= Catenoid existence threshold for equal rings
{title2=$R/H\geq\sinh A,\quad A=\coth A$}
An axisymmetric smooth <connected> soap film spanning equal radius-$R$ rings at heights $\pm H$ has stationary profile $r=k^{-1}\cosh(kz)$, $k>0$. The boundary equation is $R/H=\cosh u/u$, $u=kH$. Its unique minimum occurs at $u\tanh u=1$, because the <derivative> numerator is $u\sinh u-\cosh u$. If $A$ is this root, the minimum is $\cosh A/A=\sinh A$. There are no catenoids below that ratio, one at equality, and two above it. Existence of stationary <connected> films does not assert global minimality against pinching or disconnected competitors.
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