  Distant star-forming galaxies show a correlation between their star formation
rates (SFR) and stellar masses, and this has deep implications for galaxy
formation. Here, we present a study on the evolution of the slope and scatter
of the SFR-stellar mass relation for galaxies at $3.5\leq z\leq 6.5$ using
multi-wavelength photometry in GOODS-S from the Cosmic Assembly Near-infrared
Deep Extragalactic Legacy Survey (CANDELS) and Spitzer Extended Deep Survey. We
describe an updated, Bayesian spectral-energy distribution fitting method that
incorporates effects of nebular line emission, star formation histories that
are constant or rising with time, and different dust attenuation prescriptions
(starburst and Small Magellanic Cloud). From $z$=6.5 to $z$=3.5 star-forming
galaxies in CANDELS follow a nearly unevolving correlation between stellar mass
and SFR that follows SFR $\sim$ $M_\star^a$ with $a = 0.54 \pm 0.16$ at $z\sim
6$ and $0.70 \pm 0.21$ at $z\sim 4$. This evolution requires a star formation
history that increases with decreasing redshift (on average, the SFRs of
individual galaxies rise with time). The observed scatter in the SFR-stellar
mass relation is tight, $\sigma(\log \mathrm{SFR}/\mathrm{M}_\odot$ yr$^{-1})<
0.3\ - $ 0.4 dex, for galaxies with $\log M_\star/\mathrm{M}_\odot > 9$ dex.
Assuming that the SFR is tied to the net gas inflow rate (SFR $\sim$
$\dot{M}_\mathrm{gas}$), then the scatter in the gas inflow rate is also
smaller than 0.3$-$0.4 dex for star-forming galaxies in these stellar mass and
redshift ranges, at least when averaged over the timescale of star formation.
We further show that the implied star formation history of objects selected on
the basis of their co-moving number densities is consistent with the evolution
in the SFR-stellar mass relation.
