\section{A family containing oscillator and pair-annihilation ladders}
\label{app:oscillatorfamily}

This appendix varies both relative transition strengths and average-state
populations. It proves a strict gap above NH for all measurements
throughout the family. The oscillator slice also admits the exact
optimal readout derived below; the remaining shapes' exact risks and
the family's outcome classification are not determined.

\subsection{States, moments, and full rank}

Take \(0\le u\le1/3\), \(1\le k\le2\), and \(0<\eta\le1/6\).
The upper ladder \(C_u\) has positive entries \(\ell_1,\ell_2,1\)
defined in the main text. For this family only, set
\begin{equation}
\begin{split}
 p&=\left[2+2k-\frac{\ell_2^2(k-1)}{(1+k)^2}\right]^{-1},\qquad q=kp,\\
 t&=q-\frac{\ell_2^2p(k-1)}{(1+k)^2},\qquad S=\operatorname{diag}(p,p,q,t),\\
 K_-&=\operatorname{diag}(p,p,q,0),\qquad K_+=\operatorname{diag}(0,0,0,t),\\
 B&=C_uK_++K_-C_u,\qquad
 B_1=(B+B^\dagger)/2,\quad B_2=(B-B^\dagger)/(2i).
\end{split}
 \label{eq:familymoments}
\end{equation}
These quantities are distinct from the fixed model's moments.
The four equiprobable preparations and labels are
\begin{equation}
 \rho_{s,j}=S+s\eta B_j,\qquad \theta_{s,j}=s e_j,
 \quad s=\pm1,\ j=1,2.
 \label{eq:familypreparations}
\end{equation}
Their actual first moments are \(gB_j\), where \(g=\eta/2\),
and their second prior moment is \(\lambda=1\).

Here \(\ell_1^2\in[1/9,1/3]\),
\(\ell_2^2\in[4/9,2/3]\), and
\(\ell_2^2-\ell_1^2=1/3\).
The denominator defining \(p\) lies in \([4,6]\): subtracting 4 gives
\((k-1)[2-\ell_2^2/(1+k)^2]\), while subtracting it from 6 gives
\(2(2-k)+\ell_2^2(k-1)/(1+k)^2\). Both are nonnegative. Also
\[
 t-p=p(k-1)\left[1-\frac{\ell_2^2}{(1+k)^2}\right]\ge0.
\]
Thus \(2p+q+t=1\), \(1/6\le p\le1/4\), and \(p\le t\le q\).
For \(k>1\) the average state is nonscalar; \(k=1\) is the
scalar boundary \(S=I_4/4\).

Write the three upper entries of \(B\) as
\(\beta=(\ell_1p,\ell_2p,q+t)\), and put \(\beta_0=\beta_4=0\).
Every adjacent sum \(\beta_{i-1}+\beta_i\) is at most 1:
the two interior sums are bounded by \(2p\le1\) and
\(\ell_2p+q+t=1-(2-\ell_2)p\le1\), respectively.
Consequently \(\|B_j\|\le1/2\), and
\(\rho_{s,j}\succeq(p-\eta/2)I_4\succeq I_4/12\).
An explicit preparation decomposition is
\begin{equation}
 \rho_{s,j}=\sum_{i=1}^4 r_i|i\rangle\langle i|
 +\sum_{i=1}^3\eta\beta_i
     |\chi_i^{s,j}\rangle\langle\chi_i^{s,j}|,
 \label{eq:familymixture}
\end{equation}
where
\[
 r_i=S_{ii}-\tfrac\eta2(\beta_{i-1}+\beta_i)\ge\tfrac1{12},
 \qquad
 |\chi_i^{s,j}\rangle=(|i\rangle+s\mathrm{i}^{j-1}|i+1\rangle)/\sqrt2.
\]
The weights sum to one. The random preparation record is not supplied
to the receiver. These balanced adjacent-level superpositions are not
Glauber coherent states.

\subsection{Weak commutativity and the global NH minimum}

The actual posterior-mean operators have upper entries \(g h_i\)
and \(-ig h_i\), with
\(h=(\ell_1/2,\ell_2p/(p+q),1)\). They satisfy
\begin{equation}
 \Tr(S[S_1,S_2])
 =2ig^2\left[\frac{(p-q)\ell_2^2p^2}{(p+q)^2}+q-t\right]=0.
 \label{eq:familyweak}
\end{equation}
The cancellation follows from Eq.~\eqref{eq:familymoments};
it is not an approximation near the scalar boundary.

