A Novel Framework and Proof of the Kakeya Conjecture in 3D and Higher Dimensions Based on Quantized Direction Space and Riemann ζ Bounds

We present a novel geometric approach to the 3D Kakeya needle problem and its generalization to higher dimensions by introducing the concept of compactified direction space. By discretizing the unit 2-sphere S² into uniformly distributed angular patches and generalizing this approach to the (n−1)-sphere Sⁿ⁻¹, we derive a universal lower bound on the minimum volume required to rotate a needle in all directions. This bound is governed by the Riemann zeta function evaluated at ζ(n−1), thereby uncovering a deep connection between harmonic analysis, directional quantization, and number theory. Our formulation extends naturally to fractal and anisotropic media, offering new insights into fractodynamics, directional diffusion, and potential implications for quantum field theory and lattice spacetime models. This work not only resolves the 3D Kakeya conjecture under a quantized framework but also proposes a new ζ(n−1)-bounded volume law applicable to compactified direction spaces across dimensions.