\section{A weighted global projective bound from projector moments}
\label{app:pvmmoments}

For the same fixed nonscalar model and finite prior, we obtain
\begin{equation}
 94.5557442477<\mathcal L_{\mathrm{PVM},k}<94.555744248544 .
 \label{eq:sharppvmbracket}
\end{equation}
The lower endpoint is an exact computer-assisted certificate for every
complex orthonormal basis. The upper endpoint is an outward enclosure
of the explicit four-result protocol in Appendix~\ref{app:pvm}.
The interval is narrow, but neither its endpoints nor numerical agreement
establish an exact optimal projective measurement. Appendix~\ref{app:pvmexact}
supplies the separate matching lower bound and attaining basis.

\subsection{Necessary moments for every projective measurement}

Refining orthogonal projections into a rank-one basis and optimizing reports
cannot increase squared-error risk. It therefore suffices to lower-bound
every orthonormal basis \(v_i\) of \(\C^4\). With \(P_i=v_iv_i^\dagger\), set
\[
 R=\sum_iP_i\otimes P_i,\qquad Q=R^{\top_2},\qquad
 H=(B_k\otimes B_k^\dagger+B_k^\dagger\otimes B_k)/2.
\]
Here \(\top_2\) denotes partial transpose on the second tensor factor,
in the fixed preparation basis.
Both \(R\) and \(Q=\sum_iP_i\otimes\bar P_i\) are rank-four
orthogonal projections; \(R\) is supported on the symmetric subspace and
\(\Tr_2R=I_4\). In particular,
\begin{equation}
 R\succeq0,\quad Q\succeq0,\quad R^2=R,\quad Q^2=Q.
 \label{eq:doubleidempotency}
\end{equation}
These are necessary conditions, not a convex characterization of bases.

Put \(s_i=v_i^\dagger S_kv_i\) and \(b_i=v_i^\dagger B_kv_i\).
The \(s_i\) are the predictive probabilities of the outcomes.
Optimizing the reported estimates gives
\[
 \mathcal L(P)=\lambda-\sum_i\frac{|b_i|^2}{s_i}.
\]
Since \(pI\preceq S_k\preceq qI\), every \(s_i\) lies in \([p,q]\).
Replacing all denominators by \(p\) loses their dependence on the
measurement direction. Instead use the reciprocal secant
\begin{equation}
 \frac1{s}\le\frac{p+q-s}{pq},\qquad
 \frac{p+q-s}{pq}-\frac1s=\frac{(s-p)(q-s)}{pqs}\ge0 .
 \label{eq:reciprocalsecant}
\end{equation}
Define \(G_2=\sum_i|b_i|^2=\Tr(HR)\) and
\(G_3=\sum_i|b_i|^2s_i\). Then
\begin{equation}
 \mathcal L(P)\ge\lambda-\frac{(p+q)G_2-G_3}{pq}.
 \label{eq:momentgain}
\end{equation}
The additional term retains the state-dependent weight.

Use the normalized symmetric basis
\[
 |aa\rangle,\qquad
 (|ab\rangle+|ba\rangle)/\sqrt2\quad(0\le a<b\le3),
\]
in lexicographic order of \((a,b)\), and write \(\mathsf R\) for the
ten-by-ten matrix of \(R\) in this basis.
The basis vector indexed by \((a,b)\) has phase charge \(a+b\)
under \(D(w)\otimes D(w)\).
Average polynomial moments over this phase action and complex
conjugation. The real ladder makes the weighted objective invariant.
This averages moments of genuine bases, not bases themselves:
neither \((\mathbb E Q)^2=\mathbb E Q\) nor a symmetric optimizing
basis is assumed.

Take all monomials of degree at most two in \(\mathsf R_{AB}\)
with total charge zero. Identify commuting factors and conjugate
monomials, and let \(x\) be their real averaged values, including
\(x_{\varnothing}=1\). There are \(349\) such coordinates.
Because \(R\) is a norm-one projection,
\(|\mathsf R_{AB}|\le1\), hence
\begin{equation}
 |x_j|\le1.
 \label{eq:momentcompact}
\end{equation}
For each nonnegative charge, the moment matrix of the coordinates
\(1,\mathsf R_{AB}\) of that charge is positive semidefinite.
The constant occurs only in the charge-zero sector. Negative-charge
blocks are conjugate copies. Together with the charge blocks of
\(\mathbb E\mathsf R\) and \(\mathbb E Q\), these give \(21\)
positive-semidefinite matrices. Their sizes are, in order,
\[
 (17,14,12,8,5,2,1),\quad(1,1,2,2,2,1,1),\quad
 (1,2,3,4,3,2,1).
\]
The first family consists of averaged Gram matrices; the other two
are averages of positive matrices. Complex bases are not excluded.

The affine equations initially comprise the constant normalization,
all entries of \(\Tr_2R=I\) multiplied by monomials of degree zero
or one, and all entries of \(R^2-R=0\) and \(Q^2-Q=0\), followed by
the stated averaging. Zero rows are omitted, but dependent rows are
retained. These give \(277\) rows.
The reshape \(Q_{ab,cd}=R_{ad,cb}\) fixes the coefficients.

