How avoid empty slide with TikZ-overlay footnote in beamer

Question

Motivation

I want to maintain a footnote in beamer through some slides. To avoid jumping I tried to place them with TikZ (which works) but the height of the inline math causes empty slides.

Question

How can I maintain my awfully high footnote line without any empty pages?

MWE.tex

\documentclass{beamer}
\usepackage{tikz}
\mode<presentation>
\setbeamertemplate{footline}[frame number]
\begin{document}
\begin{frame}{Intro}
  Hello
\end{frame}
\begin{frame}{Slide Title}
    \begin{theorem}
    Some theorem \footnotemark[1] to break the ice
    \end{theorem}
    % footnote
    % https://tex.stackexchange.com/questions/6185/absolute-positioning-in-beamer?rq=1
    \begin{tikzpicture}[remember picture,overlay]
      \node at (current page.south west){%
      \footnotetext[1]{
          % {\footnotesize
              where $E =M \Big\lvert_{y =\zeta( x )}C^2$ 
          }
      };
    \end{tikzpicture}
\end{frame}
\begin{frame}{Proof Slide}
    \begin{proof}
    Its proof ... where I want to keep the footnote
    \end{proof}
    % footnote
    \begin{tikzpicture}[remember picture,overlay]
        \node at (current page.south west){%
        \footnotetext[1]{
                where $E =M \Big\lvert_{y =\zeta( x )}C^2$ 
            }
        };
    \end{tikzpicture}
\end{frame}
\end{document}

Answer 1

• I don't see any jumping in the footnote, so I'm not convinced that the absolute positioning with tikz is necessary. And even if, I doubt that a tikz picture wrapped around the outside of \footnotetext would be able to influence the position of the footnote text...

• instead of manually adding the footnote to subsequent frames, you could use a single frame and replace the theorem with the proof in overlays.

• a quick hack to avoid the empty pages: hide the height of the math with \smash{...} (to move the footnote rule up, you can then manually increase the height of the footnote with an invisible rule)

\documentclass{beamer}
\setbeamertemplate{footline}[frame number]
\begin{document}
\begin{frame}
\frametitle<1>{Slide Title}
\frametitle<2>{Proof Slide}
    \begin{theorem}<only@1>
    Some theorem where\footnote{\rule{0pt}{3ex}\smash{$E =M \Big\lvert_{y =\zeta( x )}C^2$}}  to break the ice
    \end{theorem}
    \begin{proof}<only@2>
    Its proof ... where I want to keep the footnote
    \end{proof}
\end{frame}
\end{document}