\textit{Beyond a single ladder}---To test whether the failure depends
on the transition ratio \(1:2:3\), we vary the ladder \(C_u\), with
entries just above the diagonal
\[
 \begin{gathered}
 \ell_1=\sqrt{(1-2u)/3},\qquad
 \ell_2=\sqrt{(2-2u)/3},\\
 \ell_3=1,\qquad 0\le u\le\tfrac13 .
 \end{gathered}
\]
Appendix~\ref{app:oscillatorfamily} constructs four equiprobable states
for each \(u\), population ratio \(k=S_{33}/S_{11}\in[1,2]\), and
contrast \(\eta\in(0,1/6]\). The labels are \(\pm e_1,\pm e_2\), every
state satisfies \(\rho\succeq I_4/12\), and the average state is
nonscalar for \(k>1\). Weak commutativity holds throughout this family,
but
\begin{equation}
 \mathcal L_{\min}(u,k,\eta)-\mathcal L_{\mathrm{NH}}(u,k,\eta)
 \ge\gamma\eta^2>0
 \label{eq:familygap}
\end{equation}
with \(\gamma=1/16416\). The effects required for NH equality obey
\(E_{22}=\ell_2^2E_{33}\), so \(W_u=\operatorname{diag}(0,1,-\ell_2^2,0)\)
is a normalization witness with \(\Tr W_u=1-\ell_2^2=(1+2u)/3\).
Appendix~\ref{app:familyquantitative} proves its robust
form~\eqref{eq:robustwitness} with \(\kappa^2=19/p\), \(p=S_{11}\), for
reports rescaled by \(2/\eta\). Equation~\eqref{eq:witnessgap}, times
the factor \(\eta^2/4\) from this rescaling, gives the coefficient
\(p(1-\ell_2^2)^2/304\ge\gamma\). At least \(1/3420\) of the risk
reduction predicted by NH is unattainable by any measurement; both
constants are conservative.

At \(u=0\), \(C_u\) is the truncation (compression) of the oscillator
annihilator \(\hat a/\sqrt3\) to the first four Fock levels. At
\(u=1/(m+3)\), with integer \(m\ge0\), it is the normalized compression
of the pair annihilator \(\hat a\hat b\), a standard optical SU(1,1)
generator \cite{Fabre2023}, to \(\{|n+m,n\rangle:0\le n\le3\}\). The
correspondence concerns the ladders only: the preparations are
mixtures of number states and balanced adjacent-level superpositions,
the label selects the mixture rather than a displacement or an
interferometric phase, and full rank refers to the four-level space.
On the oscillator slice \(u=0\), a two-radius, eight-effect POVM attains
the exact minimum over all measurements for every \(k\in[1,2]\) and
\(\eta\in(0,1/6]\) (Appendix~\ref{app:oscillatorfamily}). The minimal
number of outcomes and the projective optimum are established only for
the fixed model above.