\section{The exact projective optimum}
\label{app:pvmexact}

We now match a lower bound over every complex orthonormal basis with an
explicit four-result measurement. Throughout this appendix, \(k=1000001/1000000\),
the preparation space is the original \(\C^4\), and the prior and loss are
those of Eq.~\eqref{eq:principalprior}. The tensor operators below encode
one readout basis; they do not supply additional encoded copies.

\subsection{An attaining basis and its exact risk}

Use \(\beta_1,\beta_2,\beta_3,D_0,D_2,\delta\) as defined in the main text. The equation
\begin{equation}
 (\beta_1D_0-D_2\beta_3h)^2(1+h^2)-\beta_2^2\delta^2h^2=0,
 \qquad h>\frac{\beta_1D_0}{D_2\beta_3},
 \label{eq:pvmquartic}
\end{equation}
selects a unique positive root \(h_*\). Indeed, on this interval the
equation is equivalent to
\((D_2\beta_3-\beta_1D_0/h)\sqrt{1+h^2}=\beta_2\delta\).
Its left side increases strictly from zero to infinity, whereas the
right side is positive. The ancillary \texttt{candidate.json} also
specifies a rational isolating interval for this root.

Set \(x_*=h_*/\sqrt{1+h_*^2}\) and
\(y_*=1/\sqrt{1+h_*^2}\). The four vectors displayed in the main text
are orthonormal: \(x_*^2+y_*^2=1\) and
\(1+\omega+\omega^2=0\) give their norms and inner products directly.
For \(P_j=v_jv_j^\dagger\), \(j=0,1,2\),
\[
 \Tr(S_kP_j)=\frac{\tau_*}{3},\qquad
 \Tr(B_kP_j)=\frac{A_*\omega^j}{3},
\]
whereas \(\Tr(B_kP_3)=0\). Completing the square separately at each
outcome gives the optimal reports stated above and the attained risk
\begin{equation}
 \mathcal L_*=100-g_*,\qquad g_*=A_*^2/\tau_*.
 \label{eq:pvmattainedrisk}
\end{equation}
No stationarity or restricted-family argument is used as a global lower
bound in this step.

All quantities needed by the bound lie in the real algebraic field
\(K=\Q[h_*]\). In particular,
\[
 y_* =\frac{D_2\beta_3h_*-\beta_1D_0}{\beta_2\delta h_*},\qquad x_*=h_*y_*.
\]
The selected branch makes \(y_*>0\), and Eq.~\eqref{eq:pvmquartic}
gives \(x_*^2+y_*^2=1\). Thus the square roots in the basis introduce
no additional field for the risk or the certificate coefficients.
The exact enclosure of the final optimum is
\begin{equation}
\begin{split}
 94.555744248543241682944116234978
 &<\mathcal L_{\mathrm{PVM},k}\\
 &<94.555744248543241682944116234979.
\end{split}
 \label{eq:pvmexactenclosure}
\end{equation}
The enclosure describes the algebraic value in Eq.~\eqref{eq:mainpvmexact};
equality of upper and lower risks follows from the proof below, not from
the width of this interval.

\subsection{A denominator bound with exact contact}

For any rank-one PVM write \(s_i=\Tr(S_kP_i)\) and
\(w_i=|\Tr(B_kP_i)|^2\). Since \(S_k\succeq pI\), \(s_i\ge p>0\).
For \(\alpha=\tau_*/3\), define
\[
 H_\alpha(s)=\frac1\alpha-\frac{s-\alpha}{\alpha^2}
                +\frac{(s-\alpha)^2}{p\alpha^2}.
\]
The exact identity
\begin{equation}
 H_\alpha(s)-\frac1s
 =\frac{(s-\alpha)^2(s-p)}{p\alpha^2s}\ge0
 \label{eq:pvmcontactmajorant}
\end{equation}
holds for every competing outcome. With
\[
 G_2=\sum_iw_i,\quad D_3=\sum_iw_i(s_i-p),\quad
 D_4=\sum_iw_i(s_i-p)^2,
\]
it gives
\begin{equation}
 \sum_i\frac{w_i}{s_i}
 \le \frac{G_2}{p}-\frac{2\alpha-p}{p\alpha^2}D_3
                         +\frac{D_4}{p\alpha^2}.
 \label{eq:pvmcontactgain}
\end{equation}
At the proposed basis the three signal probabilities equal \(\alpha\),
and the remaining \(w_i\) vanishes. Thus this inequality has exact
contact there, unlike the reciprocal secant.

