= Uniqueness of the packed spherical bilayer radius
{title2=$R^3-r^3=3(v/a)(R^2+r^2)$}
In an incompressible, noninterdigitating <lipid bilayer> with negligible head-group thickness, let the inner radius be $r>0$, molecular tail volume $v>0$ and head-group area $a>0$. Counting heads on both <membrane leaflets> and conserving tail volume gives $R^3-r^3=3(v/a)(R^2+r^2)$. For $R>r$, the <function> $q(R)=(R^3-r^3)/(R^2+r^2)$ has <derivative> $(R^4+3R^2r^2+2Rr^3)/(R^2+r^2)^2>0$, starts at zero and tends to infinity. Thus exactly one candidate outer radius exists in this geometric model. A maximum tail length $l$ additionally requires $r_m-r\leq l$ and $R-r_m\leq l$, where $r_m^3=r^3+3(v/a)r^2$. Uniqueness does not establish compatibility with arbitrary molecular lengths.
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