
The high-dimensional Kakeya conjecture and the Fourier restriction conjectureare two central problems in harmonic analysis with deep intrinsic connections. Thecomplete resolution of the three-dimensional Kakeya conjecture reveals a cross-scalemechanism of grain condensation in directional constraint networks. This papergeneralises this mechanism to n-dimensional Euclidean space, establishing a fibrecondensation structure on the Grassmann manifold for the directional constraintnetwork of a high-dimensional Kakeya set, thereby opening a path for anomalysuppression for the Fourier restriction conjecture. The core result shows that whenthe richness of directional constraints exceeds the critical threshold characterised bymetric entropy, the frequency support of the n-dimensional tube bundle becomesincompressible into lower-dimensional submanifolds, forcing the restriction estimatefor the Fourier transform on paraboloids to hold. By introducing a slice-projectionduality, we achieve inductive closure, transforming the traditional problem of accumulating precision loss in scale induction into a geometric series absorption oferrors by the condensation operator. In the case n = 4, we present a completetemplate proof from the three-dimensional base case to four-dimensional full dimensionality, establishing sufficient conditions for the non-degeneracy of constantsin the induction step.
