  The origin of fractional quantum Hall effect (FQHE) at 4/11 and 5/13 has
remained controversial. We make a compelling case that FQHE is possible here
for fully spin polarized composite fermions, but with an unconventional
underlying physics. Thanks to a rather unusual interaction between composite
fermions, FQHE here results from the suppression of pairs with relative angular
momentum {\em three} rather than one, confirming the exotic mechanism proposed
by W\'ojs, Yi and Quinn [Phys. Rev. B {\bf 69}, 205322 (2004)]. We predict that
the 4/11 state reported a decade ago by Pan {\em et al.} [Phys. Rev. Lett. {\bf
90}, 016801 (2003)] is a conventional partially spin polarized FQHE of
composite fermions, and estimate the Zeeman energy where a phase transition
into the unconventional fully spin polarized state will occur.
