\section{A scalar-average illustration}
\label{app:scalarillustration}
We begin with a maximally mixed average state, for which weak
commutativity holds automatically. This simpler model introduces the
comparison matrix used to solve the nonscalar model in the main text.
Let the only nonzero entries of a weighted ladder \(C\) be
\begin{equation}
 C_{12}=1,\qquad C_{23}=2,\qquad C_{34}=3.
 \label{eq:ladder}
\end{equation}
Its Hermitian components \(X=(C+C^\dagger)/2\) and
\(Y=(C-C^\dagger)/(2i)\) are finite-level quadratures, not canonical oscillator
position and momentum.
Put
\begin{equation}
\begin{split}
 U&=\operatorname{diag}(-1,-1,-1,1),\\
 K_\pm&=(\1_4\pm U)/8,\qquad B=CK_++K_-C,\\
 B_1&=(B+B^\dagger)/2,\qquad B_2=(B-B^\dagger)/(2i).
\end{split}
 \label{eq:shiftmoments}
\end{equation}
Thus \(B\) has superdiagonal entries \(1/4,1/2,3/2\).
For \(s=\pm1\) and \(j=1,2\), prepare one copy of
\begin{equation}
 p_{s,j}=\tfrac14,\qquad
 \theta_{s,j}=10s e_j,\qquad
 \rho_{s,j}=\1_4/4+s B_j/5.
 \label{eq:shiftprior}
\end{equation}
The classical label is not given to the receiver.
Both \(B_j\) are traceless and Hermitian; diagonal dominance gives
\(\rho_{s,j}\succeq\1_4/20\).
The average and first moments are exactly \(S=\1_4/4,B_j\), with
\(\lambda=100\). The unique posterior means are \(S_j=4B_j\).
Weak commutativity follows from \(\Tr[S_1,S_2]=0\), although the operators
themselves need not commute. Indeed, a scalar average state makes this trace
test automatic: the obstruction concerns what that test fails to detect.

\begin{theorem}[Exact Bayesian risk and strict gap]
For the ensemble \eqref{eq:shiftprior} and the loss \eqref{eq:risk},
\begin{equation}
 \mathcal L_{\mathrm{NH}}=\frac{377}{4},\qquad
 \mathcal L_{\min}=100-4f(u_*,v_*)=\frac{377}{4}+\Gamma_*.
 \label{eq:gap}
\end{equation}
The algebraic pair and rational function \(f\) are defined below and in
Appendix~\ref{app:optimal}. The exact value satisfies
\(0.03592634498992<\Gamma_*<0.03592634498993\).
An explicit eight-effect POVM attains the minimum. The matching lower bound
holds for every POVM and measurable estimator, with no bound on outcome
number or estimate range.
\end{theorem}

The terminating decimals bound an algebraic value; they are not its exact
specification. For comparison, the Fourier projectors onto
\((1,w,w^2,w^3)^T/2\), \(w\in\{1,i,-1,-i\}\), with reports \(9w/4\),
have risk \(1519/16\) and are therefore not optimal.

\subsection{Minimizing the NH expression}
For a Hermitian contraction \(-\1\preceq R\preceq\1\), define
\(K_\pm(R)=S^{1/2}(\1\pm R)S^{1/2}/2\) and
\[
 q_R(Z)=\Tr[ZK_+(R)Z^\dagger+Z^\dagger K_-(R)Z].
\]
The quadratic plus trace norm in \eqref{eq:nh} is \(\sup_R q_R(Z)\),
where \(Z=V_1+iV_2\).
For \eqref{eq:ladder},
\[
 iS^{1/2}[X,Y]S^{1/2}
 =\operatorname{diag}(-1,-3,-5,9)/8,
\]
so \(U\) is the active sign and \eqref{eq:shiftmoments} is its stationarity
condition. For every perturbation \(E\), the active branch minus the linear
term changes by \(q_U(E)\ge0\). The supremum is no smaller than that branch
and agrees with it at \(C\). This proves a \emph{global} NH minimum, without
uniqueness. In particular,
\begin{equation}
 T_0=CK_+C^\dagger+C^\dagger K_-C
     =\operatorname{diag}(0,1/4,13/4,9/4),
 \label{eq:t0}
\end{equation}
and \(\mathcal L_{\mathrm{NH}}=\lambda-\Tr T_0=377/4\).

