\documentclass[aps,prl,reprint,superscriptaddress,nofootinbib]{revtex4-2} \usepackage{amsmath,amssymb,amsthm,bm} \usepackage{microtype} \usepackage{tikz} \usepackage[hidelinks]{hyperref} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\supp}{supp} \newcommand{\dd}{\,\mathrm d} \newcommand{\1}{I} \newcommand{\C}{\mathbb C} \newcommand{\Q}{\mathbb Q} \newtheorem{theorem}{Theorem} \newtheorem{corollary}{Corollary} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \hyphenation{OpenAI} \begin{document} \title{Bayesian Weak Commutativity Does Not Guarantee Attainability of the Nagaoka--Hayashi Bound} \author{Qihang Wang} \email{2501110049@stu.pku.edu.cn} \affiliation{School of Mathematical Sciences, Peking University, Beijing 100871, China} \hypersetup{ pdftitle={Bayesian Weak Commutativity Does Not Guarantee Attainability of the Nagaoka-Hayashi Bound}, pdfauthor={Qihang Wang}, pdfsubject={Bayesian quantum estimation and measurement incompatibility}, pdfkeywords={Bayesian estimation, Nagaoka--Hayashi bound, normalization witness, precision gap, POVM} } \begin{abstract} Bayesian weak commutativity has been suggested as the most likely condition for reaching the Nagaoka--Hayashi (NH) bound in multiparameter estimation. We show that it is not sufficient. For a full-rank four-level ensemble obeying it exactly, the effects required for NH equality cannot sum to the identity. We find the optimal risk exactly: it lies strictly above the bound and needs seven outcomes, while projective readout on the four levels adds over seven times the NH gap. A robust normalization witness bounds the gap from below across a family including truncated oscillator ladders. \end{abstract} \maketitle \textit{Introduction}---When a quantum probe must reveal several parameters at once, such as the components of a vector field \cite{Kaubruegger2023}, the optimal measurements for the individual parameters are generally incompatible \cite{Helstrom1976,Holevo2011,Ragy2016,DemkowiczDobrzanski2020}. With collective measurements on many copies, the quantum Cram\'er--Rao bound is asymptotically attainable exactly when the symmetric logarithmic derivatives commute on average in the probe state, the weak-commutativity condition \cite{Ragy2016}. For single-copy, locally unbiased estimation, the Nagaoka--Hayashi (NH) bound extends Nagaoka's two-parameter bound \cite{Nagaoka1989,Nagaoka1991} to more parameters \cite{Hayashi1999,Conlon2021}; it is a semidefinite program \cite{Conlon2021} with a Bayesian counterpart \cite{Suzuki2024}. In the local setting, a vanishing averaged commutator does not guarantee that NH is reached: a rank-three four-level example misses the bound \cite{Conlon2025}. Bayesian estimation asks a different question: the parameters follow a known prior, estimates may be biased, and one minimizes the prior-averaged squared error \cite{Personick1971,Rubio2020,Suzuki2024,Albarelli2026}. Albarelli, Branford, and Rubio \cite{Albarelli2026} showed that the Bayesian NH bound relaxes one requirement, namely that the first and second moments of the estimates come from a single normalized measurement. They proved that the bound is reached for two-parameter qubit models and suggested Bayesian weak commutativity as the most likely general condition for this; all their examples satisfy it and appear to reach the bound. Neither the local counterexample nor recent related results settle this: a different Bayesian bound attainable for qubits \cite{Zhao2026}, numerical NH saturation for a qubit channel \cite{Bia2026}, and bounds on the number of outcomes an optimal measurement needs \cite{Yamagata2026}. In this Letter, we show that the condition is not sufficient. A receiver is given one copy of a four-level system, prepared in one