\section{Preparation noise and stability of the precision hierarchy}
\label{app:robustness}

This section concerns the fixed four-state ensemble of
Eq.~\eqref{eq:principalprior}, with the same labels, equal prior and
identity-weight quadratic loss. We suppress the subscript \(k\).
There are two different statements: a specified noise family preserves
weak commutativity exactly, whereas arbitrary small preparation errors
preserve the risk separations but need not preserve weak commutativity.

\subsection{Unheralded replacement by the average state}

Let \(0<\zeta\le1\) and apply the quantum channel
\[
 \mathcal N_\zeta(\rho)=\zeta\rho+(1-\zeta)S .
\]
No flag identifying the replacement branch is supplied to the receiver.
The replacement state is the fixed nonscalar average \(S\), not \(I_4/4\).
Since the prior is centered, the resulting moments are
\(S_\zeta=S\), \(B_{j,\zeta}=\zeta B_j\) and \(\lambda=100\).

For any normalized POVM \(P\), use the finite scalar measure
\(\mu(\dd\omega)=\Tr P(\dd\omega)\) and its positive matrix density
\(P_\omega\). Set \(s(\omega)=\Tr SP_\omega\) and
\(b_j(\omega)=\Tr B_jP_\omega\). Full rank makes \(s>0\)
almost everywhere. Completing the square in the reports gives
\[
 \inf_a\mathcal L(P,a;\zeta)
 =100-\zeta^2\int\frac{b_1^2+b_2^2}{s}\,\dd\mu ,
 \qquad a_j^{\mathrm{opt}}=\zeta b_j/s .
\]
The posterior means lie in the convex hull of the four labels, so these
integrals are finite. This argument uses densities, not a finite-outcome
assumption. It also covers randomized reports by conditional averaging.
Consequently, for either the unrestricted POVM class or the original-space
PVM class, including parameter-independent classical mixtures,
\begin{equation}
 \mathcal L_{\mathcal C}(\zeta)
 =100+\zeta^2[\mathcal L_{\mathcal C}(1)-100].
 \label{eq:replacementscaling}
\end{equation}
The NH expression has the same scaling: substitute
\(V_j=\zeta W_j\) into Eq.~\eqref{eq:nh}, a bijection on Hermitian
pairs for positive \(\zeta\). Every quadratic and trace-norm term then
has the common factor \(\zeta^2\).
The symmetric posterior means become \(\zeta S_j\), so their averaged
commutator remains zero.

For positive \(\zeta\), maximizing the measurement gain is the same
problem as before. The optimal effects are unchanged, all optimal reports
scale by \(\zeta\), and the complete reported-measurement face in
Appendix~\ref{app:seven} is carried to its scaled copy.
Seven nonzero finite outcomes remain necessary and sufficient.
At \(\zeta=0\), all measurement effects can instead attain risk \(100\)
with zero reports; seven-outcome minimality does not persist.

\subsection{Arbitrary errors around the original ensemble}

Let \(\widetilde\rho_a\) be four density matrices satisfying
\begin{equation}
 \max_a\|\widetilde\rho_a-\rho_a\|_{\mathrm{op}}
 \le\epsilon\le2^{-16}.
 \label{eq:preparationerror}
\end{equation}
This neighborhood is around the original, unattenuated ensemble, not
uniformly around all members of the replacement-noise family.
Write \(\delta\rho_a=\widetilde\rho_a-\rho_a\).
Then \(\Tr\delta\rho_a=0\),
\(\|\delta\rho_a\|_1\le4\epsilon\), and
\(\widetilde\rho_a\succeq(1/20-\epsilon)I_4\).

Project any report onto the closed convex hull of the four labels.
Euclidean metric projection cannot increase the distance to any label,
so it cannot increase either ensemble's risk and stays in the same
measurement class. The hull has diameter \(20\).
For the clipped report, the loss effect for each label satisfies
\[
 A_a=\int\|a(\omega)-\theta_a\|^2P(\dd\omega),
 \qquad 0\preceq A_a\preceq400I_4.
\]
The trace-zero perturbation therefore obeys
\[
 |\Tr\delta\rho_a A_a|
 =|\Tr\delta\rho_a(A_a-200I_4)|
 \le200\|\delta\rho_a\|_1\le800\epsilon .
\]
Averaging and taking infima over the identical set of clipped procedures
gives the uniform comparison
\begin{equation}
 |\widetilde{\mathcal L}_{\mathcal C}-\mathcal L_{\mathcal C}|
 \le800\epsilon .
 \label{eq:riskstability}
\end{equation}
It holds for both classes above, including arbitrary outcome spaces and
classical mixtures; an attained class minimum is not assumed.

