\section{An algebraic optimum and an attaining measurement}
\label{app:optimal}

We determine the optimum for the scalar-average model by matching a
measurement lower bound with an explicit measurement. The construction
starts with a small rational system whose relevant root is specified
exactly below. The nonscalar model is treated separately. All terminating decimals used as interval endpoints denote
rational numbers.

\subsection{Exact branch and certificate}

Use \(G(u,v),d,a,b,f\) from Eq.~\eqref{eq:gramfamily}.
The two possibly nonzero off-diagonal row products of \(G\) are
\(u(-m_0-b)+av\) and \(a(w_0+u)-b(t_0+v)\); both vanish identically.
Thus \(GG^T\) is diagonal and \(G+G^T=-(B+B^\dagger)\) whenever \(d\ne0\).
The following fixes the pair without reference to a numerical optimizer.

\begin{lemma}[Algebraic branch certificate]
In the open rectangle
\[
 \mathcal R=(-.06554650,-.06554649)\times(-.75260404,-.75260402)
\]
there is exactly one solution of \(\partial_u f=\partial_v f=0\).
At that solution all rational functions below are defined, and the radius
and normalization predicates stated in this section hold exactly.
\end{lemma}

Here is an explicit algebraic representation and its verification.
Let \(H(\tau)=\sum_{j=0}^{19}h_j\tau^{19-j}\), with descending coefficients
\[
\begin{split}
(h_0,\ldots,h_{19})={}&(
131072,-1245184,11370496,-64700416,322347008,\\
&-1258181632,4206531584,-11721978624,27448818560,\\
&-53716943712,88531411546,-122973008797,149294842212,\\
&-170720691580,204085237692,-217657066176,156759120176,\\
&-65808870640,14355966240,-1244703600).
\end{split}
\]
There is exactly one root in \((.26218596,.26218600)\).
The certificate isolates that root more narrowly by
\[
 \tau_*=\text{the root of }H\text{ in }
 (n/10^{85},(n+1)/10^{85}),
\]
where
\[
\begin{split}
 n={}&262185980717760078672813858628124444009858668228893460\\
 &0183539745432024070490218894321
 \quad\text{(concatenation of the two displayed integer strings).}
\end{split}
\]
Set \(u_*=-\tau_*/4\) and \(v_*=\mathsf N(u_*)/\mathsf D(u_*)\).
The complete integer coefficient arrays of the degree-18 polynomials
\(\mathsf N,\mathsf D\), in descending powers of \(u\), are given by
\texttt{v\_numerator\_u\_desc} and \texttt{v\_denominator\_u\_desc}
in the ancillary file \texttt{P3349-optimal-certificate.json}.
That file also gives \(H\) and the two narrow interval endpoints as exact
rational strings. Together these four fields specify the branch fully.

A Sturm sequence over the rationals gives root count one in both intervals.
Substitution in the rational system gives \(\partial_u f=\partial_v f=0\)
by polynomial reduction modulo \(H\). The resulting rational univariate
representation (RUR) expresses the pair in terms of one algebraic root.
Interval evaluation maps
the root into \(\mathcal R\), with \(\mathsf D(u_*)\ne0\) and \(d(u_*,v_*)\ne0\).
The Hessian satisfies \(f_{uu}>0\) and
\(f_{uu}f_{vv}-f_{uv}^2>0\) throughout the closed rectangle.
The denominator \(d\) stays away from zero there, so \(f\) is smooth and
strictly convex there. This proves local uniqueness of the stationary pair,
not global minimization of \(f\) outside the rectangle.

The ancillary checker \texttt{P3349-optimal-checker.py} reproduces these exact
predicates. Its module \texttt{optimal\_exact\_arithmetic.py} implements
rational polynomial division, Sturm variations,
interval arithmetic, and first and second derivative jets.
Inverses in the quotient algebra are verified by the Euclidean algorithm
and rejected if they are nonunits; no unproved irreducibility of \(H\) is
needed. Rational intervals reject every denominator containing zero.
Thus the branch, rather than a rounded stationary point, supplies the inputs.

\subsection{The two report radii and effect directions}

Equality in the measurement lower bound requires the effects to lie in
the kernel of the comparison matrix. We now find these directions and
their allowed reports. All quantities refer to the selected pair above.
Reordering \(G_*\) by indices \((1,3,2,4)\) gives
\(\begin{pmatrix}0&R\\T&0\end{pmatrix}\), with
\[
 R=\begin{pmatrix}u_*&a_*\\-m_0-b_*&v_*\end{pmatrix},\qquad
 T=\begin{pmatrix}-w_0-u_*&b_*\\-a_*&-t_0-v_*\end{pmatrix}.
\]
Define
\[
 \sigma=16\Tr(RT),\qquad \kappa=256\det(RT),\qquad
 h(x)=x^2-\sigma x+\kappa.
\]
Exact interval evaluation proves
\(\sigma>0,\ \kappa>0,\ \sigma^2-4\kappa>0\).
Let \(x_-<x_+\) be the two roots, and \(r_s=\sqrt{x_s}>0\).
The block determinant identity is
\[
 \det(rI/2+2G_*)=h(r^2)/16.
\]
Put \(c_i=4(G_*G_*^T)_{ii}\).
For \(x=r^2\) equal to either root, define
\[
\begin{split}
 \nu_1(x)&=1,\\
 \nu_2(x)&=\frac{x/4+c_1}{r w_0},\\
 \nu_3(x)&=\frac{(x/4+c_2)(x/4+c_1)-xw_0^2}{xw_0m_0},\\
 \nu_4(x)&=\frac{(x/4+c_3)\nu_3(x)-m_0(x/4+c_1)/w_0}{r t_0}.
\end{split}
\]
The radial matrix \(N(r)=F(r)-A_*\) is tridiagonal, with diagonal
\(x/4+c_i\) and off-diagonal entries \(-r(w_0,m_0,t_0)\).
The first three row equations give precisely these recurrences; the final
one reduces to zero modulo \(h(x)\). Explicitly, with
\(n_2=x/4+c_1\) and
\(n_4=(x/4+c_3)\nu_3-m_0n_2/w_0\), that reduction is
\[
 (x/4+c_4)n_4-xt_0^2\nu_3=0\pmod{h(x)}.
\]
Consequently \(N(r_s)\nu(x_s)=0\).
These are left-kernel vectors of \(rI/2+2G_*\), not an assumed right kernel.
Write \(\nu_s=\nu(x_s)\).

