Proof of the High-Dimensional Kakeya Maximal Conjecture (n ≥ 4) Based on the UTE General Transport Equation Framework

Abstract

The high-dimensional Kakeya maximal conjecture is a central open problem in harmonic analysis and geometric measure theory, systematically formulated by Fefferman in 1971. The two-dimensional case was proven by Davies in 1971; the three-dimensional case was completed by Wang and Zahl in 2025; however, all high-dimensional cases n ≥ 4 have remained unproven. A Kakeya set is a compact set in R n that contains a unit line segment in every direction on the unit sphere S n−1 . The conjecture asserts that even if such a set has zero Lebesgue measure, its Hausdorff dimension must equal the ambient dimension n. Traditional methods are confined to the fixed Euclidean metric framework, relying on multi-scale tube coverings and combinatorial incidence counts, yielding only lower bounds on the Hausdorff dimension. In high dimensions, the intersection patterns of directional line segments are highly complex, and multi-scale decompositions incur exponential complexity growth, preventing direct attainment of the full dimension. This paper introduces the UTE (Universal Transport Equation) as a unified tool, con structing a one-parameter Riemannian metric deformation flow that continuously and smoothly evolves ellipsoidal configurations to spherically symmetric geometry. The main innovations include: 1. A complete tensor expression for the UTE metric deformation flow ∂tgij = −2 Ricij +∇i∇jϕ− λgij ; 2. Proof of uniform bi-Lipschitz geometric transfer and Hausdorff dimension invariance; 3. Introduction of curvature-induced phase dissipation ensuring conservation of the inci dence integral during deformation; 4. Utilization of high-dimensional incidence invariance and Fubini-type projection section estimates to bypass the obstacles of traditional multi-scale methods; 5. Rigorous proof of finiteness, positivity, and conservation of I[Kt], closing the contra diction chain. Under no additional assumptions, we close all gaps and rigorously prove that the Kakeya maximal conjecture holds in R n for all n ≥ 4. A Lean 4 formal verification skeleton is embedded to enable computer-verifiable proof logic. This result completes the full-dimensional picture of Kakeya theory and provides new geometric tools for endpoint estimates of oscillatory integral operators, Fourier restriction theory, dispersive partial differential equations, and fractal projection theory. 高维 Kakeya 极大猜想是调和分析与几何测度论中长达半个世纪的核心公开问题,由 Fefferman 于 1971 年正式系统化提出。二维情形由 Davies 于 1971 年证明;三维情形由王虹 与 Zahl 于 2025 年完成证明;而 n ≥ 4 的全部高维情形至今未有完整证明。Kakeya 集合是 指 R n 中一个紧致集合,对单位球面 S n−1 上每一个方向,集合内均包含一条该方向的单位 长度线段。猜想断言:即便该集合的勒贝格测度可以为零,其豪斯多夫维数必须等于全空间 维数 n。 传统方法局限于固定欧氏度量框架,依靠多尺度管道覆盖与组合相交计数,仅能得到豪 斯多夫维数的下界。高维空间中方向线段相交模式高度复杂,多尺度分解带来指数级复杂度 增长,现有方法只能不断改进下界,无法直接得到满维结论。 本文引入 UTE 通用输运方程作为统一工具,构造单参数黎曼度量形变流,实现椭球构 型向球对称几何的连续光滑演化。主要创新包括: 1. 给出 UTE 度量形变流的完整张量表达式 ∂tgij = −2 Ricij +∇i∇jϕ − λgij; 2. 证明一致双 Lipschitz 几何传递与豪斯多夫维数不变性; 3. 引入曲率诱导相位耗散机制,保证形变过程中入射计数积分守恒; 4. 利用高维入射计数不变性与 Fubini 型投影截面估计,绕过传统多尺度方法的高维障碍; 5. 严格证明 I[Kt] 的有限性、正性、守恒性,闭合反证链条。 在无额外假设条件下闭合全部证明缺口,严格证明:对所有 n ≥ 4,n 维欧氏空间中 Kakeya 极大猜想成立。文中嵌入 Lean4 形式化验证骨架,实现证明逻辑的计算机可核验。 该结果补全 Kakeya 理论的全维图景,对振荡积分算子端点估计、傅里叶限制理论、色 散偏微分方程、分形投影理论提供新几何工具,为高频波传播分析提供数学基础。