Fourier Energy Strata for the Cubic Schrödinger Equation

We prove a geometric hierarchy of Fourier layer energies for the weakly nonlinear cubic Schrodinger equation on ${\mathbb{T}}_{\pi }\times {\mathbb{T}}_1$, retaining its nontrivial resonant dynamics. For initial Fourier support in a rectangle, form nested rectangles with half-widths $(2j+1)$ times the initial ones; layers are the differences between consecutive rectangles. On $0\le t\le T{\delta }^{-1}$, the energy in layer $j$, measured by its squared $H^1$ norm, is at most $C_N{\delta }^{2j}$ for $1\le j\le N$; the energy outside the $N$th rectangle is at most $C_N{\delta }^{2N+2}$. These bounds hold at every fixed depth $N$ under a uniform $H^s$ bound, $s>2$, with constants depending on $N$. The Sobolev bound holds for every fixed $T$ in the defocusing case. We also obtain layer estimates for support bounded in one coordinate and for geometrically decaying initial Fourier tails. The proof uses a filtered Poincare-Dulac normal form: input layers select which nonresonant interactions are removed, while the same layer weights control the interactions left in the equation. Finite coordinate changes and stability estimates transfer this hierarchy to the exact solution.