For a Hermitian contraction \(R\), use the quadratic form \(q_R\)
defined in the main text, now with this \(S\). The NH objective is
\(1+\sup_R q_R(Z)-2g\Re\Tr(B^\dagger Z)\).
The contraction \(U=\operatorname{diag}(-1,-1,-1,1)\) is active
at \(Z=gC_u\): indeed
\[
 i[X_u,Y_u]=-\tfrac12\operatorname{diag}
 (\ell_1^2,\ell_2^2-\ell_1^2,1-\ell_2^2,-1),
\]
where \(X_u,Y_u\) are the Hermitian components of \(C_u\).
The first three displayed diagonal magnitudes are strictly positive.
The relation \(B=C_uK_++K_-C_u\) gives, for every matrix \(E\),
\[
 q_U(gC_u+E)-2g\Re\Tr[B^\dagger(gC_u+E)]
 =-g^2\Tr T_0+q_U(E),
\]
where
\(T_0=C_uK_+C_u^\dagger+C_u^\dagger K_-C_u
=\operatorname{diag}(0,p\ell_1^2,p\ell_2^2+t,q)\).
Since \(q_U(E)\ge0\), this supporting branch bounds the objective
globally and is attained at \(gC_u\). Therefore
\begin{equation}
 \mathcal L_{\mathrm{NH}}(u,k,\eta)=1-g^2\Tr T_0.
 \label{eq:familyNH}
\end{equation}
No uniqueness of the NH minimizer is needed.

\subsection{Every complex report and every normalized POVM}

For \(w\in\C\) define
\(F(w)=|w|^2S-\bar wB-wB^\dagger+T_0\).
For every \(\psi\in\C^4\),
\begin{equation}
\begin{split}
 \psi^\dagger F(w)\psi={}&p|w\psi_1-\ell_1\psi_2|^2
 +p|w\psi_2-\ell_2\psi_3|^2\\
 &+q|w\psi_3-\psi_4|^2+t|\bar w\psi_4-\psi_3|^2.
\end{split}
 \label{eq:familyallcomplex}
\end{equation}
This proves positivity on the entire complex report plane. If \(w\ne0\)
and a nonzero \(\psi\) lies in the kernel, then \(\psi_3\ne0\):
otherwise the four residual equations force every component to vanish.
The last two equations give \(|w|=1\), and the second gives
\(|\psi_2|^2=\ell_2^2|\psi_3|^2\).
At \(w=0\) the kernel is \(\operatorname{span}(e_1)\), which
obeys the same relation. Thus every positive effect supported in any
of these kernels satisfies
\begin{equation}
 E_{22}=\ell_2^2 E_{33}.
 \label{eq:familypopulation}
\end{equation}

For the actual model the residual at a report \(z\) is
\(F_g(z)=g^2F(z/g)\). Push any measurement and its reporting rule
forward to a normalized POVM \(M\) on \(\C\). Risk expansion yields
\[
 \mathcal L(M)-\mathcal L_{\mathrm{NH}}
 =\int_{\C}\Tr[F_g(z)M(\dd z)].
\]
For finite risk, integrability follows from \(S\succeq I_4/6\);
infinite-risk procedures cannot attain NH. Relative to
\(\mu(E)=\Tr M(E)\), the POVM has a positive matrix density \(A(z)\).
Equality of risks would force \(\Tr[F_g(z)A(z)]=0\) almost
everywhere, hence \(\operatorname{range}A(z)\subseteq\ker F_g(z)\).
Equation~\eqref{eq:familypopulation}, integrated against \(\mu\),
would give \(1=\ell_2^2\), whereas \(\ell_2^2\le2/3\).
This excludes equality for arbitrary countably additive POVMs,
including continuous or singular report distributions and randomized
reporting. No reality, phase covariance or finite-support assumption
has entered the argument.

\input{oscillator-gap-quantitative}

\subsection{Optical instances and the meaning of the compression}

Let \(\Pi_4\) project onto the first four single-mode Fock states.
At \(u=0\),
\[
 C_0=\left.\Pi_4\hat a\Pi_4/\sqrt3\right|_{\operatorname{range}\Pi_4}.
\]
For an integer \(m\ge0\), let \(\Pi_{m,4}\) project onto
\(\{|n+m,n\rangle:0\le n\le3\}\).
Since \(\hat a\hat b|n+m,n\rangle
=\sqrt{n(n+m)}|n-1+m,n-1\rangle\),
\[
 C_{1/(m+3)}=
 \left.\frac{\Pi_{m,4}\hat a\hat b\Pi_{m,4}}{\sqrt{3(m+3)}}
 \right|_{\operatorname{range}\Pi_{m,4}}.
\]
The normalized \(1:2:3\) ladder occurs at \(m=0\); its matrix
coefficients approach the oscillator shape as \(m\) grows. Only this
discrete sequence is identified with integer number-difference sectors.
Other \(u\) interpolate the relative transition strengths. The standard
uncompressed SU(1,1) pair realization is given in Ref.~\cite{Fabre2023},
Eqs.~(2.1)--(2.5).

Compression changes commutation relations at the highest retained
level. It does not produce a finite-dimensional unitary SU(1,1)
representation or canonical position and momentum. The prepared states
are full rank on the encoded space, not on the entire Fock space.
Likewise, an original-space PVM restriction cannot be imposed on an
unrestricted optical apparatus: unused levels may implement a projective
dilation of an effective POVM. No optical PVM disadvantage or
experimental resource advantage is inferred from this identification.

The two checkers in \texttt{anc/oscillator-family/} verify the
multivariate identities by separate symbolic-polynomial implementations.
The first uses SymPy; the second uses standard-library rational sparse
polynomials and imports none of the first. Their algebraic checks
complement, rather than replace, the kernel and quantitative
matrix-measure arguments above.

\input{oscillator-optimal-details}