2D FFT of k field (spatial frequencies)

We present a numerical framework for solving the Ricci flow equation ∂ₜ g₍μν₎ = −2 R₍μν₎ on discrete ten-dimensional manifolds using an adaptive sparse hash-grid architecture (Version 3). The metric tensor is represented as a dynamical field with inertia via a Heavy Ball integrator (β = 0.90 adaptive), motivated by an analogy with classical electrodynamics. Compact topology T⁴ₗ is encoded via periodic boundary conditions on four dimensions.