\section{Global optimum, arbitrary outcomes, and extensions} \label{app:globalcomparison} We first prove that the NH minimum and the measurement lower bound hold without restricting the outcome space. Here \(S,B_j,C,U,T_0,F,G,A\) refer to the scalar weighted-ladder model in Appendix~\ref{app:scalarillustration}. Appendix~\ref{app:nonscalaroptimal} treats the nonscalar model used in the main text. \subsection{The NH minimum and measurements with arbitrary outcomes} The NH expression is a supremum of quadratic-minus-linear functions. We identify one that is minimized at \(C\) and equals the supremum there. For \(Z=V_1+iV_2\), expansion gives \[ q_R(Z)=\Tr S(V_1^2+V_2^2) +\Tr\!\left[R\,iS^{1/2}[V_1,V_2]S^{1/2}\right]. \] The dual representation of the Hermitian trace norm proves \eqref{eq:nh}. Each \(q_R\) is a positive-semidefinite quadratic because its two weights are positive. At \(C\), the \(U\) branch is active. The equation \(B=CK_++K_-C\) cancels the terms linear in an arbitrary complex perturbation \(E\): \[ q_U(C+E)-2\operatorname{Re}\Tr[B^\dagger(C+E)] =q_U(C)-2\operatorname{Re}\Tr(B^\dagger C)+q_U(E). \] The supremum over \(R\) is at least this branch at every \(C+E\), and agrees with it at \(C\). Thus \(C\) minimizes the full NH expression, not just a stationary branch. Stationarity and quadratic homogeneity give \(q_U(C)=\operatorname{Re}\Tr(B^\dagger C)=\Tr T_0\). The positive dual matrix has the signed decomposition \[ F(z)=(zI-C)K_+(\bar zI-C^\dagger) +(\bar zI-C^\dagger)K_-(zI-C)\succeq0. \] This yields \eqref{eq:globalf}. To compare with a general measurement, group its outcomes by the reported estimate. A randomized reporting rule can likewise be included in the POVM through its classical Markov kernel. The finite scalar measure \(\mu(E)=\Tr M(E)\) has total mass four; write \(M(\dd z)=M_z\mu(\dd z)\), with \(M_z\succeq0\) almost everywhere. Finite risk and the finite prior second moment imply a finite predictive estimate second moment by \(|z|^2\le2|z-\theta|^2+2|\theta|^2\). For every full-rank average \(S\succeq s_*I\), \(s_*>0\), the predictive measure dominates \(s_*\mu\), controlling its estimate second moment. In the simplest model \(s_*=1/4\); Cauchy--Schwarz controls the first moment. Expansion of squared loss is therefore legitimate and proves \eqref{eq:globaldual}. For the global factor, use \(D(w)BD(w)^\dagger=\bar w B\). The diagonal matrices \(S,T_0,A,GG^T\) commute with \(D(w)\), and \[ Q(r)Q(r)^T=r^2I/4+r(G+G^T)+4GG^T =r^2I/4-r(B+B^\dagger)+T_0-A. \] This proves \eqref{eq:factor}, including the origin. Positivity implies \(\Tr[M_z(F(z)-A)]\ge0\) almost everywhere. The fixed matrix \(A\) is bounded and integrable, even though it is indefinite. Normalization gives its integral as \(\Tr A\). The resulting lower bound does not require a finite outcome set or the existence of a minimizing measurement. Nor does it impose a first-moment constraint on the measurement. No density relative to Lebesgue measure is assumed: the argument covers atomic, continuous and singular measures, including uncountable supports. A parameter-independent ancilla and normalized sequential instruments on the same received copy induce a normalized effective POVM. In adjoint form, their positive outcome maps give \(P(E)=\Phi_E^*(I)\), and trace preservation of their total instrument gives \(P(\mathcal O)=I\). Adaptive feedforward and classical randomization are therefore included. All failure outcomes remain normalized. Conditional postselected loss or another parameter-encoded resource defines a different experiment. \subsection{A nonoptimal Fourier benchmark} For the four fourth roots of unity \(w\), the vectors \(\psi_w=(1,w,w^2,w^3)^T/2\) are orthonormal, by the finite geometric sum of \((w\bar v)^j\), \(0\le j\le3\). Hence \(\sum_w\psi_w\psi_w^\dagger=I\). They satisfy \(\psi_w^\dagger B\psi_w=9w/16\), \(\psi_w^\dagger S\psi_w=1/4\), and \(\psi_w^\dagger T_0\psi_w=23/16\). Reports \(z_w=9w/4\) give \[ \sum_w\psi_w^\dagger F(z_w)\psi_w=11/16. \] This proves the nonsharp excess-risk upper bound \(11/16\) for every realization with the same \(S,B,T_0\). The finite-ensemble risk is \(1519/16\); the square-prior risk is \(\lambda-\Tr T_0+11/16\). Neither the hidden label nor a parameter-dependent measurement is used. Appendix~\ref{app:optimal} supplies a strictly better attaining protocol; these projectors are not optimal. \subsection{Replacing the maximally mixed average} For Eq.