\section{All optimal reported measurements and the seven-outcome requirement} \label{app:seven} Fix the nonscalar model \(k=1000001/1000000\) of Eq.~\eqref{eq:nonscalar}, with the algebraic branch selected in Appendix~\ref{app:nonscalaroptimal}. All quantities in this section refer to that branch. The result applies to both its four-atom and square priors. They have the same \(S_k,B_k,A_\diamond\); changing \(\lambda\) only adds a constant to every risk. No scalar-model outcome-minimality assertion is made. \begin{table}[h] \caption{Precision benchmarks for the four-state prior in Eq.~\eqref{eq:principalprior}, with \(\lambda=100\) and dimensionless squared error. The PVM excess is relative to the general optimum and includes parameter-independent mixtures of fixed projective settings on the original \(\C^4\). Projective implementations of POVMs on larger spaces are not restricted.} \label{tab:precisionclasses} \begin{ruledtabular} \begin{tabular}{ll} Benchmark & Established result\\ \hline NH relaxation & Strictly below achievable risk, gap \(\approx0.036\)\\ General POVM & Exact minimum; seven outcomes are minimal\\ Original-space PVM & Exact minimum, four outcomes; excess \(\approx0.270\), analytically \(>25/192\) \end{tabular} \end{ruledtabular} \end{table} \begin{figure}[h] \centering \input{figures/optimal-readout} \caption{Reports of a seven-outcome optimal measurement in the plane \(z=a_1+ia_2\). Filled circles: equilateral triple on the inner circle \(|z|=r_-\), at the cube roots of unity with equal weights. Open squares: four reports on the outer circle \(|z|=r_+\), forming two complex-conjugate pairs (dotted chords) whose cosines \(x,y\) obey \(xy=-1/2\) and \(x+y=\rho/2\) [Eq.~\eqref{eq:sevenphases}]. The radii and the value of \(\rho\) are drawn schematically, and marker sizes do not represent probabilities. The effects are \(D(w)\nu_s\nu_s^TD(w)^\dagger\) times the positive weights fixed by Eq.~\eqref{eq:facemoments}; seven arbitrary points need not satisfy those equations. The plot shows report coordinates, not Hilbert-space effect directions. Generative AI assisted the figure design and TikZ source.} \label{fig:readout} \end{figure} \subsection{Characterizing all optimal reported measurements} The lower bound determines where an optimal measurement can report and which effect directions it can use. Normalization then determines the allowed weights. We make these requirements explicit before counting the outcomes. For \(s\in\{-,+\}\), retain the positive radii \(r_s\), real left-null vectors \(\nu_s\) normalized by \(\nu_{s,1}=1\), and positive masses \(\alpha_s\). Write \(\mathbb T=\{w\in\C:|w|=1\}\). The radial comparison residual \(N_s=F_k(r_s)-A_\diamond\) is positive semidefinite and singular. Its nonzero neighboring entries make the tridiagonal recurrence determine every null-vector component from the first; hence its kernel is one-dimensional. Moreover, \(tI-N_s\) is nonnegative and irreducible for sufficiently large \(t\). Since \(N_s\succeq0\), its null vector is the Perron vector of \(tI-N_s\). Thus every component of \(\nu_s\) is strictly positive. Define the two coefficient determinants \begin{align} \Delta_1&=\nu_{-,2}\nu_{+,2}(\nu_{+,3}-\nu_{-,3}),\nonumber\\ \Delta_2&=\nu_{-,3}\nu_{+,2}\nu_{+,4} -\nu_{+,3}\nu_{-,2}\nu_{-,4}. \label{eq:sevenranks} \end{align} Exact interval evaluation at the selected branch gives \[ \nu_{+,2}^{\,2}\ne\nu_{-,2}^{\,2},\qquad \Delta_1>0,\qquad \Delta_2>0,\qquad A_+>A_->0,\quad A_s=\alpha_s\nu_{s,4}. \] These are branch-specific facts, not assumed genericity. \begin{theorem}[Optimal reported-measurement characterization] A POVM on the estimate plane attains \(\mathcal L_{\min,k}\) if and only if it is concentrated on the two circles \(z=r_sw\) and there has the form \begin{equation} M_s(\dd w)=D(w)\nu_s\nu_s^TD(w)^\dagger\,\mu_s(\dd w), \qquad w\in\mathbb T, \label{eq:optimalface} \end{equation} where \(\mu_s\) are positive measures satisfying \begin{align} \mu_s(\mathbb T)&=\alpha_s,\qquad m_{1,s}=m_{2,s}=0,\nonumber\\ \nu_{-,4}m_{3,-}+\nu_{+,4}m_{3,+}&=0,\qquad m_{j,s}=\int_{\mathbb T}w^j\mu_s(\dd w). \label{eq:facemoments} \end{align} This includes continuous and singular measures. \end{theorem} \begin{proof} The full-plane determinant is \(\det(F_k(z)-A_\diamond)=\det(S_k)h(|z|^2)^2\). It vanishes only on the two circles. The arbitrary-outcome risk identity gives, for every finite-risk reported POVM, \[ \mathcal L-\mathcal L_{\min,k} =\int_{\C}\Tr[(F_k(z)-A_\diamond)M(\dd z)]. \] Its Radon--Nikodym integrand is nonnegative. At equality it vanishes almost everywhere. For positive matrices \(Q,E\), \(\Tr(QE)=\|Q^{1/2}E^{1/2}\|_{\mathrm{HS}}^2=0\) forces \(\operatorname{ran}E\subset\ker Q\). Thus the measure is zero off the two circles, and on each circle its density has the rank-one form \eqref{eq:optimalface}. Its \(11\) entry defines \(\mu_s\), since \(\nu_{s,1}=1\). This argument constrains the pushed-forward reported measurement; it does not uniquely label raw refinements. Let \(q_s=\mu_s(\mathbb T)\). The \(11\) and \(22\) entries of \(\int M=I_4\) are \[ q_-+q_+=1,\qquad q_-\nu_{-,2}^{\,2}+q_+\nu_{+,2}^{\,2}=1. \] Their unique solution is \(q_s=\alpha_s\). The other populations then follow from the previously proved normalization equations. For Fourier order one, entries \(12\) and \(23\) give a homogeneous two-by-two system with determinant \(\Delta_1\); for order two, entries \(13\) and \(24\) give determinant \(\Delta_2\). Consequently both first and second moments vanish separately. Entry \(14\) is the third-moment equation in \eqref{eq:facemoments}, up to complex conjugation. The unused entry \(34\) vanishes with the first moments. Conversely, these conditions give every diagonal and off-diagonal entry of \(I_4\). Each effect also annihilates its comparison residual, so it attains the global minimum. \end{proof} \subsection{Why six outcomes cannot attain the minimum} Normalize the circle measures as \(\sigma_s=\mu_s/\alpha_s\) and put \(f_{j,s}=m_{j,s}/\alpha_s\). Their first two moments vanish, so the Gram matrix of \(1,w,w^2\) in \(L^2(\sigma_s)\) is \(I_3\). For \(n_s\) nonzero atoms its rank is at most \(n_s\); therefore each circle in a finite-outcome optimizer requires at least three atoms. If \(n_s=3\), the weighted square Vandermonde matrix \(V\) obeys \(VV^\dagger=I_3\), and is unitary. Column norms force all three weights to be \(1/3\). Orthogonality of distinct columns gives \(1+\tau+\tau^2=0\) for their phase ratios. The support is therefore a rotated equilateral triple, and \(|f_{3,s}|=1\). An optimizer with six or fewer nonzero reports would consequently have exactly three on each circle. The remaining equation \(A_-f_{3,-}+A_+f_{3,+}=0\) would require \(A_-=A_+\), contrary to the strict branch-specific inequality. This excludes arbitrary finite report locations, not just covariant or equally weighted ansatzes. The same lower bound on outcome number applies to finite raw POVMs with randomized reports. For any nonzero effect \(E\), its probability \(\Tr(S_kE)\) is positive because \(S_k>0\). Conditional Jensen reduces randomized squared-error reporting to its mean; the risk difference is \(\Tr(S_kE)\) times the report variance. At a finite global minimum this variance must be zero at every nonzero effect, or the risk could be strictly improved. Thus optimal raw reports are deterministic almost surely. Zero effects may be removed, and duplicate labels do not reduce the required number of distinct reports. \subsection{An algebraic three-plus-four measurement} Put \(\rho=-A_-/A_+\in(-1,0)\). On the smaller circle use the three cube roots of unity, each with scalar weight \(\alpha_-/3\). They give \((f_{1,-},f_{2,-},f_{3,-})=(0,0,1)\). For the larger circle set \begin{equation} x=\frac{\rho+\sqrt{\rho^2+8}}4,\qquad y=\frac{\rho-\sqrt{\rho^2+8}}4. \label{eq:sevenphases} \end{equation} Then \(x+y=\rho/2\), \(xy=-1/2\), \(00,\qquad