  A central line of inquiry in condensed matter science has been to understand
how the competition between different states of matter give rise to emergent
physical properties. Perhaps some of the most studied systems in this respect
are the hole-doped LaMnO$_3$ perovskites, with interest in the past three
decades being stimulated on account of their colossal magnetoresistance (CMR).
However, phase segregation between ferromagnetic (FM) metallic and
antiferromagnetic (AFM) insulating states, which itself is believed to be
responsible for the colossal change in resistance under applied magnetic field,
has until now prevented a full atomistic level understanding of the orbital
ordered (OO) state at the optimally doped level. Here, through the detailed
crystallographic analysis of the hole-doped phase diagram of a prototype
system, we show that the superposition of two distinct lattice modes gives rise
to a striped structure of OO Jahn-Teller active Mn$^{3+}$ and charge disordered
(CD) Mn$^{3.5+}$ layers in a 1:3 ratio. This superposition leads to an exact
cancellation of the Jahn-Teller-like oxygen atom displacements in the CD layers
only at the 3/8th doping level, coincident with the maximum CMR response of the
manganties. Furthermore, the periodic striping of layers containing
Mn$^{3.5+}$, separated by layers of fully ordered Mn$^{3+}$, provides a natural
mechanism though which long range OO can melt, a prerequisite for the emergence
of the FM conducting state. The competition between insulating and conducting
states is seen to be a key feature in understanding the properties in highly
correlated electron systems, many of which, such as the CMR and high
temperature superconductivity, only emerge at or near specific doping values.
