The Gradient Energy with a Mode Cutoff

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $\varphi:H^{-1}\to\mathbb{R}$ be twice differentiable along the modes, with first mode derivatives $\partial_{k}\varphi$, and let $x\in H^{-1}$. Sums over the cube $\Gamma_{N}$ are sums over a finite index set.

1. (At cutoff $N$)¶ For $N\in\mathbb{N}$, the gradient energy at cutoff $N$ of $\varphi$ at $x$ is

$$|D_{N}\varphi(x)|^{2}=\sum_{k\in\Gamma_{N}}\bigl(\partial_{k}\varphi(x)\bigr)^{2}.$$

2. (Uncut)¶ If the sequence $(|D_{N}\varphi(x)|^{2})_{N\in\mathbb{N}}$ converges, $|D\varphi(x)|^{2}$ is said to be defined, and denotes the limit, which is unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences.