Lemma A.2 — normalization singles out $p=2$

\emphFor a fixed normalized $\psi$ and orthogonal decomposition $\mathcal H=\bigoplus_iP_i\mathcal H$, $\sum_i\|\psi_i\|^p=1$ holds for every such decomposition if and only if $p=2$. Proof. By Parseval/Pythagoras, $\sum_i\|\psi_i\|^2=\|\psi\|^2=1$ for every orthogonal decomposition, so $p=2$ works. For $p\ne2$, $\sum_i\|\psi_i\|^p$ depends on the decomposition (e.g. for $\psi$ an equal superposition of $d$ orthonormal terms and $P_i$ the corresponding rank-1 projectors, $\sum_i\|\psi_i\|^p=d^{1-p/2}\ne1$ for $p\ne2$), so $w_i:=\|\psi_i\|^p$ is not a context-independent, automatically normalized weight for any $p\ne2$. $\blacksquare$

Hence the squared-norm measure is singled out by the correct mechanism (automatic normalization, Lemma A.2), together with local additivity (Lemma A.1) — not by tensor-product-composition stability, which does not distinguish $p=2$ from any other exponent.