\subsection{Exact optimal readout on the oscillator slice}
\label{app:oscillatoroptimal}

The oscillator endpoint admits a stronger result than a gap estimate.
For \(u=0\), we determine the exact minimum risk and an attaining
eight-effect POVM for every \(1\le k\le2\) and \(0<\eta\le1/6\).
The lower bound covers all measurements; only the attaining construction
uses four equally spaced phases at each of two radii.

Retain the moments in Eq.~\eqref{eq:familymoments}, now with
\(\ell_1=1/\sqrt3\), \(\ell_2=\sqrt{2/3}\), and put
\[
 w_0=p/\sqrt3,\qquad m_0=p\sqrt{2/3},\qquad n_0=q+t.
\]
For real \(\xi,\zeta\), define
\begin{align*}
 d&=p^2\zeta(n_0+\zeta)-qt\xi(w_0+\xi),\\
 A&=\frac{\xi m_0pt(n_0+\zeta)}{d},\qquad
 D=\frac{\xi m_0qt(w_0+\xi)}{d},\\
 R_k(\xi,\zeta)&=\begin{pmatrix}
 0&\xi&0&A\\
 -w_0-\xi&0&D&0\\
 0&-m_0-D&0&\zeta\\
 -A&0&-n_0-\zeta&0
 \end{pmatrix},\qquad
 c_i=(R_kS^{-1}R_k^T)_{ii}.
\end{align*}
These are the weighted-Gram formulas of
Appendix~\ref{app:nonscalaroptimal} with different, explicitly stated
ladder entries. Their algebra does not require that appendix's fixed
value of \(k\). Indeed, \(R_k+R_k^T=-(B+B^\dagger)\), and
the only possible off-diagonal weighted row products are
\[
 -\frac{\xi(m_0+D)}p+\frac{A\zeta}t=0,\qquad
 \frac{A(w_0+\xi)}p-\frac{D(n_0+\zeta)}q=0.
\]
Consequently, with \(A_0=T_0-\operatorname{diag}(c_i)\),
\begin{equation}
 F(r\omega)-A_0=
 D(\omega)(rS+R_k)S^{-1}(rS+R_k)^TD(\omega)^\dagger\succeq0
 \label{eq:oscillatorglobalfactor}
\end{equation}
for every \(r\ge0\), \(|\omega|=1\), including the origin.
Here \(D(\omega)=\operatorname{diag}(1,\omega,\omega^2,\omega^3)\),
distinct from the scalar entry \(D\) in \(R_k\).

We specify the exact branch through normalization, not stationarity.
Reorder \(R_kS^{-1}\) as \(1,3,2,4\), obtaining off-diagonal blocks
\(K,L\), and set
\[
 \sigma=\Tr(KL),\qquad \kappa=\det(KL),\qquad
 h(x)=x^2-\sigma x+\kappa.
\]
Apply the tridiagonal null-vector recurrence in
Appendix~\ref{app:nonscalaroptimal} with these \(w_0,m_0,n_0,p,q,t,c_i\).
At the roots of \(h\), write the squared components as
\(\nu_i(x)^2=E_i+X_i x\), with \(\nu_1=1\).
The explicit rational formulas there give
\(y=(1-E_2)/X_2\) and the two equations
\begin{equation}
 G_3(k;\xi,\zeta)=E_3+X_3y-1=0,\qquad
 G_4(k;\xi,\zeta)=E_4+X_4y-1=0.
 \label{eq:oscillatornormalization}
\end{equation}
All their denominators are nonzero on the boxes used below. On each box,
\[
 \sigma>0,\quad\kappa>0,\quad\sigma^2>4\kappa,
 \quad X_2\ne0,\quad h(y)<0.
\]
Thus the roots \(x_-<x_+\) are positive and distinct, and
\(\alpha_+=(y-x_-)/(x_+-x_-)\), \(\alpha_-=1-\alpha_+\)
are positive. With \(r_s=\sqrt{x_s}\) and the real vectors
\(\nu_s=\nu(x_s)\), the effects and physical reports are
\begin{equation}
 M_{s,\omega}=\frac{\alpha_s}{4}
 D(\omega)\nu_s\nu_s^TD(\omega)^\dagger,
 \qquad z_{s,\omega}=\frac\eta2 r_s\omega,
 \quad \omega\in\{1,i,-1,-i\}.
 \label{eq:oscillatorattaining}
\end{equation}
Equation~\eqref{eq:oscillatornormalization} gives every diagonal entry
of \(\sum M\) equal to one; summing the four phases removes all
off-diagonal entries. The vectors are left-null vectors:
\((r_sS+R_k)^T\nu_s=0\). They therefore annihilate the residual
in Eq.~\eqref{eq:oscillatorglobalfactor}.

