Global Regularity of the Three-Dimensional Incompressible Navier–Stokes Equations in a Compactified Prism Framework Complete Proof and Lean 4 Formalization Skeleton for the Projection of the Universal Transport Equation onto the Three-Sphere

Within the Prism Unified Field Theory (PUFT) framework, this paper gives a com plete proof of global regularity for the three-dimensional incompressible Navier–Stokes equations on the compactified geometry 𝕊 3 . This paper is an explicit expansion of the “Navier–Stokes existence and smoothness” coordinate chart ΣNS in the PUFT paper (DOI: 10.5281/zenodo.22317006). Three core results: 1. Projection access: We explicitly restrict the PUFT projection operator onto 𝕊 3 , ob taining the fluid projection equation of UTE; through the welding map defined in Definition 4.3, we align this equation with the PUFT welding compatibility theorem. 2. Curvature dissipation: Using the constant-curvature property Ric = 2𝑔 on 𝕊 3 , the Bochner identity gives exponential decay at the 𝐿 2 level: ‖𝑢(𝑡)‖2 𝐿2 ≤ ‖𝑢0 ‖ 2 𝐿2𝑒 −4𝜈𝑡 . 3. Global regularity: At the 𝐻1 level, we introduce the higher-order dissipation term induced by the welding map, and close the energy cascade via the Ladyzhenskaya and Young inequalities, obtaining a global a priori bound, and hence a global smooth solution via the BKM criterion. We also give a Lean 4 formalization skeleton, in which spectral flow conservation and welding compatibility reference the formalization modules of PUFT v2; the analytic kernel (Gagliardo–Nirenberg interpolation and the energy cascade) is marked as unproven target. 本文在棱镜统一场论(Prism Unified Field Theory, PUFT)框架下,给出三维 不可压缩 Navier–Stokes 方程在紧致化几何 S 3 上全局正则性的完整证明。本文是 PUFT 论文(DOI: 10.5281/zenodo.22317006)中“Navier–Stokes 存在性与光滑性” 坐标卡 ΣNS 的显式展开。 三个核心结果: 1. 投影接入:将 PUFT 的投影算子 ΠΣNS 显式限制在 S 3 上,得到 UTE 的流体 投影方程;通过定义 4.3 中的焊接映射 Θblowup,使该方程与 PUFT 的焊接相 容性定理(PUFT 定理 4.3)一致。 2. 曲率耗散:利用 S 3 上 Ric = 2g 的常曲率性质,Bochner 恒等式给出 L 2 层面 的指数衰减 ∥u(t)∥ 2 L2 ≤ ∥u0∥ 2 L2 e −4νt。 3. 全局正则性:在 H1 层面引入 Θblowup 诱导的高阶耗散项 A(u) = α Θ∗ blowup∆MΘblowupu, 通过 Ladyzhenskaya 不等式与 Young 不等式闭合能量级联,得到全局先验界, 进而由 BKM 判据给出全局光滑解。 本文同时给出 Lean 4 形式化骨架,其中谱流守恒与焊接相容性引用 PUFT v2 的形式化模块;解析内核(Gagliardo–Nirenberg 插值与能量级联)标注为待证目标。