of four known full-rank states, and estimates two real coordinates; we minimize the prior-averaged squared error over all positive operator-valued measures (POVMs) and estimators. Weak commutativity holds exactly, but every effect required for NH equality has the same fixed ratio between two level populations, and the identity does not, so these effects can never form a complete measurement. A normalization witness, an observable that is nonpositive on all such effects but has positive trace, captures this mismatch, and its robust form makes the risk gap quantitative. We find the exact optimal risk and a seven-outcome measurement attaining it, and prove that fewer outcomes cannot. Projective measurements (PVMs) on the same four levels have a strictly larger exact minimum, attained with four outcomes. The obstruction is not tied to one ladder: across a family that contains truncated oscillator and pair-annihilation ladders, a robust witness shows that at least \(1/3420\) of the risk reduction predicted by NH is unattainable, and on the oscillator slice we find the optimal measurement exactly. \textit{The estimation task}---A prior \(\pi\) selects a dimensionless label \(\theta\in\mathbb R^2\) and a state \(\rho_\theta\) on \(\C^4\). For unweighted squared error, only three prior moments matter: the average state \(S=\mathbb E_\pi\rho_\theta\), the label--state correlations \(B_j=\mathbb E_\pi(\theta_j\rho_\theta)\), and \(\lambda=\mathbb E_\pi\|\theta\|^2\), which is the risk of always reporting zero. A measurement is a normalized, countably additive POVM \(P\) on an arbitrary outcome space \(\mathcal O\), followed by a measurable report \(a:\mathcal O\to\mathbb R^2\). Its risk is \begin{equation} \mathcal L(P,a)=\mathbb E_\pi\int_{\mathcal O} \|a(\omega)-\theta\|^2\Tr[\rho_\theta P(\dd\omega)]. \label{eq:risk} \end{equation} We write \(\mathcal L_{\min}\) for the infimum over all measurements and reports. The symmetric posterior-mean operators \(S_j\) are defined by \(SS_j+S_jS=2B_j\) \cite{Personick1971}, and Bayesian weak commutativity is the condition \begin{equation} \Tr(S[S_1,S_2])=0. \label{eq:weak} \end{equation} It constrains only this average; the two operators themselves need not commute. For Hermitian \(V_1,V_2\), the two-parameter NH bound reads \begin{equation} \begin{split} \mathcal L_{\mathrm{NH}}=\lambda+\inf_{V_1,V_2}\bigl\{ &\Tr S(V_1^2+V_2^2)-2\sum_j\Tr B_jV_j\\ &+\|iS^{1/2}[V_1,V_2]S^{1/2}\|_1\bigr\}. \end{split} \label{eq:nh} \end{equation} The last term is a trace norm, and, unlike in the local bound, the pair \(V_1,V_2\) is not constrained by unbiasedness. Equations~\eqref{eq:nh} and \eqref{eq:weak} are Eqs.~(45) and (75) of Ref.~\cite{Albarelli2026}, whose Sec.~6 suggests the latter as the most likely attainability condition. \textit{A four-level counterexample}---Let \(C=|1\rangle\langle2|+2|2\rangle\langle3|+3|3\rangle\langle4|\) be a weighted ladder operator, and let \(X=(C+C^\dagger)/2\) and \(Y=(C-C^\dagger)/(2i)\) be its two finite-level quadratures. At the maximally mixed average \(S=I/4\) the trace in Eq.~\eqref{eq:weak} vanishes for every ensemble, so we make the populations unequal. With the population ratio \(k=S_{33}/S_{11}=1000001/1000000\), define \begin{equation} \begin{split} p&=\left[2+2k-\frac{4(k-1)}{9(1+k)^2}\right]^{-1},\\ q&=kp,\quad t=q-\frac{4p(k-1)}{9(1+k)^2},\\ S_k&=\operatorname{diag}(p,p,q,t). \end{split} \label{eq:nonscalar} \end{equation} Unless stated otherwise, \(k\) is fixed at this rational value. Set \(U=\operatorname{diag}(-1,-1,-1,1)\), \(K_{\pm,k}=S_k(I\pm U)/2\), and \(B_k=CK_{+,k}+K_{-,k}C\). The nonzero entries of \(B_k\), just above the diagonal, are \(\beta_1=p\), \(\beta_2=2p\), and \(\beta_3=3(q+t)\); we write \(B_k=B_{1,k}+iB_{2,k}\) with