Power-of-two normalization never collapses a row: some coordinate keeps ≥ 1/n of the energy

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$, $\htmlData{mx=5}{x : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{Z}}}$ and $\htmlData{mx=1}{k \in \htmlData{mx=0}{\mathbb{N}}}$, if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}$ and $\htmlData{mx=1}{4^{\htmlData{mx=0}{k}}} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{\sum_{i} \htmlData{mx=5}{\htmlData{mx=4}{\htmlData{mx=2}{x}(\htmlData{mx=3}{i})}^{2}}}$, then