\subsection{A lower bound for every measurement}
Group the outcomes of \(P\) by their reported estimates \(a\). This gives
a POVM \(M\) on the complex estimate plane \(z=a_1+ia_2\).
The positive comparison matrix is
\begin{equation}
 F(z)=|z|^2S-\bar zB-zB^\dagger+T_0,
 \label{eq:globalf}
\end{equation}
and direct risk expansion yields
\begin{equation}
 \mathcal L(P,a)-\mathcal L_{\mathrm{NH}}
   =\int\Tr[M(\dd z)F(z)].
 \label{eq:globaldual}
\end{equation}
There is no requirement that the measurement's first moments equal \(X,Y\).
Finite risk suffices for integrability; infinite-risk procedures obey the
bound trivially. Operator-dual formulations go back to
Ref.~\cite{Holevo1973} and are used for the Bayesian NH bound in
Ref.~\cite{Albarelli2026}, alongside distinct conic formulations of
local bounds \cite{Hayashi2023}.

A simple rational witness is the real matrix
\begin{equation}
 G=\begin{pmatrix}
 0&-1/16&0&2/47\\
 -3/16&0&1/94&0\\
 0&-24/47&0&-3/4\\
 -2/47&0&-3/4&0
 \end{pmatrix}.
 \label{eq:g}
\end{equation}
It satisfies \(G+G^T=-(B+B^\dagger)\), and \(GG^T\) is diagonal.
Consequently
\begin{equation}
 \begin{aligned}
 A&=T_0-4GG^T\\
  &=\frac1{141376}\operatorname{diag}(-3233,15399,-6080,-1024),\\
 \Tr A&=\gamma=\frac{2531}{70688}.
 \end{aligned}
 \label{eq:a}
\end{equation}
For \(z=rw\), \(r\ge0\), \(|w|=1\), put
\(D(w)=\operatorname{diag}(1,w,w^2,w^3)\) and \(Q(r)=r\1_4/2+2G\).
The following identity holds over the entire estimate plane:
\begin{equation}
 F(rw)-A=D(w)Q(r)Q(r)^TD(w)^\dagger\succeq0.
 \label{eq:factor}
\end{equation}
At the origin take \(r=0,w=1\).
This is a polynomial coefficient identity, not a sampled positivity test.
Although \(A\) is indefinite, POVM normalization gives
\[
 \int\Tr[M(\dd z)A]=\Tr A.
\]
Integrating \eqref{eq:factor} in \eqref{eq:globaldual} proves the simple
rational lower bound \(\mathcal L_{\min}-\mathcal L_{\mathrm{NH}}\ge\gamma\).
The number \(\gamma\) is a certified lower bound, not the exact excess risk.
Its scale can be compared with the NH-allowed gain
\(\lambda-\mathcal L_{\mathrm{NH}}=23/4\), giving a missed fraction of at
least \(\gamma/(23/4)\). This is a relaxation error, not a sensing advantage
or an incompatibility measure invariant under arbitrary changes of loss.
Parameter-independent ancillas, classical randomization, and sequential
instruments on the same received copy induce effective POVMs and are
included. Every abort outcome must remain part of the normalized experiment;
either conditional postselection or the use of extra parameter-encoded
copies changes the task.

\subsection{A matching optimal measurement}
The rational comparison can be improved until it is met by a measurement.
Write \((w_0,m_0,t_0)=(1/4,1/2,3/2)\), and define
\begin{equation}
\begin{split}
 d&=v(t_0+v)-u(w_0+u),\\
 a&=u m_0(t_0+v)/d,\qquad b=u m_0(w_0+u)/d,\\
 G(u,v)&=\begin{pmatrix}
 0&u&0&a\\-w_0-u&0&b&0\\
 0&-m_0-b&0&v\\-a&0&-t_0-v&0
 \end{pmatrix}.
\end{split}
 \label{eq:gramfamily}
\end{equation}
Put \(f(u,v)=\|G(u,v)\|_F^2\).
The selected pair \((u_*,v_*)\) is the unique stationary point of \(f\) in
the rational rectangle
\[
 \begin{gathered}
 -.06554650<u<-.06554649,\\
 -.75260404<v<-.75260402.
 \end{gathered}
\]
Appendix~\ref{app:optimal} fixes its algebraic branch and proves existence,
nonvanishing denominators and the exact normalization identities.
Set \(G_*=G(u_*,v_*)\), \(c_i=4(G_*G_*^T)_{ii}\),
\(A_*=T_0-\operatorname{diag}(c_i)\), and
\(\Gamma_*=\Tr A_*=23/4-4f(u_*,v_*)\).
The same coefficient identity gives, for every \(z=rw\),
\begin{equation}
 F(rw)-A_*=D(w)(rI/2+2G_*)(rI/2+2G_*)^TD(w)^\dagger\succeq0.
 \label{eq:optimalfactor}
\end{equation}
Hence every experiment has risk at least \(100-4f(u_*,v_*)\).