Hermitian \(B_{j,k}\). The four equiprobable preparations are \begin{equation} \theta_{s,j}=10s e_j,\qquad \rho_{s,j}=S_k+sB_{j,k}/5,\quad s=\pm1,\ j=1,2 . \label{eq:principalprior} \end{equation} They have average state \(S_k\), first moments \(B_{j,k}\), and \(\lambda=100\). Diagonal dominance gives \(\rho_{s,j}\succeq I_4/20\), so every preparation has full rank. The posterior-mean operators have upper entries \(h_j\) and \(-ih_j\), with \(h_1=1/2\), \(h_2=2p/(p+q)\), and \(h_3=3\). Thus \begin{equation} \Tr(S_k[S_{1,k},S_{2,k}]) =2i[(p-q)h_2^2+9(q-t)]=0 . \label{eq:nonscalarwc} \end{equation} The choice of \(t\) makes the two terms cancel; since the populations are unequal, this is a genuine constraint rather than an identity. We use the phase symmetry of the moments only to construct measurements, never to restrict competing ones. This single task has three distinct precision levels: \begin{equation} \mathcal L_{\mathrm{NH},k} <\mathcal L_{\min,k} <\mathcal L_{\mathrm{PVM},k}. \label{eq:riskhierarchy} \end{equation} Here \(\mathcal L_{\mathrm{PVM},k}\) is the infimum over projective measurements on the original \(\C^4\) with arbitrary reporting rules. The optimum over general measurements and its distance from NH are \begin{equation} \mathcal L_{\min,k}=\lambda-\sum_i c_{i,k},\qquad \Gamma_k=\Tr T_{0,k}-\sum_i c_{i,k}>1/32 , \label{eq:nonscalargap} \end{equation} where \(T_{0,k}=\operatorname{diag}(0,p,4p+9t,9q)\) and the algebraic numbers \(c_{i,k}\) are defined by the construction below and in Appendix~\ref{app:nonscalaroptimal}. Their certified enclosures give \(\mathcal L_{\min,k}\approx94.2859248008006880\) and \(\Gamma_k\approx0.0359263355222758\), and the exact projective minimum of Eq.~\eqref{eq:mainpvmexact} is \(\mathcal L_{\mathrm{PVM},k}\approx94.5557442485432417\). The gap \(\Gamma_k\) is thus about \(0.6\%\) of the risk reduction predicted by NH, \(\lambda-\mathcal L_{\mathrm{NH},k}=\Tr T_{0,k}\approx5.75\), whereas the extra cost of projective readout, \(\mathcal L_{\mathrm{PVM},k}-\mathcal L_{\min,k}\approx0.270\), is more than seven times larger [Fig.~\ref{fig:hierarchy}(b)]. Table~\ref{tab:precisionclasses} in Appendix~\ref{app:seven} summarizes the three benchmarks. \begin{figure}[t] \centering \input{figures/precision-mechanism} \caption{Why weak commutativity is not enough, for the ensemble of Eq.~\eqref{eq:principalprior}. (a) Two diagonal entries of an effect \(E\). Every effect allowed by NH equality, and every sum of them, lies on the ray \(9E_{22}=4E_{33}\), where \(\Tr WE=0\) for the witness \(W=\operatorname{diag}(0,9,-4,0)\) (arrow, normal to the ray). The identity lies in the shaded region \(\Tr WE>0\) (\(\Tr W=5\)) and cannot be reached. (b) NH bound (dashed) and exact optima over all POVMs and over PVMs on \(\C^4\), rounded, on a truncated axis (no error bars); arrows mark \(\Gamma_k\) and the projective excess. Generative AI assisted the figure design and TikZ source.} \label{fig:hierarchy} \end{figure} \textit{Why the NH bound cannot be reached}---For a Hermitian contraction \(R\), put \(K_\pm(R)=S_k^{1/2}(I\pm R)S_k^{1/2}/2\) and \(q_R(Z)=\Tr[ZK_+(R)Z^\dagger+Z^\dagger K_-(R)Z]\). The quadratic and trace-norm terms of Eq.