  The proportions of oxygen, carbon and major rock-forming elements (e.g. Mg,
Fe, Si) determine a planet's dominant mineralogy. Variation in a planet's
mineralogy subsequently affects planetary mantle dynamics as well as any deep
water or carbon cycle. Through thermodynamic models and high pressure diamond
anvil cell experiments, we demonstrate the oxidation potential of C is above
that of Fe at all pressures and temperatures indicative of 0.1 - 2 Earth-mass
planets. This means that for a planet with (Mg+2Si+Fe+2C)/O > 1, excess C in
the mantle will be in the form of diamond. We model the general dynamic state
of planets as a function of interior temperature, carbon composition, and size,
showing that above a critical threshold of $\sim$3 atom% C, limited to no
mantle convection will be present assuming an Earth-like geotherm. We assert
then that in the C-(Mg+2Si+Fe)-O system, only a very small compositional range
produce habitable planets. Planets outside of this habitable range will be
dynamically sluggish or stagnant, thus having limited carbon or water cycles
leading to surface conditions inhospitable to life as we know it.