\subsection{Universal moments of complex projective measurements}

Let \(R=\sum_iP_i\otimes P_i\) and \(Q=R^{\top_2}\), with the
partial transpose convention of Appendix~\ref{app:pvmmoments}.
Both are orthogonal projections. Moreover,
\[
 \Tr_2R=I,\qquad
 T_3=R_{12}R_{23}=\sum_iP_i^{\otimes3},\qquad
 T_4=R_{12}R_{23}R_{34}=\sum_iP_i^{\otimes4}.
\]
These facts hold for arbitrary complex basis vectors. The latter tensors
are positive, and their row and column indices are separately symmetric.
They express \(D_3\) and \(D_4\) as polynomials of degree two and three
in the entries of \(\mathsf R\), the matrix of \(R\) in the normalized
symmetric basis of Appendix~\ref{app:pvmmoments}.

Average these polynomial moments over
\(D(e^{i\phi})\otimes D(e^{i\phi})\) and complex conjugation.
The real ladder and diagonal \(S_k\) make all three objective components
invariant. This is an average of moment vectors of genuine complex bases,
not an assumption that a competitor is real or covariant.
For lexicographically ordered pairs \(A=(a,b)\), \(a\le b\), put
\[
 m_A=\begin{cases}1,&a=b,\\2,&a<b,\end{cases}
 \qquad X_{AB}=\frac{\mathsf R_{AB}}{\sqrt{m_Am_B}}.
\]
The ancillary model lists the real averaged monomials in these coordinates,
including the empty monomial of value one. Entries of its matrices and
equalities have rational coefficients.

The positive matrices \(M_\ell\) are principal submatrices and positive
diagonal congruences of the following universal constructions: charge
blocks of the Gram matrix of \(1,\mathsf R_{AB}\); averages of \(R\)
and \(Q\); symmetric tensor-coordinate blocks of \(T_3,T_4\); and
localizations of \(R\) and \(I-R\) by constant or diagonal coordinate
features. The additional two-by-two localizations use an arbitrary
complex coordinate feature on each diagonal and the corresponding
product in the off-diagonal entry. Positivity follows by restricting
\(R\) or \(I-R\) to those two feature-weighted coordinates, then
averaging. Taking the real part preserves positivity of these real
symmetric blocks. Explicitly, for complex coordinate features \(v,w\),
let \(V=[\bar v e_a,w e_b]\). Then
\[
 V^\dagger\mathsf R V=
 \begin{pmatrix}
 |v|^2\mathsf R_{aa}&vw\mathsf R_{ab}\\
 \overline{vw\mathsf R_{ab}}&|w|^2\mathsf R_{bb}
 \end{pmatrix}\succeq0.
\]
The same congruence applies to \(I-\mathsf R\). Neither feature must
be real, and the selected coordinates need not differ.

The equations \(A_r=0\) consist of the partial-trace identities,
\(R^2=R\), \(Q^2=Q\), and the separate permutation symmetries and
partial traces of \(T_3,T_4\), together with their listed monomial
localizations. Multiplication precedes averaging. In particular, these
are not idempotency equations for the averaged matrices.
The supplied model contains \(3,198\) positive matrix blocks and
\(17,538\) equations. Its reconstruction checks every matrix entry,
every equation and all three weighted objective components.
Only necessary conditions for projective measurements are used; no
claim that this finite relaxation characterizes all bases is required.

\subsection{An exact positive identity and global matching}

Each \(M_\ell\) is positive for every physical basis, and each \(A_r\)
vanishes for every such basis. The matrices \(N_\ell\) form congruences
of these positive blocks; the supplied \(Q_\ell\) weight their traces.
If \(M_\ell\) has size \(d_\ell\), then \(N_\ell\) is
\(d_\ell\) by \(r_\ell\) and \(Q_\ell\) is \(r_\ell\) by \(r_\ell\);
the factor file specifies these dimensions.
For these matrices \(N_\ell\) and symmetric \(Q_\ell\) over
\(K\), and multipliers \(f_r\in K\), the literal coefficient identity is
\begin{equation}
\begin{split}
 g_* -\frac{G_2}{p}+\frac{2\alpha-p}{p\alpha^2}D_3
                 -\frac{D_4}{p\alpha^2}
 &=\sum_\ell\Tr\!\left[Q_\ell N_\ell^T M_\ell N_\ell\right]
   +\sum_r f_r A_r .
\end{split}
 \label{eq:pvmexactidentity}
\end{equation}
Every coefficient is checked exactly modulo the quartic, without a
coefficient residual. Every nonempty \(Q_\ell\) is strictly positive
at the selected real root. Directed rational interval LDL factorization
certifies all \(4,551\) pivots as positive; integer-dyadic and Fraction
backends give the same result. The matrices in this positivity check are
the same matrices appearing in the exact identity.
A separate implementation also checks the complete identity and all
\(4,551\) signs. It uses a four-coefficient rational quotient,
Gaussian inverses, rational Sturm isolation and exact algebraic LDL,
without importing the preceding interval-Schur implementation.

For a physical basis, \(A_r=0\), \(M_\ell\succeq0\), and
\(N_\ell^TM_\ell N_\ell\succeq0\). Each trace on the right is therefore
nonnegative. Equations~\eqref{eq:pvmcontactgain} and
\eqref{eq:pvmexactidentity} give \(\sum_iw_i/s_i\le g_*\), and hence
\(\mathcal L(P)\ge100-g_*\) for every complex orthonormal basis.
The explicit basis attains this value, proving Eq.~\eqref{eq:mainpvmexact}.
Refinement and conditional-mean reporting extend the lower bound to
coarse projectors and randomized reports. Parameter-independent classical
mixtures of fixed settings cannot lower it, since their risks are averages
of risks already bounded below. No restriction on projective dilations
of POVMs in a larger space is inferred.

\subsection{Reproducing the proof}

This is a computer-assisted exact proof. The analytical constructions above
establish that its constraints hold for every complex basis; exact arithmetic
establishes the coefficient identity and the signs of its Gram matrices.
The full certificate data and verification code are part of the proof,
not an additional numerical test. The PDF states the identity's form rather
than printing every coefficient. The directory \texttt{anc/pvm-global/}
contains the rational model, algebraic factors, direct coefficient identity
and root specification.
The verifier \texttt{audit\_model.py} reconstructs the universal positive
operators and checks the physical identities. The independent
\texttt{verify\_identity.py} reconstructs the weighted target and every
raw coefficient, without an equality-elimination transcript. The model
audit uses the supplied operator generator; the analytic projector,
tensor and feature-congruence constructions above establish why its
constraints apply to every complex basis.
The two modes of \texttt{verify\_psd.py} certify the ordered-field signs
on that same certificate. Finally, \texttt{verify\_attainment.py}
multiplies all sixteen pairs of the four explicit projectors, checks
normalization and all four optimal reports, and encloses the achieved
risk with rational arithmetic. These checks require no numerical
optimization software. File hashes and the commands are supplied in
\texttt{SHA256SUMS.txt} and \texttt{README.txt} in that directory.
The additional \texttt{independent\_exact.py} requires GMP rational
arithmetic and implements the separate algebraic check just described.
All enclosure receipts are bound to their arithmetic dependencies.