% Current Gaps
\begin{frame}{Existing methods can handle either continuous or discrete, but not both}

  \begin{center}
    \begin{tikzpicture}[scale=0.9, transform shape]
      % Formal Methods circle
      \draw[thick, fill=red!20, opacity=0.7] (-2,0) circle (2.5cm);
      \node at (-2, 2) {\textbf{Formal Methods}};
      \node at (-2, 1.2) {\small (HARDENS)};
      \node[align=center, font=\small] at (-2, 0.3) {
        Discrete \checkmark
      };
      \node[align=center, font=\small, text=red] at (-2, -0.5) {
        Continuous \textbf{$\times$}
      };

      % Control Theory circle
      \draw[thick, fill=blue!20, opacity=0.7] (2,0) circle (2.5cm);
      \node at (2, 2) {\textbf{Control Theory}};
      \node[align=center, font=\small] at (2, 0.3) {
        Continuous \checkmark
      };
      \node[align=center, font=\small, text=red] at (2, -0.5) {
        Discrete \textbf{$\times$}
      };

      % The Gap
      \node[align=center, font=\Large, text=orange] at (0, -3) {
        \textbf{THE GAP}
      };
    \end{tikzpicture}
  \end{center}

  %SPEAKER NOTES: See comments below
%
    \textbf{Formal Methods (HARDENS):}
    Can verify discrete logic (requirements → verified binaries). Achieved in 9 months, low cost.
    But: Cannot handle continuous dynamics.

    \textbf{Control Theory:}
    Can verify continuous stability (Lyapunov, LQR, robust control).
    But: Cannot verify discrete transitions or mode switching.

    \textbf{THE GAP:} Hybrid system verification with formal guarantees spanning both continuous and discrete.

    \textbf{HARDENS Achievement:} Complete RTS verification (discrete only), TRL 3-4, no experimental validation.

    \textbf{We need to bridge this gap.}
% (End of speaker notes)
\end{frame}