~\eqref{eq:nh} together equal \(\sup_R q_R(Z)\), with \(Z=V_1+iV_2\). At \(Z=C\) the supremum is attained at \(R=U\), for which \(K_\pm(U)=K_{\pm,k}\). Keep \(R=U\) fixed and perturb \(Z=C+\Delta\). By the definition of \(B_k\), the terms linear in \(\Delta\) cancel, leaving the nonnegative \(q_U(\Delta)\). Taking the supremum over \(R\) can only raise the objective, and leaves it unchanged at \(C\). Hence \(Z=C\) is a global NH minimizer, the optimal NH operators are the quadratures, \(V_1=X\) and \(V_2=Y\), and \(\mathcal L_{\mathrm{NH},k}=\lambda-\Tr T_{0,k}\). To compare with actual measurements, group the outcomes by their complex report \(z=a_1+ia_2\); this defines a POVM \(M\) on the complex plane. With \[ F_k(z)=|z|^2S_k-\bar zB_k-zB_k^\dagger+T_{0,k}, \] expanding the risk gives \begin{equation} \mathcal L-\mathcal L_{\mathrm{NH},k} =\int_{\C}\Tr[F_k(z)M(\dd z)] . \label{eq:principaldual} \end{equation} This is the standard Bayesian operator-duality comparison \cite{Holevo1973,Albarelli2026}; local conic bounds \cite{Hayashi2023} use a different objective. Because \(S_k\) has full rank, Eq.~\eqref{eq:principaldual} holds for every finite-risk measurement, including continuous or singular ones with unbounded reports, and it does not require the first moments of \(M\) to match the NH optimizer. Since \(F_k(z)=(z-C)K_{+,k}(\bar z-C^\dagger)+(\bar z-C^\dagger)K_{-,k}(z-C)\succeq0\), every report adds nonnegative excess risk, and reaching NH requires all of it to vanish: the density of \(M\) with respect to the scalar measure \(\mu=\Tr M\) must have its range in the kernel of \(F_k(z)\) for \(\mu\)-almost every report \(z\) (Appendix~\ref{app:globalcomparison}). A kernel vector satisfies \(z\psi_1=\psi_2\), \(z\psi_2=2\psi_3\), \(z\psi_3=3\psi_4\), and \(\bar z\psi_4=3\psi_3\): the first three components of \(C\psi=z\psi\) and the last component of \(C^\dagger\psi=\bar z\psi\) hold. Nonzero solutions exist only at \(z=0\) and on the circle \(|z|=3\), and all obey \(9|\psi_2|^2=4|\psi_3|^2\). Every effect built from them inherits this balance, whereas the identity has equal populations on both levels; summing such effects to the identity would require \(9=4\), however many outcomes are used. No POVM, finite or continuous, attains NH. \textit{Normalization witnesses}---This population mismatch can be detected by a single observable, much as an entanglement witness detects entanglement \cite{Terhal2000}. Consider any comparison identity of the form~\eqref{eq:principaldual}, valid for every finite-risk measurement, with \(F(z)\succeq0\). We call a Hermitian \(W\) a normalization witness if \(\Tr W>0\) while \(\psi^\dagger W\psi\le0\) for every kernel vector \(\psi\) of every \(F(z)\). If a POVM attained NH, its density \(M_z\) would lie in these kernels, giving the contradiction \(\Tr W=\int\Tr(WM_z)\,\mu(\dd z)\le0\). In our example \(W=\operatorname{diag}(0,9,-4,0)\) measures the population imbalance: \(\psi^\dagger W\psi=0\) on every kernel vector, whereas \(\Tr W=5\) [Fig.~\ref{fig:hierarchy}(a)]. Conversely, if no witness exists, NH is attained. Suppose \(F\) is continuous and \(F(z)\succ0\) for large \(|z|\), as here. The projectors onto unit kernel vectors then form a compact set of unit-trace matrices, which generates a closed convex cone. Without a witness, convex separation places the identity in this cone, and Carath\'eodory's theorem yields an NH-attaining POVM with at most \(16\) rank-one effects, the real dimension of \(4\times4\) Hermitian matrices. Thus NH is attained if and only if no witness exists. For rank-one effects, membership in the cone is a weighted tight-frame condition \cite{Eldar2002}. A witness can also bound the size of the gap. Call \(W\) robust if \(\Tr W>0\) and \begin{equation} \psi^\dagger W\psi\le \kappa\,\|\psi\|\,[\psi^\dagger F(z)\psi]^{1/2} \quad\text{for all } z,\psi . \label{eq:robustwitness} \end{equation} Applying Eq.