curvature–entropy minimizing-movement program is developed separately and is not part of this archival version. No new unconditional proof of the Poincaré conjecture is claimed. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords normalized Ricci flow; spectral geometry; Laplace spectrum; relative entropy; Wasserstein distance; spectral crossings; Riesz-projectors; Perelman entropy; shrinking solitons; Cheeger–Gromov convergence; Sobolev space