Spectral Dissipation and Round Convergence for Ricci-Like Flows on Closed Three-Manifolds

Description We develop a spectral–geometric framework for normalized Ricci-like flows on closed three-manifolds using probability laws constructed from the scalar Laplace spectrum. The basic energy combines relative entropy with quadratic Wasserstein distance, and its restriction to the metric-induced spectral curve leads to an intrinsic projected spectral gradient. Repeated eigenvalues and spectral crossings are treated without choosing individual eigenbranches. Symmetric moment coordinates resolve the round multiplicity cone, while Riesz-projector traces provide complete-cluster observables that remain well defined through internal crossings. The principal result concerns a fixed crossing-safe infinite probability law on the positive spectrum, built from reflected Gaussian kernels and an exponentially decaying spectral weight flattened at the round eigenvalues. Under the stated normalized Ricci-like hypotheses, compact renormalized-scale control, tame forcing, and a compact $C^{2,\alpha}$ Cheeger–Gromov trajectory envelope, we derive a trajectory-wide differential restricted spectral-dissipation estimate $$\frac{d}{dt}E_{\mathrm{can}}(t) \le -\eta_{\mathrm{glob}}\widehat G(t)+\varepsilon_{\mathrm{raw}}(t), \qquad \varepsilon_{\mathrm{raw}}\in L^1(0,\infty).$$ A mixed entropy–spectral Polyak–Łojasiewicz estimate then yields a Duhamel decay law for the canonical energy. Reflected-Gaussian identifiability together with Weyl asymptotics recovers the normalized positive spectrum, and compact moduli separation converts convergence of the canonical spectral law into entry into the round Cheeger–Gromov regime. The paper also develops an independent geometric route for the exact volume-normalized Ricci flow. Under explicit noncollapsing, Sobolev, scalar, and negative-scalar form hypotheses, current three-dimensional noncollapsed Ricci-flow limit theory yields smooth late precompactness; Perelman’s entropy framework and three-dimensional shrinking-soliton rigidity then give exponential convergence to the round metric. For arbitrary compactly supported finite parameter boxes, the corresponding global contraction theorem remains conditional on an additional restricted spectral-dissipation hypothesis. Berger-sphere calculations illustrate why ordered spectral blocks and several natural pointwise spectral quantities do not provide a universal global monotonicity principle. The release includes Supplementary Material S1, RSD_CANONICAL_HEAD_CERT_v1.0_2026-09-24.zip, containing the deterministic 256-bit MPFR directed-rounding verification used for the finite-head numerical certificate in the canonical infinite-law argument. The higher spectral tail is controlled analytically. 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; Polyak–Łojasiewicz inequality; Cheeger–Gromov convergence; shrinking solitons; MPFR verification