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

This preprint develops an entropy–spectral framework for normalized Ricci-like flows on closed three-manifolds. Spectral data from the scalar Laplacian are encoded as filtered probability laws, and the resulting geometric evolution is studied through an entropy–transport functional combining relative entropy and quadratic Wasserstein distance. The principal spectral result concerns a fixed crossing-safe infinite law on the positive Laplace spectrum. Repeated eigenvalues are handled without selecting eigenbranches: symmetric moments resolve round spectral multiplicities, while Riesz projectors provide complete-cluster observables through internal crossings. For a reflected Gaussian spectral kernel with an exponentially decaying weight flattened at the round eigenvalues, the paper derives a trajectory-wide differential spectral inequality and a mixed entropy–spectral Polyak–Lojasiewicz estimate, yielding a Duhamel-type decay law for the canonical spectral energy. This canonical result does not assume a global restricted spectral-dissipation inequality for the chosen infinite law. The manuscript also develops a separate exact normalized Ricci-flow branch. Under a uniform Euclidean lower volume bound at every radius below a fixed scale, a uniform Sobolev inequality, an upper bound for the averaged scalar-curvature normalization coefficient, and a negative-scalar form bound, recent three-dimensional noncollapsed Ricci-flow limit theory is used to obtain smooth late Cheeger–Gromov precompactness. Perelman’s lambda/nu entropy dichotomy then yields, on the closed simply connected fixed-volume branch, finite-time entry into the positive-lambda regime, selection of a compact shrinking Ricci-soliton limit, and exponential convergence modulo diffeomorphisms to the normalized round three-sphere. For general Ricci-like perturbations, the compact trajectory-envelope and tame-forcing hypotheses are retained. The paper further gives a conditional arbitrary-parameter-box extension under a global restricted spectral-dissipation hypothesis. Version v5.0 is a major mathematical revision of v4.0. It incorporates a substantially strengthened exact-flow argument based on noncollapsed Ricci-flow limit spaces, removes several previous auxiliary assumptions from that exact-flow theorem, and includes a full theorem-facing, notation, bibliography, and submission-hygiene regression. No new unconditional proof of the Poincaré conjecture is claimed. Changes in v5.0 Major revision relative to v4.0. Added a new exact normalized Ricci-flow theorem based on uniform local volume noncollapse and three-dimensional noncollapsed Ricci-flow limit-space theory. Derived full late smooth Cheeger–Gromov precompactness in the exact-flow branch from the retained local-volume, Sobolev, averaged-scalar, and negative-scalar-form hypotheses. Replaced the previous selected-time/good-slice compactness route in the principal exact-flow theorem by an F-limit tangent-rigidity argument. Removed the compact trajectory envelope (H4) and normalized-scale compactness (NS) from the exact-flow theorem. Removed the windowed traceless-Ricci input (T3b) from the exact-flow theorem; only the averaged-scalar component (T3a) remains there. Removed recurrent (L^p)-regularity, APN recurrence, and a priori compactness of Perelman minimizing scales from the exact-flow proof. Strengthened the nonpositive-(\lambda) argument using smooth late precompactness and compact steady/expanding soliton rigidity. Hardened the positive-(\lambda) shrinker-selection argument, including an exact upper bound for the (\nu)-minimizing scale from its scale-stationarity equation. Integrated and source-sealed the Fang–Li noncollapsed Ricci-flow limit-space interface and the complete three-dimensional shrinking-soliton classification. Reorganized the Introduction and theorem map to distinguish the canonical spectral theorem, the exact Ricci-flow theorem, the compact-envelope entropy route, and the arbitrary-PB conditional extension. Eliminated a notation collision between the spectral counting measure (\nu_t) and Perelman minimizing probability measures; the latter are now denoted (d\eta_t) or (d\eta_f). Performed a full cross-reference, bibliography, terminology, and journal-facing sentence audit. Corrected a hidden broken LaTeX reference and normalized bibliography metadata. Final manuscript: 159 pages. The principal canonical spectral-dissipation framework of v4.0 is retained and further integrated with the revised geometric theorem architecture. Keywords Ricci flow; spectral geometry; Wasserstein geometry; optimal transport; Perelman entropy; Ricci solitons; spectral dissipation; Laplace spectrum; Cheeger–Gromov convergence; noncollapsed Ricci flow; gradient Ricci shrinkers; entropy methods Related identifiers For the v4.0 → v5.0 version relationship, do not manually add a related identifier if v5.0 is created with Zenodo’s New version button from the existing v4.0 record. Zenodo automatically links version DOIs inside the same version chain. Previous specific version:v4.0 — DOI 10.5281/zenodo.22919817 Use the new v5.0 version-specific DOI when citing this revised manuscript after publication. License Prefer retaining the same license as v4.0 for continuity. If v4.0 was released under CC BY 4.0, retain Creative Commons Attribution 4.0 International for v5.0.