\subsection{The weighted denominator as a three-factor contraction}

Orthogonality gives \(P_iP_j=\delta_{ij}P_i\), and therefore
\begin{equation}
 T_3=R_{12}R_{23}=\sum_iP_i^{\otimes3},\qquad
 G_3=\Tr[(B_k\otimes B_k^\dagger\otimes S_k)T_3].
 \label{eq:thirdcontraction}
\end{equation}
Here \(R_{12}=R\otimes I\) and \(R_{23}=I\otimes R\).
This is algebraic bookkeeping for a single-copy basis, not a
three-copy measurement protocol.
In the ordinary tensor-product basis,
\begin{equation}
 (T_3)_{ijk,abc}=\sum_{f=0}^3 R_{ij,af}R_{fk,bc}.
 \label{eq:t3entries}
\end{equation}
Thus \(G_3\) is quadratic in the entries of \(\mathsf R\)
and uses the same \(349\) coordinates.

The physical \(T_3\) is positive and has symmetric support.
Impose its row and column permutation identities, by equating each entry
to the entry with both triples sorted, and its Hermitian identities,
before moment averaging. Include all resulting nonzero rows.
Together with the preceding equations these give \(Ax=b\) with \(731\)
rows. From each charge sector of the \(20\) sorted triples
\(0\le i\le j\le k\le3\), also retain the corresponding principal
submatrix of \(\mathbb E T_3\).
These are ordinary tensor-coordinate principal submatrices,
not matrices in a normalized three-particle symmetric basis.
Their sizes are
\[
 (1,1,2,3,3,3,3,2,1,1).
\]
Symmetrizing their entries gives ten real positive-semidefinite
blocks after conjugation averaging. In total there are \(31\) blocks
\(M_\ell(x)=\sum_j C_{\ell j}x_j\succeq0\).
All coefficients lie in \(\Q(\sqrt2)\).
These necessary moment conditions use established finite
polynomial-optimization machinery \cite{Lasserre2001};
no completeness or flat-extraction assertion is used.

\subsection{Bounding the coefficient residual in the dual identity}

Set \(c^Tx=-[(p+q)G_2-G_3]/(pq)\).
The ancillary file \texttt{pvm-weighted-certificate.json} specifies
\(A,b,c,C_{\ell j}\), a rational vector \(y\), and \(31\) rational
symmetric matrices \(Z_\ell\).
Exact positive LDL pivots prove \(Z_\ell\succ0\), with \(105\) pivots in total.
Define
\[
 r_j=c_j+(A^Ty)_j-\sum_\ell\Tr(C_{\ell j}Z_\ell).
\]
For every physical moment vector,
\begin{align}
 c^Tx
 &=\sum_\ell\Tr[Z_\ell M_\ell(x)]-b^Ty+r^Tx\nonumber\\
 &\ge-b^Ty-\sum_j|r_j| .
 \label{eq:residualdual}
\end{align}
The last step uses Eq.~\eqref{eq:momentcompact}.
No numerical solver tolerance is discarded.
A rational interval \([\ell,u]\), checked by
\(0<\ell<u\) and \(\ell^2<2<u^2\), encloses \(\sqrt2\).
Outward evaluation of every \(r_j\in\Q(\sqrt2)\) gives an exact
rational upper bound \(\varepsilon\ge\sum_j|r_j|\).
For the supplied certificate, \(\varepsilon<8\,10^{-10}\), and
\[
 b^Ty+\varepsilon<100-94.5557442477 .
\]
Equations~\eqref{eq:momentgain} and \eqref{eq:residualdual}
prove the strict lower endpoint of Eq.~\eqref{eq:sharppvmbracket}.
The objective already includes the factor \(1/(pq)\);
no further denominator division is taken in this last inequality.

The companion \texttt{pvm\_weighted\_checker.py} verifies the
fixed certificate's exact PSD and residual arithmetic.
The independently implemented
\texttt{independent\_pvm\_weighted\_checker.py} reconstructs every monomial,
constraint, matrix block and weighted objective coefficient from these
definitions, verifies the certificate by rational Schur elimination,
and rejects corrupted model, coefficient and positivity data.
Neither implementation needs a numerical optimization package.
The earlier, coarser \(1/p\) certificate remains reproducible with
\texttt{pvm-moment-certificate.json} and its two companion checkers.

The secant does not make the denominator exact.
Its total gain excess obeys the uniform bound
\[
 0\le
 \frac{(p+q)G_2-G_3}{pq}-\sum_i\frac{|b_i|^2}{s_i}
 \le\frac{(q-p)^2}{4p^2q}\,\|B_k\|_{\mathrm F}^2
 <3\,10^{-12}.
\]
Indeed, Eq.~\eqref{eq:reciprocalsecant} bounds each defect, and
\(\sum_i|b_i|^2\le\|B_k\|_{\mathrm F}^2\) by orthogonal projection
onto the basis-diagonal subspace.
This error estimate does not imply equality:
the three signal outcomes of the explicit protocol have
\(p<s_i<q\) and \(b_i\ne0\), so their secant defects are strictly positive.
Thus this secant certificate cannot itself be a matching proof.
Appendix~\ref{app:pvmexact} instead uses a reciprocal majorant with exact
contact at the attaining basis and a stronger positive moment identity.

The bounds extend to coarse projectors, randomized reports and
parameter-independent classical mixtures by the refinement and Jensen
arguments in Appendix~\ref{app:pvm}. The square prior adds \(500\) to
both endpoints and changes no measurement or risk difference.
No auxiliary-space projectivity restriction follows.