\subsection{Choosing weights so the effects sum to the identity}

Since \(\kappa>0\), reduction modulo \(h\) uses
\(x^{-1}=(\sigma-x)/\kappa\).
The squared components reduce to
\(\nu_i(x)^2=a_i^{\mathrm{sq}}+b_i^{\mathrm{sq}}x\).
For the second component,
\[
 a_2^{\mathrm{sq}}=8c_1+\frac{16c_1^2\sigma}{\kappa},
 \qquad b_2^{\mathrm{sq}}=1-\frac{16c_1^2}{\kappa}.
\]
Interval evaluation excludes zero from \(b_2^{\mathrm{sq}}\).
Set \(y=(1-a_2^{\mathrm{sq}})/b_2^{\mathrm{sq}}\).
Exact reduction gives the two additional identities
\[
 a_3^{\mathrm{sq}}+b_3^{\mathrm{sq}}y=1,\qquad
 a_4^{\mathrm{sq}}+b_4^{\mathrm{sq}}y=1.
\]
For clarity, these coefficients are not the Bayesian moments \(B_1,B_2\).
An exact rational enclosure gives \(h(y)<0\), hence \(x_-<y<x_+\).
The weights
\[
 \alpha_+=\frac{y-x_-}{x_+-x_-},\qquad
 \alpha_-=\frac{x_+-y}{x_+-x_-}
\]
are strictly positive and satisfy
\(\alpha_-x_-+\alpha_+x_+=y\).
All four squared-component balances in
Appendix~\ref{app:scalarillustration} follow.
The zero reductions for the kernel and the three normalization identities,
and the strict interval predicates, are explicitly tested by the ancillary
checker. They do not rely on floating-point equality.

For \(i\ne j\), the phase sum in Eq.~\eqref{eq:optimalpovm} contains
\(\sum_{w^4=1}w^{i-j}=0\), since \(1\le|i-j|\le3\).
Its diagonal sum is one by those balances.
Thus the eight nonzero rank-one effects resolve the identity exactly.
Their supporting vectors lie in
\(\ker[F(r_sw)-A_*]\), as follows by phase conjugation of the radial kernel.
The NH equality kernels \(\ker F(z)\) are a different locus.

\subsection{The minimum risk and ensembles sharing it}

Integrating Eq.~\eqref{eq:optimalfactor} with the arbitrary-outcome argument
of the previous appendix gives
\[
 \mathcal L\ge\lambda-\sum_i c_i=\lambda-4f(u_*,v_*).
\]
For the eight effects the residual trace vanishes term by term, giving
equality. Thus this is a global optimum over all POVMs and estimators, not
only over two-seed strategies. Exact rational interval evaluation proves
\[
 .03592634498992<\Gamma_*<.03592634498993,\qquad
 94.28592634498992<\mathcal L_{\min}<94.28592634498993
\]
for the four-label ensemble. The exact risk is the algebraic expression
\(100-4f(u_*,v_*)\); the displayed decimals are only outward bounds.

Every finite-second-moment realization with the same \(S,B_j\) has
\(\mathcal L_{\min}=\lambda-4f(u_*,v_*)\) and the same attaining protocol.
This includes the connected square and the four-square density realization.
For the separate scalar depolarization path, keep the effects and multiply
all reports by \(\eta>0\); then
\[
 \mathcal L_{\min,\eta}=\lambda-4\eta^2f(u_*,v_*),\qquad
 \mathcal L_{\min,\eta}-\mathcal L_{\mathrm{NH},\eta}=\eta^2\Gamma_*.
\]
At \(\eta=0\) the zero report is optimal.

For the centered spectator lift, use comparison \(qA_*\oplus0\).
The attaining effects are \(M_{s,w}\oplus0\) and one further effect
\(0\oplus I_{N-4}\) reported at zero. The core effects are not multiplied by
the mixing weight \(q\): state probabilities already supply that factor.
They normalize to \(I_N\) and meet the full-matrix lower bound, yielding
\[
 \mathcal L_{\min,N}=\lambda-4qf(u_*,v_*),\qquad
 \mathcal L_{\min,N}-\mathcal L_{\mathrm{NH},N}=q\Gamma_*.
\]
Coupled-block competing measurements obey the same comparison.
This scalar calculation does not determine the nonscalar four-level optimum,
which is treated separately in Appendix~\ref{app:nonscalaroptimal}.
No optimizer uniqueness or outcome minimality is inferred.