  We study the centimeter- to millimeter-wavelength synchrotron spectrum of the
core of the radio galaxy M87 at $\lesssim0.8\,{\rm mas}~\sim110R_{s}$ spatial
scales using four years of fully simultaneous, multi-frequency VLBI data
obtained by the Korean VLBI Network (KVN). We find a core spectral index
$\alpha$ of $\gtrsim-0.37$ ($S\propto \nu^{+\alpha}$) between 22GHz and 129GHz.
By combining resolution-matched flux measurements from the Very Long Baseline
Array (VLBA) at 15GHz and taking the Event Horizon Telescope (EHT) 230GHz core
flux measurements in epochs 2009 and 2012 as lower limits, we find evidence of
a nearly flat core spectrum across 15GHz and 129GHz, which could naturally
connect the 230GHz VLBI core flux. The extremely flat spectrum is a strong
indication that the jet base does not consist of a simple homogeneous plasma,
but of inhomogeneous multi-energy components, with at least one component with
the turn-over frequency $\gtrsim100$GHz. The spectral shape can be
qualitatively explained if both the strongly (compact, optically thick at
$>$100GHz) and the relatively weakly magnetized (more extended, optically thin
at $<$100GHz) plasma components are colocated in the footprint of the
relativistic jet.