~\eqref{eq:robustwitness} to the eigenvectors of each \(M_z\), and the Cauchy--Schwarz inequality first to \(\Tr(WM_z)\) and then to the \(\mu\) integral, whose total mass is \(\Tr I_4=4\), gives \begin{equation} \mathcal L-\mathcal L_{\mathrm{NH}}\ge\frac{(\Tr W)^2}{4\kappa^2} \label{eq:witnessgap} \end{equation} for every measurement. \textit{The optimal measurement}---To find the exact optimum we sharpen the comparison by a constant matrix. Appendix~\ref{app:nonscalaroptimal} constructs an algebraic real matrix \(R_\diamond\) with \(R_\diamond+R_\diamond^T=-(B_k+B_k^\dagger)\) for which \(R_\diamond S_k^{-1}R_\diamond^T\) is diagonal; its diagonal entries are the \(c_{i,k}\). Put \(A_\diamond=T_{0,k}-\operatorname{diag}(c_{i,k})\) and \(D(w)=\operatorname{diag}(1,w,w^2,w^3)\). For every \(r\ge0\) and \(|w|=1\), \begin{equation} \begin{split} F_k(rw)-A_\diamond={}&D(w)(rS_k+R_\diamond)S_k^{-1}\\ &\times(rS_k+R_\diamond)^TD(w)^\dagger\succeq0 . \end{split} \label{eq:nonscalaroptimalfactor} \end{equation} Integrating against any POVM and using normalization gives \(\int\Tr[A_\diamond M]=\Tr A_\diamond\), even though \(A_\diamond\) need not be positive. Equation~\eqref{eq:nonscalaroptimalfactor} therefore proves the lower bound in Eq.~\eqref{eq:nonscalargap} with no restriction on the measurement or the estimator. Ancillas and sequential instruments on the received copy enter through their effective POVMs; extra encoded copies or postselection would change the task. The same factorization shows how to attain the bound. The matrix \(F_k(z)-A_\diamond\) is singular only on two circles, with radii \(r_-\mathcal L_{\min,k}+\frac{25}{192}. \label{eq:mainpvmpenalty} \end{equation} The proof in Appendix~\ref{app:pvm} covers all complex bases, coarse-grained projectors, reporting rules, and parameter-independent classical mixtures of single-copy settings. Related tools include normal completions \cite{Kaliuzhnyi2009,Wasson2013}, the normal Procrustes problem \cite{Bierly2026}, and Bayesian readout optimization \cite{Kaubruegger2023}. The exact projective optimum has a simple form. With the entries \(\beta_i\) of \(B_k\), put \(D_0=p+q+t\), \(D_2=2p+q\), and \(\delta=t-p\). The mixing ratio \(h_*\) is the unique root on the indicated branch: \[ \begin{gathered} (\beta_1D_0-D_2\beta_3h)^2(1+h^2)=\beta_2^2\delta^2h^2,\\ D_2\beta_3h>\beta_1D_0. \end{gathered} \] Define \(x_*=h_*/\sqrt{1+h_*^2}\), \(y_*=1/\sqrt{1+h_*^2}\), and \(\omega=e^{2\pi i/3}\). An optimal basis is then \[ \begin{gathered} v_j=\frac{(x_*,\omega^j,\omega^{2j},y_*)^T}{\sqrt3},\quad j=0,1,2,\\ v_3=(y_*,0,0,-x_*)^T. \end{gathered} \] Writing \(\tau_*=p+q+px_*^2+ty_*^2\) and \(A_*=\beta_2+\beta_1x_*+\beta_3y_*\), its optimal reports are \(z_j=(A_*/\tau_*)\omega^j\) and \(z_3=0\). We prove \begin{equation} \mathcal L_{\mathrm{PVM},k}=100-\frac{A_*^2}{\tau_*}. \label{eq:mainpvmexact} \end{equation} To show that no basis does better, write \(s_i=\Tr(S_kP_i)\) for the prior-averaged probability of outcome \(i\) of any orthonormal basis and \(w_i=|\Tr(B_kP_i)|^2\); with optimal reports the risk is \(100-\sum_iw_i/s_i\). Since every \(s_i\ge p\), we replace \(1/s\) on \(s\ge p\) by a quadratic majorant tangent to it at \(s=\tau_*/3\), the value at the three signal outcomes of the attaining basis. The projectors enter the resulting polynomial through \(R=\sum_iP_i\otimes P_i\), which obeys \(R^2=R\), \((R^{\top_2})^2=R^{\top_2}\), and \((R\otimes I)(I\otimes R)=\sum_iP_i^{\otimes3}\), with \(\top_2\) the partial transpose on the second factor. From these identities, Appendix~\ref{app:pvmexact} builds a computer-assisted positive algebraic identity that bounds the gain for every complex basis; its coefficients, Gram matrices, and rational-arithmetic verifiers in the ancillary files are part of the proof. Appendices~\ref{app:pvm} and \ref{app:pvmmoments} give separate analytic and moment bounds. \input{oscillator-family-main} \textit{Preparation noise and other priors}---The gaps survive two kinds of preparation noise. Under the unheralded replacement channel \(\rho\mapsto\zeta\rho+(1-\zeta)S_k\), \(0<\zeta\le1\), the optimal POVM and PVM risks and the NH bound all obey \(\mathcal L(\zeta)=100+\zeta^2[\mathcal L(1)-100]\); weak commutativity, the optimal effects (with reports scaled by \(\zeta\)), and the seven-outcome minimum are preserved. Replacing each state of Eq.