Integrating that positive factor against any normalized POVM gives the
global lower bound \(1-g^2\sum_i c_i\), where \(g=\eta/2\).
The displayed measurement attains it exactly. Hence
\begin{equation}
 \boxed{\displaystyle
 \mathcal L_{\min}(0,k,\eta)
 =1-\frac{\eta^2}{4}\sum_{i=1}^4 c_i(k).}
 \label{eq:oscillatorexactrisk}
\end{equation}
This is a matching lower bound and measurement, not a numerical
agreement or an optimum within a two-orbit ansatz. The arbitrary-outcome
integration from the preceding subsections applies unchanged.

\paragraph{Exact branch selection on the full interval.}
The ancillary directory \texttt{anc/oscillator-optimal/} supplies
20\,000 closed parameter intervals
\[
 I_j=[1+j/20000,1+(j+1)/20000],\qquad 0\le j<20000.
\]
Their union is exactly \([1,2]\). For each \(I_j\), a listed rational
center \(b_j\in\Q^2\) and radius \(\varepsilon=7/10^6\) define
\(\mathcal B_j=b_j+[-\varepsilon,\varepsilon]^2\).
For every \(k\in I_j\), select the unique root of
Eq.~\eqref{eq:oscillatornormalization} in \(\mathcal B_j\).
The two certificate files contain the intervals and centers; no
floating-point approximation is used as the definition of the root.

Here is the existence and uniqueness argument verified on every closed
box. An invertible rational preconditioner \(C_j\) obeys
\(\|I-C_j\partial_{(\xi,\zeta)}G(\mathcal B_j,I_j)\|_\infty<1\).
The centered image
\[
 b_j-C_jG(b_j,k)+
 [I-C_j\partial_{(\xi,\zeta)}G(\mathcal B_j,I_j)]
       (\mathcal B_j-b_j)
\]
lies strictly inside \(\mathcal B_j\), uniformly for \(k\in I_j\).
To bound its parameter dependence, the implementations use the mean-value
identity centered at the midpoint \(k_j\) of \(I_j\):
\[
 G(b_j,k)=G(b_j,k_j)+
       \partial_kG(b_j,I_j)(k-k_j),
\]
where the last expression denotes an interval enclosure.
The contraction theorem then proves a root for every real parameter
in the interval, not merely at its midpoint. It also proves uniqueness
within the stated root box and nonsingularity of the normalization
Jacobian. Pole exclusion and all the strict radius/mixture inequalities
are checked on the entire same box.
At a shared endpoint either adjoining box may be used: both constructions
attain a universal lower bound, so their minimum-risk values agree.
No uniqueness of the optimal measurement is asserted.

\texttt{verify\_oscillator.py} uses outward dyadic rational intervals
and forward derivatives. \texttt{independent\_oscillator.py} imports none
of it, instead using decimal-rational intervals, a separately assembled
weighted matrix and block determinants. The reduced null-vector square
coefficients are independently derived by quadratic quotient recurrence
in \texttt{matching\_identities.py}. Exact interval arithmetic verifies
the same full coverage and strict predicates in both implementations.
For algebraic \(k\), the isolated roots, measurement entries and risk
are algebraic; Eq.~\eqref{eq:oscillatorexactrisk} also defines the exact
optimum for arbitrary real \(k\in[1,2]\).
The result does not determine the minimum number of outcomes or the
projective optimum on this slice, nor the exact risks for all \(u\).