  We aim to obtain a measure of the curvature of time-resolved spectra that can
be compared directly to theory. This tests the ability of models such as
synchrotron emission to explain the peaks or breaks of GBM prompt emission
spectra. We take the burst sample from the official Fermi GBM GRB time-resolved
spectral catalog. We re-fit all spectra with a measured peak or break energy in
the catalog best-fit models in various energy ranges, which cover the curvature
around the spectral peak or break, resulting in a total of 1,113 spectra being
analysed. We compute the sharpness angles under the peak or break of the
triangle constructed under the model fit curves and compare to the values
obtained from various representative emission models: blackbody,
single-electron synchrotron, synchrotron emission from a Maxwellian or
power-law electron distribution. We find that 35% of the time-resolved spectra
are inconsistent with the single-electron synchrotron function, and 91% are
inconsistent with the Maxwellian synchrotron function. The single temperature,
single emission time and location blackbody function is found to be sharper
than all the spectra. No general evolutionary trend of the sharpness angle is
observed, neither per burst nor for the whole population. It is found that the
limiting case, a single temperature Maxwellian synchrotron function, can only
contribute up to $58^{+23}_{-18}$% of the peak flux. Our results show that even
the sharpest but non-realistic case, the single-electron synchrotron function,
cannot explain a large fraction of the observed GRB prompt spectra. Because of
the fact that any combination of physically possible synchrotron spectra added
together will always further broaden the spectrum, emission mechanisms other
than optically thin synchrotron radiation are likely required in a full
explanation of the spectral peaks or breaks of the GRB prompt emission phase.