~\eqref{eq:principalprior} by any density matrix within operator-norm distance \(2^{-16}\), with the same labels, prior, and loss, leaves \(\widetilde{\mathcal L}_{\min}-\widetilde{\mathcal L}_{\mathrm{NH}}>1/64\) and \(\widetilde{\mathcal L}_{\mathrm{PVM}}-\widetilde{\mathcal L}_{\min}>1/10\). Even the unmodified seven-outcome readout stays more than \(1/10\) below every PVM reoptimized for the perturbed states, although such errors need not preserve weak commutativity or exact optimality (Appendix~\ref{app:robustness}). Nor is the finite prior essential. The affine family \(\rho_\theta=S_k+(\theta_1B_{1,k}+\theta_2B_{2,k})/300\) with a uniform prior on \([-30,30]^2\) has the same \(S_k,B_{j,k}\) and \(\rho_\theta\succeq I_4/20\); only \(\lambda\) changes, to \(600\), adding \(500\) to every risk. The scalar average \(S=I/4\) is treated in Appendices~\ref{app:scalarillustration}--\ref{app:optimal}. An earlier real-entry example over \(\Q(\sqrt2)\) \cite{Wang2026preliminary}, as corrected, shows only that NH-equality effects cannot be normalized, not a gap for all POVMs. \textit{Conclusion}---Bayesian weak commutativity does not guarantee that the NH bound can be reached. In our full-rank four-level ensemble the effects that would saturate NH cannot sum to the identity; the true optimum needs seven outcomes, and projective readout on \(\C^4\) costs more than seven times the NH gap. Normalization witnesses turn this obstruction into a test: for a continuous comparison matrix that is positive definite at large reports, NH is attained exactly when no witness exists, and a robust witness bounds the gap from below, here across a ladder family containing truncated oscillators. These results concern ideal single-copy readout of finite-level systems, not hardware or interferometric advantages. Open questions include exact risks at other ladder shapes, the family's projective penalty, and witnesses for phase or displacement encodings. \begin{acknowledgments} The author acknowledges Gewu Intelligence Lab for providing the collaborative research environment in which this project was developed. \textit{Use of generative AI}---OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, Grok 4.6, OpenAI GPT-6.1 Sol, and OpenAI GPT-6 Astra contributed to the development and preparation of this manuscript. Their combined work included model and example construction, analytical derivations, proof and certificate development, exact-arithmetic checking, literature analysis, physical interpretation, manuscript organization, LaTeX drafting, and revision. Anthropic Claude Opus 5.5 revised the final text line by line, checked the citations against primary records, and redrew both figures. The author directed the work, verified its content, and takes responsibility for it; every computational step can be rechecked with the exact-arithmetic replay checkers provided as ancillary files. \end{acknowledgments} \textit{Data availability}---All exact matrices, certificates, and rational-arithmetic replay checkers are included as ancillary files. No experimental data were used. \input{r17_bayesian_arxiv-references} \clearpage \onecolumngrid \setcounter{secnumdepth}{1} \appendix \input{scalar-illustration} \input{gap-details} \input{optimal-details} \input{nonscalar-optimal-details} \input{seven-outcome-details} \input{pvm-penalty-details} \input{pvm-moment-details} \input{pvm-exact-details} \input{robustness-details} \input{oscillator-family-details} \end{document}