Theorem: Retained-order visibility test

Fix the instrument and scale. Let $N\ge 1$ and $q\ne 0$, and suppose, for each fixed real $\ell$,

If

in trace norm for every fixed $\ell$, then $B_y=C_y=0$ for all $y$.

In particular, for the third-order-labelled coefficient, $N=3$, absence of response through the sixth encoding order forces erasure of this whole amplitude family.

Proof. The leading order in each block is

so a nonzero fixed $\ell$ forces $B_y=0$. After that exact matrix vanishes, the leading term is

using $\mathrm{sech}u=1-u^2/2+O(u^4)$. This forces $C_y=0$.