  One of the major challenges for theoretical modeling of Type Ia supernova
explosions is to explain the diversity of these events and the empirically
established correlation between their peak luminosity and light curve shape. In
the framework of the so-called Chandrasekhar mass models, the progenitor's
carbon-to-oxygen ratio has been suggested to be a principal source of peak
luminosity variations due to a variation in the production of radioactive
$^{56}$Ni during the explosion. The underlying idea is that an enhanced carbon
mass fraction should result in a more vigorous explosion since here the energy
release from nuclear reactions is increased. It was suspected that this would
produce a higher amount of $^{56}$Ni in the ejecta. In this letter we describe
a mechanism resulting from an interplay between nucleosynthesis and turbulent
flame evolution which counteracts such an effect. Based on three-dimensional
simulations we argue that it is nearly balanced and only minor differences in
the amount of synthesized $^{56}$Ni with varying carbon mass fraction in the
progenitor can be expected. Therefore this progenitor parameter is unlikely to
account for the observed variations in Type Ia supernova luminosity. We discuss
possible effects on the calibration of cosmological measurements.