~\eqref{eq:nonscalar}, the only nonzero entries of \(B_k\) are \[ B_k{}_{12}=p,\qquad B_k{}_{23}=2p,\qquad B_k{}_{34}=3(q+t), \qquad T_{0,k}=\operatorname{diag}(0,p,4p+9t,9q). \] Thus \(B_{j,k}\) are traceless and Hermitian. The posterior means have zero diagonal, respective upper entries \(h_j\) and \(-ih_j\), and conjugate lower entries. Their Sylvester equations hold entrywise. Their commutator gives Eq.~\eqref{eq:nonscalarwc}; the first edge contributes zero since the first two average eigenvalues are both \(p\). The other contributions cancel by the chosen \(t\), not by an automatic trace identity for nonscalar averages. The signed commutator at the unchanged \(C\) is diagonal, with entries \(-p/2,-3p/2,-5q/2,9t/2\). All \(p,q,t\) are positive, so its active sign remains \(U\). The weights \(K_{\pm,k}\) are positive and obey \(B_k=CK_{+,k}+K_{-,k}C\). The active-branch proof applies to every complex perturbation, not only weighted ladders, yielding the unrestricted global NH minimum \(\lambda-\Tr T_{0,k}\). The four equally weighted preparations \(S_k\pm B_{j,k}/5\) at \(\pm10e_j\) have moments \(S_k,B_{j,k}\). The square realization follows from covariance \(300I_2\). For the specified rational \(k\), exact diagonal dominance gives \(\rho_\theta\succeq I/20\) in both cases, throughout the square in the continuous case. The independent real and imaginary coherence directions preserve identifiability and smoothness on a neighborhood of the support. To cover every reporting rule, let \(K\) be the closed convex hull of the prior support. Euclidean metric projection \(\Pi_K\) satisfies \(\|\Pi_K(a)-\theta\|\le\|a-\theta\|\) for every \(\theta\in K\), by its variational inequality. Replacing each report by \(\Pi_K(a)\), also in a randomized rule, never increases risk. Thus every original strategy has risk at least that of a strategy reporting in \(K\). There is no conditioning or truncation of the POVM. The finite-prior hull has \(|z|\le10\); the square has \(|z|^2\le1800\) and \(|z|\le45\). Let \(d_S=\|S_k-S\|_{\mathrm{op}}\), \(d_B=\|B_k-B\|_{\mathrm{op}}\), and \(d_T=\|T_{0,k}-T_0\|_{\mathrm{op}}\). The first and last differences are diagonal, and the middle is a weighted shift whose norm is its largest entry modulus. Exact rational comparisons give \[ d_S\le1/7000000,\qquad d_B\le3/4000000,\qquad d_T\le5/4000000. \] Uniformly on the projected region, \[ \|F_k(z)-F(z)\|_{\mathrm{op}}\le |z|^2d_S+2|z|d_B+d_T. \] Take \(e=100d_S+20d_B+d_T\) for the atoms or \(e=1800d_S+90d_B+d_T\) for the square. Since \(F(z)\succeq A\), we obtain \(F_k(z)\succeq A-eI\). Normalized integration over all outcomes gives \[ \mathcal L-\mathcal L_{\mathrm{NH},k}\ge\Tr(A-eI) =\gamma-4e\ge1/32. \] The rational replay checks these outward inequalities. The same bound holds before projection and for the unrestricted infimum. Continuity gives sufficiently nearby \(k\) within this tuned weak-commutativity family, not arbitrary preparation stability. \subsection{A separate scalar depolarization path} For the scalar-average realization and \(0<\eta\le1\), \(S_\eta=S\), \(B_\eta=\eta B\), and \(C_\eta=\eta C\). The active sign remains \(U\), so \(T_{0,\eta}=\eta^2T_0\) and \(\mathcal L_{\mathrm{NH},\eta}=\lambda-\eta^2\Tr T_0\). The global identity \[ F_\eta(z)=\eta^2F(z/\eta)\succeq\eta^2 A \] gives the elementary gap at least \(\eta^2\gamma\). The