Perelman Elevation Scheme — Functorial Elevation from the Ricci-Flow–Geometrization Program to the 𝒯⁶⁴ Topological Ontology

Perelman’s proof of the Poincaré conjecture and Thurston’s geometrization conjecture constitutes a core achievement of three-dimensional topology and geometric analysis. This paper proposes a functorial elevation scheme: the Ricci-flow–geometrization program defined on the category of three-manifolds is elevated to an evolution–decomposition structure defined on the 64-dimensional compact-torus topological ontology \(\mathcal{T}^{64}\). The paper supplies the formal framework of the elevation: a functor \(\Phi\) between the category \(\mathcal{M}_3\) of three-manifolds and the category \(\mathcal{F}_{64}\) of projected fibres over \(\mathcal{T}^{64}\); the image of the Ricci-flow equation under \(\Phi\); the classification of the eight three-dimensional model geometries as fibre-cell types under \(\Phi\); and a monotonicity lemma for the evolution functional obtained as the image of the entropy-monotonicity formula under \(\Phi\). Further deepened results are given: the variational origin \(S_{\mathrm{fib}}\) of the non-flat correction term \(\Lambda_{AB}=\kappa\theta\omega_{AB}\); the coordinate form of \(\Lambda_{AB}\); the explicit \(\Gamma_2\) expansion of the Bakry–Émery condition; the functoriality of the sheafified \(\Phi^+\); a Lean-4-compilable skeleton together with a fill-in checklist; and the logical implication graph of P1–P7 together with joint manifestation completeness \(P_{\mathrm{joint}}^{\mathrm{logic}}\approx0.840\). The methodological foundation consists of three core principles: the Fingerprint Principle, the Meta-language Criterion and the Manifestation Boundary. The paper does not modify Perelman’s proof itself; it supplies a re-location of that proof on the \(\mathcal{T}^{64}\) ontology. 佩雷尔曼对庞加莱猜想与瑟斯顿几何化猜想的证明,是三维拓扑与几何分析的核心成果。本文提出一个函子式提升方案:把定义在三维流形范畴上的里奇流—几何化纲领,提升为定义在 \(\mathcal{T}^{64}\) 64 维紧致环面拓扑本体上的演化—分解结构。 本文给出该提升的形式框架:三维流形范畴 \(\mathcal{M}_3\) 与 \(\mathcal{T}^{64}\) 上的投影纤维范畴 \(\mathcal{F}_{64}\) 之间的函子 \(\Phi\);里奇流方程在 \(\Phi\) 下的像;8 种三维标准几何结构在 \(\Phi\) 下的纤维胞腔分类;熵单调公式在 \(\Phi\) 下的演化泛函单调性引理。 本文进一步给出深化结果:非平坦修正项 \(\Lambda_{AB}=\kappa\theta\omega_{AB}\) 的变分来源 \(S_{\mathrm{fib}}\);\(\Lambda_{AB}\) 的坐标形式;Bakry–Émery 条件的 \(\Gamma_2\) 显式展开;层化 \(\Phi^+\) 的函子性;Lean 4 可编译骨架与待填清单;P1–P7 的逻辑蕴含图与联合显影完整度 \(P_{\mathrm{joint}}^{\mathrm{logic}}\approx0.840\)。 本文的方法论地基是三条核心原则:指纹原理、元语言判据、显影边界。本文不修改佩雷尔曼证明本身;本文给出的是该证明在 \(\mathcal{T}^{64}\) 本体上的重新定位。