= Spectral inverse theorem for the matrix-valued U2 quantity
{title2=$cn/2\leq\sum_p d_{\rho_p}\leq2n/c$}
If $\|f(x)\|_{\mathrm{op}}\leq1$ and $\|f\|_{U^2}^4\geq cn>0$, select the <singular values> $\lambda_p\geq\sqrt{c/2}$ of the <matrix Fourier blocks>. Their weighted count $m=\sum_pd_{\rho_p}$ lies between $cn/2$ and $2n/c$. Scaling unit <right singular vectors> and their corresponding unit <left singular vectors> by $\sqrt{d_{\rho_p}}$ gives matrices $U(p),V(p)$ satisfying $\mathbb E_xf(x)U(p)\rho_p(x)^*=\lambda_pV(p)$ and <Hilbert-Schmidt inner product> orthogonality within each chosen <irreducible representation>. The <Schur averaging of rectangular matrices> then gives $\mathbb E_x\|UP(x)^*b\|_2^2=\|b\|_2^2$ for the concatenated $U$ and the corresponding <block diagonal matrix> $P(x)$.
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