Fourier effects and reports \(\eta9w/4\) give excess \(\eta^2 11/16\). Weak commutativity persists since the average remains scalar; full rank persists under convex mixing with \(I/4\). At \(\eta=0\), no outcome carries parameter information and the centered prior makes the zero report optimal: \(\mathcal L_{\min}=\mathcal L_{\mathrm{NH}}=\lambda\). Appendix~\ref{app:optimal} sharpens this to exact excess \(\eta^2\Gamma_*\). This is a statement about the scalar path only. \subsection{The complete equality-kernel locus of the scalar example} The four kernel equations in Appendix~\ref{app:scalarillustration} follow from the two positive residual terms. At \(z=0\) they give only \(e_1\). At \(z\ne0\), a nonzero solution has \(\psi_3\ne0\); the last two equations force \(|z|=3\), and the first three determine all components from \(\psi_1\). Conversely those components solve the equations on that circle. The second equation then gives the population identity. If equality held, the nonnegative dual integral would vanish, so \(M_z\) would be kernel-supported almost everywhere. Integrating \(9(M_z)_{22}=4(M_z)_{33}\) would contradict normalization. \subsection{An absolutely continuous prior} Each square is \(10s e_j+[-1,1]^2\); they are disjoint and each has area four. The density \(1/16\) therefore assigns each square probability \(1/4\). Write \(\theta=10s e_j+\eta\) on that square. The conditional mean of \(\eta\) is zero, with \(\mathbb E\eta_1^2=\mathbb E\eta_2^2=1/3\). Since the state is constant there, the zeroth and first state moments are unchanged, and \(\lambda=100+2/3=302/3\). Equation \eqref{eq:nh} consequently gives \(\mathcal L_{\mathrm{NH}}=1139/12\). The same dual matrix and comparison certify the same lower bound \(\gamma\) on excess risk. Appendix~\ref{app:optimal} gives its exact value \(\Gamma_*\). The extra within-square variance is not encoded in the state; it adds equally to all achievable risks and to the NH bound. Outside the prior support the state can be defined arbitrarily as a valid density matrix. No differentiability or injectivity is asserted. \subsection{A smooth identifiable family on a connected square} For \eqref{eq:smoothprior}, symmetry gives \(\mathbb E\theta_j=0\) and \(\mathbb E\theta_j\theta_k=300\delta_{jk}\). Linearity and tracelessness of \(B_j\) therefore give exactly \(\mathbb E\rho_\theta=I/4\), \(\mathbb E\theta_j\rho_\theta=B_j\), and \(\Tr\rho_\theta=1\). The maximum absolute off-diagonal row sums of \(B_1\) and \(B_2\) are both one. The triangle inequality and \(|\theta_j|\le30\) bound each off-diagonal row sum of \(\rho_\theta\) by \(1/5\). Its diagonal is \(1/4\), proving the stated uniform positive margin. That strict margin persists on an open neighborhood of the square; the affine state map is smooth there. Since \(B_1\) is real with a nonzero \(12\) entry and \(B_2\) is purely imaginary with a nonzero \(12\) entry, these two Hermitian directions are linearly independent over \(\mathbb R\), proving injectivity. The prior second moment is \(600\); all other quantities in \eqref{eq:globalf} are unchanged. Hence the NH value is \(2377/4\), and the same global factor yields the same excess-risk bound. The prior is uniform on its compact square, not a claimed globally smooth positive density on all of \(\mathbb R^2\). \subsection{Strict separation in every higher Hilbert-space dimension} Each prior realization above is centered, with \(\mathbb E_\pi\theta=0\). For \(N>4\), choose \(00\), proving the strict gap for all POVMs, including ones that couple the blocks. The posterior means are \(S_j\oplus0\), with weak commutator \(q\Tr S[S_1,S_2]=0\), and every preparation remains full rank. For example, \(q=4/(N+1)\) and \(\tau=I_{N-4}/(N-4)\) give average-state eigenvalues \(1/(N+1)\) and \((N-3)/[(N+1)(N-4)]\), which are distinct. Thus the higher-dimensional conclusion is not confined to scalar average states.