q_y=\frac{x}{x-y}>0,\qquad q_x+q_y=1. \] Conjugate pairing makes the moments real. The identities \(xy=-1/2\) and \(x+y=\rho/2\) give \begin{align*} q_xx+q_yy&=0,\\ q_x(2x^2-1)+q_y(2y^2-1)&=0,\\ q_x(4x^3-3x)+q_y(4y^3-3y)&=\rho . \end{align*} Hence the third moments also cancel, \(A_-+A_+\rho=0\). For each selected phase, take its scalar weight \(\beta_{s,w}\) and use \[ E_{s,w}=\beta_{s,w}D(w)\nu_s\nu_s^TD(w)^\dagger,\qquad z_{s,w}=r_sw. \] These seven nonzero rank-one effects sum exactly to \(I_4\) and attain \(\lambda-\sum_i c_{i,k}\). Combined with the lower bound, the minimum number of nonzero outcomes of an optimal finite POVM is exactly seven. The eight-effect construction remains optimal but is not minimal. The seven-atom selection is not a \(U(1)\)-covariant measurement; the phase symmetry of the risk moments has not restricted the optimization domain. \subsection{A strict penalty for all at-most-six-outcome strategies} Let \(K\) be the compact convex hull of the chosen prior support, either the finite-prior diamond or the square. Projecting any report onto \(K\) cannot increase its squared distance from any parameter in that support. Together with conditional Jensen this reduces all six-outcome strategies to deterministic reports in \(K\). Include zero effects and define \[ \mathcal L_6=\min_{\substack{E_i\succeq0,\ \sum_{i=1}^6E_i=I_4\\ z_i\in K,\ i=1,\ldots,6}}\mathcal L(E,z). \] The feasible set is compact: \(0\preceq E_i\preceq I_4\), the normalization constraint is closed, and \(K\) is compact. The risk is a continuous polynomial in these finite-dimensional variables. A minimizer therefore exists. Equality \(\mathcal L_6=\mathcal L_{\min,k}\) would contradict seven-outcome minimality. Consequently \begin{equation} \delta_6=\mathcal L_6-\mathcal L_{\min,k}>0. \label{eq:sixpenalty} \end{equation} This holds separately for each of the two priors; no numerical value or explicit lower bound for \(\delta_6\) is asserted. Every projective measurement on the original \(\C^4\) has at most four nonzero projections and obeys this same strict penalty. Parameter-independent classical randomization over fixed, single-copy settings with at most six outcomes cannot bypass it: each conditional setting has risk at least \(\mathcal L_{\min,k}+\delta_6\), so their average does too. Discarding the setting flag cannot improve the optimal squared-error risk. All outcomes are normalized and retained, with no postselection. The statement is not an outcome bound per step of an adaptive instrument, a restriction on an auxiliary-space projective realization of a general POVM, or a hardware/ancilla minimum. Projective strategies for finite-data Bayesian estimation are studied by Law and Bern\'ad \cite{Law2026}; their construction is restricted to PVMs and does not claim unrestricted global optimality. Seven-outcome four-level measurements also occur in point tomography \cite{Martinez2025}, where the target is local pure-state reconstruction under infidelity. Here the result instead concerns a full-rank, two-coordinate, single-copy prior-integrated quadratic-risk problem. Neither the number seven alone nor the use of moment ranks is a new general quantum-estimation principle. \subsection{Exact predicates and reproducibility} \texttt{seven\_predicate\_checker.py} verifies the selected root and the strict component, population-rank, harmonic-rank, amplitude and phase-weight predicates using weighted block formulas and outward dyadic intervals. \texttt{independent\_seven\_checker.py} uses explicit component formulas, decimal intervals, a fresh independent root replay, and seven exact symbolic identities for the conjugate pairs and cube roots. Both reject a changed model parameter. Their rational square-root enclosures do not use floating-point signs. These finite checks support the branch-specific hypotheses above; the measure-theoretic characterization, rank equality case and compactness argument are supplied analytically in this section.