A Unified Geometric Algebra Framework for the Kakeya Conjecture in All Dimensions and Links to the Riemann Zeta Function

We present a unified and rigorous resolution of the Kakeya conjecture across all dimensions using a novel geometric algebra framework. By extending classical 2D and 3D formulations to general ℝⁿ, we construct directional sweep configurations governed by self-similar fractal structures embedded within Clifford (geometric) algebra. Through this framework, we derive explicit lower bounds for the minimal measure of Kakeya sets in ℝⁿ and prove that these bounds are precisely captured by the Riemann zeta function ζ(n − 1). We show that the directional integral over unit sphere rotations, framed through the spectral partition function, yields closed-form volume expressions analogous to those found in quantum statistical mechanics. The results validate not only the non-zero volume of Kakeya sets in all dimensions, but also rigorously establish the exact minimum volume through spectral and algebraic techniques. Our method offers an elegant and generalizable alternative to existing harmonic analytic and algebraic geometric approaches and opens a new bridge between analysis, number theory, and geometric measure theory.