Signed two-flux composite-fermion filling (${\nu }^{\ast }=\nu /(1-2\nu )$)

Attach two statistical flux quanta ${\Phi }_0=h/e$ per electron at density $n_s$. For $B>0$, $B^{\ast }=B-2n_s{\Phi }_0=B(1-2\nu )$ and the signed effective filling is ${\nu }^{\ast }=n_s{\Phi }_0/B^{\ast }=\nu /(1-2\nu )$. Inversion gives $\nu ={\nu }^{\ast }/(2{\nu }^{\ast }+1)$. Integer ${\nu }^{\ast }=p$ gives $p/(2p+1)$, while ${\nu }^{\ast }=-p$ gives $p/(2p-1)$. Using nonnegative effective filling instead requires explicitly tracking the effective-field sign. At half filling the mean effective field vanishes.