  We present a detailed analysis of the specific star formation rate -- stellar
mass ($\mathrm{sSFR}-M_*$) of $z\le 0.13$ disk central galaxies using a
morphologically selected mass-complete sample ($M_* \ge 10^{9.5} M_{\odot}$).
Considering samples of grouped and ungrouped galaxies, we find the
$\mathrm{sSFR}-M_*$ relations of disk-dominated central galaxies to have no
detectable dependence on host dark-matter halo (DMH) mass, even where
weak-lensing measurements indicate a difference in halo mass of a factor
$\gtrsim5$. We further detect a gradual evolution of the $\mathrm{sSFR}-M_*$
relation of non-grouped (field) central disk galaxies with redshift, even over
a $\Delta z \approx 0.04$ ($\approx5\cdot10^{8}\mathrm{yr}$) interval, while
the scatter remains constant. This evolution is consistent with extrapolation
of the "main-sequence-of-star-forming-galaxies" from previous literature that
uses larger redshift baselines and coarser sampling.
  Taken together, our results present new constraints on the paradigm under
which the SFR of galaxies is determined by a self-regulated balance between gas
inflows and outflows, and consumption of gas by star-formation in disks, with
the inflow being determined by the product of the cosmological accretion rate
and a fuelling-efficiency -- $\dot{M}_{\mathrm{b,halo}}\zeta$. In particular,
maintaining the paradigm requires $\dot{M}_{\mathrm{b,halo}}\zeta$ to be
independent of the mass $M_{\mathrm{halo}}$ of the host DMH. Furthermore, it
requires the fuelling-efficiency $\zeta$ to have a strong redshift dependence
($\propto (1+z)^{2.7}$ for $M_*=10^{10.3} M_{\odot}$ over $z=0 - 0.13$), even
though no morphological transformation to spheroids can be invoked to explain
this in our disk-dominated sample. The physical mechanisms capable of giving
rise to such dependencies of $\zeta$ on $M_{\mathrm{halo}}$ and $z$ for disks
are unclear.