There are two positive radii \(r_-<r_+\) at which the radial matrix
\(F(r)-A_*\) is singular. Its real kernel vectors \(\nu_-,\nu_+\) are
fixed with first component one, and positive weights \(\alpha_-,\alpha_+\)
satisfy
\[
 \alpha_-+\alpha_+=1,\qquad
 \alpha_-\nu_{-,i}^2+\alpha_+\nu_{+,i}^2=1\quad(1\le i\le4).
\]
All these quantities are explicitly algebraic in Appendix~\ref{app:optimal}.
For each sign \(s\) and fourth root of unity \(w\), use
\begin{equation}
 M_{s,w}=\frac{\alpha_s}{4}D(w)\nu_s\nu_s^TD(w)^\dagger,
 \qquad z_{s,w}=r_s w.
 \label{eq:optimalpovm}
\end{equation}
Summing the four phases removes every off-diagonal entry, and the displayed
balance gives \(\sum_{s,w}M_{s,w}=I_4\).
Each effect is supported in \(\ker[F(z_{s,w})-A_*]\), so its comparison
residual vanishes. This protocol meets the unrestricted lower bound and
proves Eq.~\eqref{eq:gap}. Global optimality follows from this primal--dual
equality, not from stationarity or optimization within a covariant ansatz.
The two-seed construction uses established finite-group covariance
\cite{Chiribella2006}; neither uniqueness nor minimal outcome number is claimed.

\textit{What obstructs NH equality?—}
The complete kernel locus provides a complementary interpretation.
A vector \(\psi\in\ker F(z)\) obeys
\[
 z\psi_1=\psi_2,\quad z\psi_2=2\psi_3,\quad
 z\psi_3=3\psi_4,\quad \bar z\psi_4=3\psi_3.
\]
These are the NH equality kernels \(\ker F(z)\), not the optimal-cost
kernels \(\ker[F(z)-A_*]\). At zero the former kernel is
\(\operatorname{span}(e_1)\); otherwise it is
one-dimensional and exists exactly on \(|z|=3\).
Every such vector satisfies \(9|\psi_2|^2=4|\psi_3|^2\).
All effects confined to these kernels inherit \(9M_{22}=4M_{33}\), whereas
a resolution of the identity would require \(9=4\).
This excludes equality even for an overcomplete or continuous POVM.
Kernel exclusion alone need not imply a gap to the infimum;
the uniform factor \eqref{eq:factor} supplies that stronger conclusion.

\subsection{A continuous prior with the same moments}
The same moments admit a smooth, identifiable realization.
For a uniform prior on the connected square \([-30,30]^2\), let
\begin{equation}
 p(\theta)=\frac1{3600},\qquad
 \rho_\theta=\frac{\1_4}{4}
              +\frac{\theta_1B_1+\theta_2B_2}{300}.
 \label{eq:smoothprior}
\end{equation}
Its covariance is \(300\1_2\), so its state moments remain \(S,B_j\).
Diagonal dominance gives \(\rho_\theta\succeq\1_4/20\) throughout the square.
The state map is affine and injective in the two real coordinates, and is
smooth on an open neighborhood of the support.
Here \(\lambda=600\), \(\mathcal L_{\mathrm{NH}}=2377/4\), and
\(\mathcal L_{\min}=600-4f(u_*,v_*)=2377/4+\Gamma_*\). The two parameters encode
linearly independent real and imaginary coherence directions.
The prior is uniform with a boundary, not globally smooth on \(\mathbb R^2\).