A Multiscale Directional Energy Framework for Kakeya-Type Sets in R 4

This document presents a structural analysis of Kakeya-type sets based on multiscale directional distributions and associated energy functionals. The approach does not claim a resolution of the Kakeya conjecture, but introduces a coercive framework that captures persistent directional incoherence across scales via local probability measures and spectral graph constructions. A dichotomy is established between strong directional clustering and the growth of a multiscale energy functional, providing an obstruction mechanism compatible with existing methods in geometric measure theory and harmonic analysis. The final section briefly situates the construction within a broader informational–geometric perspective, without affecting the mathematical validity of the results.