Constraint Network and Anomaly Suppression Series B: Anomaly Suppression for High-Dimensional Kakeya and Fourier Restriction Conjectures

The high-dimensional Kakeya conjecture and the Fourier restriction conjecture
are two central problems in harmonic analysis with deep intrinsic connections. The
complete resolution of the three-dimensional Kakeya conjecture reveals a cross-scale
mechanism of grain condensation in directional constraint networks. This paper
generalises this mechanism to n-dimensional Euclidean space, establishing a fibre
condensation structure on the Grassmann manifold for the directional constraint
network of a high-dimensional Kakeya set, thereby opening a path for anomaly
suppression for the Fourier restriction conjecture. The core result shows that when
the richness of directional constraints exceeds the critical threshold characterised by
metric entropy, the frequency support of the n-dimensional tube bundle becomes
incompressible into lower-dimensional submanifolds, forcing the restriction estimate
for the Fourier transform on paraboloids to hold. By introducing a slice-projection
duality, we achieve inductive closure, transforming the traditional problem of ac
cumulating precision loss in scale induction into a geometric series absorption of
errors by the condensation operator. In the case n = 4, we present a complete
template proof from the three-dimensional base case to four-dimensional full di
mensionality, establishing sufficient conditions for the non-degeneracy of constants
in the induction step.