For NH we need only an upper bound on its perturbed infimum.
Keep the feasible trial \(V_1=X,V_2=Y\). Direct multiplication gives
\[
 H=i[X,Y]=\operatorname{diag}(-1/2,-3/2,-5/2,9/2),
 \qquad D=X^2+Y^2+|H|=\operatorname{diag}(1,4,9,9).
\]
For any positive average \(S'\), decomposing \(H=H_+-H_-\) and
using the trace-norm triangle inequality gives
\[
 \|S'^{1/2}HS'^{1/2}\|_1\le\Tr S'|H|.
\]
At the original diagonal \(S\) equality holds, and \(X,Y\) attain
the NH infimum. Thus, with
\(\delta S=\widetilde S-S\) and
\(\delta B_j=\widetilde B_j-B_j\),
\[
 \widetilde{\mathcal L}_{\mathrm{NH}}-\mathcal L_{\mathrm{NH}}
 \le\Tr\delta S D-2\sum_j\Tr\delta B_jV_j .
\]
Here \(\|\delta S\|_{\mathrm{op}}\le\epsilon\) and
\(\Tr\delta S=0\), whereas \(\Tr|D-5I_4|=13\).
The first term is at most \(13\epsilon\).
The four fixed labels give \(\|\delta B_j\|_{\mathrm{op}}\le5\epsilon\).
Finally \(\Tr X^2=\Tr Y^2=7\) implies
\(\|X\|_1,\|Y\|_1\le\sqrt{28}<6\).
The two linear terms cost at most \(120\epsilon\), yielding
\begin{equation}
 \widetilde{\mathcal L}_{\mathrm{NH}}
 \le\mathcal L_{\mathrm{NH}}+133\epsilon .
 \label{eq:nhstability}
\end{equation}
This is a one-sided feasible-trial estimate, not a claim that the trial
remains an optimizer.

Combining Eqs.~\eqref{eq:riskstability} and \eqref{eq:nhstability}
with the original strict bounds proves
\begin{align}
 \widetilde{\mathcal L}_{\min}-\widetilde{\mathcal L}_{\mathrm{NH}}
 &>1/32-933\epsilon\ge1115/65536>1/64,
 \label{eq:robustnhgap}\\
 \widetilde{\mathcal L}_{\mathrm{PVM}}-\widetilde{\mathcal L}_{\min}
 &>25/192-1600\epsilon\ge325/3072>1/10.
 \label{eq:robustpvmgap}
\end{align}
There is also a guarantee for the original explicit seven-outcome
protocol \((P_*,a_*)\), with both its effects and reports unchanged.
Its original optimal reports are posterior means and lie in the label
hull. The same bounded-loss comparison therefore gives
\(\widetilde{\mathcal L}(P_*,a_*)\le\mathcal L_{\min}+800\epsilon\).
Combining this with the PVM side of Eq.~\eqref{eq:riskstability} yields
\begin{equation}
 \widetilde{\mathcal L}_{\mathrm{PVM}}
 -\widetilde{\mathcal L}(P_*,a_*)
 >25/192-1600\epsilon\ge325/3072>1/10 .
 \label{eq:fixedprotocoladvantage}
\end{equation}
Thus this fixed protocol retains its advantage even when the competing
PVM is reoptimized for the perturbed preparations. No knowledge of the
individual errors is required to use the fixed protocol, provided the
stated error bound applies. Measurement effects and reporting are ideal;
errors in the readout itself are not modeled.
Neither exact weak commutativity, continued global optimality of this
protocol, exact risks, nor a seven-outcome minimum is asserted under
arbitrary preparation errors.
The tolerances are conservative mathematical bounds in the stated
dimensionless loss units, not experimental calibration specifications.
The ancillary robustness checkers reconstruct the exact matrix identities
and constants; the measure and optimization arguments are the analytical
proof above.