Entropy–Spectral Contraction and Rigidity for a Non-Surgical Ricci-Flow Route to the Poincaré Conjecture

This upload contains Version v3.2r5 of the manuscript: **Entropy-Spectral Contraction and Rigidity for a Non-Surgical Ricci-Flow Route to the Poincare Conjecture** The manuscript develops a non-surgical entropy-spectral theorem package for closed, orientable, simply-connected 3-manifolds under the standing normalized tame-flow hypotheses and parameter-box framework fixed in Section 2. The main mechanism associates to a normalized Ricci-type flow a family of filtered spectral probability measures and studies the unified energy E(μ) = KL(μ μ_∞) + α I(μ) + β W₂²(μ,μ_∞). Within the fixed parameter-box framework, the paper proves a local \(W_2\)-\(λ\) convexity estimate and combines it with a modulated EVI realization theorem for the geometric spectral curve. The resulting energy decay has the defect-absorbed form E(μₜ) ≤ E(μ₀)e^-λPBt + ∫₀ᵗ e^-λPB(t-s)R(s)\,ds. The manuscript also includes a framework-internal fixed-scale Perelman-\(μ\) route to a time-averaged logarithmic Sobolev inequality, yielding Gaussian heat-kernel bounds and spectral-tail control. These estimates feed the spectral-to-geometric bootstrap, producing convergence, up to diffeomorphisms and endpoint extraction, to a constant-curvature endpoint. In the final endgame, the topology route is separated from the spectral rigidity seal. The constant-curvature endpoint is identified directly by the standard simply-connected space-form argument, while the finite-fingerprint theorem package is retained as a spectral rigidity and consistency seal for the same round endpoint regime. Version v3.2r5 incorporates the following refinements: - synchronization of the endpoint-identification route with the space-form lemma;- preservation of the finite-fingerprint theorem as a spectral consistency seal rather than the sole endpoint identifier;- canonical parameter-box reduction for one execution of the closure theorem;- normalized-scale trapping inside the standing tame-flow framework;- a defect-ledger decomposition for the modulated EVI bridge;- first-cluster norm-equivalence and Gram-observability thickening;- public-facing tone polish, including removal of implementation/dashboard-style language. The result should be read as a theorem-level non-surgical closure package inside the standing normalized tame-flow framework, not as a claim independent of that framework. The broader remaining direction is the generation of the Section 2 standing tame-flow package from arbitrary initial data and further presentation-independent sharpening of the PB-positivity mechanism.