A Proof of the Kakeya Conjecture via High-Dimensional Symmetric Topology and Measure Shrinkage

Abstract This paper presents a novel framework to resolve the Kakeya Conjecture. By introducing a geometric constraint based on high-dimensional symmetric topology—mapping to 120-cell adjacency structures—and dynamic Hamiltonian routing equivalence, we transform this measure-theoretic problem into a continuous spatiotemporal model. We establish a rigorous mathematical proof utilizing the measure shrinkage principle of multi-dimensional compression gate circuits, demonstrating that the spatial overlap of Kakeya tubes does not cause anomalous dimension collapse. Consequently, this study proves that the Minkowski dimension of a Kakeya set in $\mathbb{R}^n$ strictly aligns with the topological dimension $n$, establishing that optimal topological connectivity prevents measure collapse.