= Free-filament variance with fixed translation and tilt
{title2=$\operatorname{Var}h(x)=\frac{k_BTL^3}{A}P(x/L)$}
For the frame fixed by zero mean height and zero first height moment, the <variance> profile is $P(u)=1/105-11u/105+13u^2/35-u^3/3-u^4/3+3u^5/5-u^6/5$. It obeys $P(1-u)=P(u)$, $P(0)=1/105$ and $P(1/2)=1/320$. Integrating a <white noise> curvature field twice, subtracting its affine projection and integrating the squared kernel proves the profile. These free-end conditions must not be confused with physically clamping the ends.
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