We describe the bisection procedure used to compute $\tau = \hat{F}_0^{-1}(1-\alpha;\mathcal{B}_0)$
in the KDE-ROLL forward pass (\Cref{sec:roll-kde-forward}).

\paragraph{Bracket initialisation.}
Because $\sigma(u;\,v) \to 0$ exponentially as $u \to -\infty$ and $\sigma(u;\,v) \to 1$
as $u \to +\infty$, with tails decaying within roughly $10/v_0$ of the data range, the
interval
\[
  \mathrm{lo} = \min(\mathbf{s}^{(0)}) - \tfrac{10}{v_0},
  \qquad
  \mathrm{hi} = \max(\mathbf{s}^{(0)}) + \tfrac{10}{v_0}
\]
is guaranteed to satisfy $\hat{F}_0(\mathrm{lo}) < 1-\alpha < \hat{F}_0(\mathrm{hi})$.

\paragraph{Bisection iterations.}
At each step we evaluate the midpoint,
\[
  \tau_{\mathrm{mid}} = \tfrac{1}{2}(\mathrm{lo} + \mathrm{hi}),
\]
and update the bracket:
\[
  \begin{cases}
    \mathrm{lo} \leftarrow \tau_{\mathrm{mid}} & \text{if } \hat{F}_0(\tau_{\mathrm{mid}};\,\mathcal{B}_0) < 1-\alpha, \\
    \mathrm{hi} \leftarrow \tau_{\mathrm{mid}} & \text{otherwise.}
  \end{cases}
\]
Iteration continues until $\lvert\hat{F}_0(\tau_{\mathrm{mid}};\,\mathcal{B}_0)-(1-\alpha)\rvert < \varepsilon$
(with $\varepsilon = 10^{-4}$), or until a maximum iteration count is reached.
The method converges unconditionally and requires no derivative evaluation, in contrast
to the Newton--Raphson alternative documented in Appendix~\ref{appendix:kde-nr}.
