\section{A quantitative penalty for projective readout} \label{app:pvm} This appendix gives an elementary explanation of the projective penalty and an earlier feasible benchmark. Appendix~\ref{app:pvmmoments} improves the lower bound; Appendix~\ref{app:pvmexact} proves the exact optimum with a different mixing ratio. We retain the fixed nonscalar model, its two priors and the identity quadratic loss. This section concerns projective measurements on the original \(\C^4\), not a projective realization of a general POVM on a larger space. Bayesian comparisons between general and restricted projective readout are established \cite{Kaubruegger2023}. Normal completions \cite{Kaliuzhnyi2009,Wasson2013} and quantitative joint-diagonalization bounds \cite{Glashoff2013} provide related matrix methods. The cost below is a degenerate weighted quadratic form, rather than the full Frobenius off-diagonal energy of a Hermitian pair. \begin{proposition}[Certified projective-risk bracket] Let \(\mathcal L_{\mathrm{PVM},k}\) be the infimum over all PVMs on the original four-level system and all reporting rules. For either fixed prior, \begin{equation} \mathcal L_{\mathrm{NH},k}+\frac{112p}{165} \le\mathcal L_{\mathrm{PVM},k} \le\lambda-\frac{\mathcal A_k^2}{\tau_k}, \label{eq:pvmbracket} \end{equation} where \[ \tau_k=p+q+\frac{p+36t}{37},\qquad \mathcal A_k=2p+\frac{p+18(q+t)}{\sqrt{37}}. \] In particular, \begin{equation} \mathcal L_{\mathrm{PVM},k}>\mathcal L_{\min,k}+\frac{25}{192}. \label{eq:pvmpenalty} \end{equation} The lower bounds include parameter-independent classical mixtures of single-copy original-space PVM settings. The upper endpoint is the exact risk of the explicit four-result PVM given below, not the exact restricted optimum. Neither endpoint is asserted sharp. \end{proposition} \subsection{How projectivity constrains the risk} Projectivity forces the matrix of complex reports to be normal. We first express the risk as a weighted quadratic cost, then show why normality prevents that cost from vanishing. For deterministic complex reports \(z_i\) at orthogonal projections \(P_i\), put \(Z=\sum_i z_iP_i\). This matrix is normal, including for degenerate projections or coincident reports. Write \[ K_+=\operatorname{diag}(0,0,0,t),\qquad K_-=\operatorname{diag}(p,p,q,0), \] and retain the ladder \(C\), \(B_k=CK_++K_-C\), and \(T_{0,k}\). For \(E=Z-C\) define \[ q_U(E)=\Tr(EK_+E^\dagger+E^\dagger K_-E). \] Expanding its cross terms gives the unrestricted identity \begin{align*} &\Tr(S_kZ^\dagger Z)-2\Re\Tr(B_k^\dagger Z) +\Tr T_{0,k}-q_U(Z-C)\\ &\hspace{18mm}=\Tr[K_-(Z^\dagger Z-ZZ^\dagger)] . \end{align*} Normality makes the right side zero. The PVM risk is \(\lambda+\Tr(S_kZ^\dagger Z)-2\Re\Tr(B_k^\dagger Z)\); hence \begin{equation} \mathcal L(P,z)-\mathcal L_{\mathrm{NH},k}=q_U(Z-C). \label{eq:pvmcost} \end{equation} No first-moment constraint on the measurement is imposed. The general optimization framework is already the normal Procrustes problem \cite{Bierly2026}: set \(X_{\mathrm{in}}=S_k^{1/2}\), \(Y_{\mathrm{in}}=B_kS_k^{-1/2}\) and minimize \(\|ZX_{\mathrm{in}}-Y_{\mathrm{in}}\|_{\mathrm F}^2\) over normal \(Z\). For normal matrices this squared norm equals \[ \mathcal L(P,z)-\lambda+\Tr(B_kS_k^{-1}B_k^\dagger). \] Eliminating reports in an orthonormal basis \(v_i\) gives the established gain \(\sum_i|v_i^\dagger B_kv_i|^2/(v_i^\dagger S_kv_i)\) \cite{Bierly2026}. For scalar \(S\), the related nearest-normal Frobenius formulation is also established \cite{Noferini2026}. Full rank bounds \(Z\), and thus its eigenvalue reports, on bounded-cost sublevel sets; it does not assume arbitrary reporting rules are bounded. These frameworks do not by themselves certify a global minimum. The result of this appendix is an elementary lower bound and a feasible benchmark. Use the block notation \[ Z=\begin{pmatrix}A&b\\c^\dagger&d\end{pmatrix},\qquad A_0=\begin{pmatrix}0&1&0\\0&0&2\\0&0&0\end{pmatrix},\qquad b_0=(0,0,3)^T, \] and set \(D=A-A_0\), \(f=b-b_0\), and \(Q=q_U(Z-C)\). The top-left block of normality yields \[ H(A,b):=AA^\dagger-A^\dagger A+bb^\dagger=cc^\dagger, \qquad H_0=H(A_0,b_0)=\operatorname{diag}(1,3,5). \] Here \(p55/224\), hence \(112p/165>1/6\). Appendix~\ref{app:nonscalaroptimal} gives \(\Gamma_k<7/192\). Subtracting this excess above NH proves \eqref{eq:pvmpenalty} because \(1/6-7/192=25/192\). In particular, this refines the simpler \(3/64\) separation without solving the restricted optimization. For every nonzero PVM effect, \(\Tr(S_kP_i)>0\). Conditional Jensen replaces a randomized report by its mean without increasing risk; infinite-risk procedures obey the bound trivially. Parameter-independent choice of a fixed single-copy PVM setting preserves the uniform barrier on averaging. Forgetting that setting cannot improve Bayes risk. Auxiliary-space projections, a per-step outcome restriction on an adaptive instrument, and postselection are not the comparison class. This penalty does not quantify the larger class of all at-most-six-outcome general POVMs in Appendix~\ref{app:seven}. \subsection{An exact four-result projective benchmark} Let \(\omega=e^{2\pi i/3}\) and choose the orthonormal basis \[ \xi=\frac{e_1+6e_4}{\sqrt{37}},\qquad \eta=\frac{6e_1-e_4}{\sqrt{37}},\qquad \psi_j=\frac{\xi+\omega^je_2+\omega^{2j}e_3}{\sqrt3} \quad(j=0,1,2), \] consisting of the three \(\psi_j\) and \(\eta\). Their rank-one projectors form a normalized four-result PVM. The predictive probabilities and complex first moments are \[ \psi_j^\dagger S_k\psi_j=\tau_k/3,\qquad \psi_j^\dagger B_k\psi_j=\mathcal A_k\omega^j/3. \] The fourth probability is \((36p+t)/37>0\), and its first moment is zero. Use posterior-mean reports \(\mathcal A_k\omega^j/\tau_k\) and \(0\), respectively. The fourth outcome is retained, not postselected away. The attained risk is exactly \begin{equation} \mathcal L_{\mathrm{trial},k} =\lambda-\mathcal A_k^2/\tau_k\in\Q(\sqrt{37}). \label{eq:pvmtrial} \end{equation} Measurement entries also contain the cube root of unity; the risk's quadratic field is not a restriction to real measurement entries. The convenient mixing ratio \(1:6\) is a strictly suboptimal benchmark; the exact optimum is given in Appendix~\ref{app:pvmexact}. For \(\lambda=100\), the exact lower endpoint is \[ \mathcal L_{\mathrm{NH},k}+\frac{112p}{165} =\frac{74780508814454731109009}{792001166000594000099} >94.419695354840734919600705, \] and two independent rational enclosures give \[ 94.555744248543274360535112 <\mathcal L_{\mathrm{trial},k} <94.555744248543274360535113. \] Thus \(94.41969535<\mathcal L_{\mathrm{PVM},k}<94.55574425\). The four Fourier vectors \((1,w,w^2,w^3)^T/2\), \(w\in\{1,i,-1,-i\}\), with posterior reports \(3(1-p)w\), instead have risk \(\lambda-9(1-p)^2>\mathcal L_{\mathrm{trial},k}\). The same projectors and reports apply to both priors; changing \(\lambda\) adds \(500\) to their total risks. This is a certified bracket, not an exact PVM optimum. Appendix~\ref{app:pvmmoments} sharpens this lower endpoint using a separate exact moment certificate. The elementary barrier and explicit upper protocol in this section remain unchanged. Appendix~\ref{app:pvmexact} matches a stronger global bound with an attaining measurement. \texttt{pvm\_bracket\_checker.py} implements exact arithmetic in \(\Q(\sqrt{37},\omega)\), constructs all four projectors and reports, and checks the risk, Young coefficients, model floor and outward enclosures. \texttt{independent\_pvm\_bracket\_checker.py} independently constructs the orthonormal basis with symbolic radicals and isolates \(\sqrt{37}\) by rational polynomial intervals, using a fresh independent root replay. The earlier \texttt{pvm\_penalty\_checker.py} and its independently implemented formal-polynomial counterpart also retain checks of the normal-risk and perturbation identities. The universal projection, Cauchy, Young and reporting arguments above are analytical. No local nonconvex numerical minimum is used as proof.