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 $gtrsim-0.37$ ($Spropto u^{+alpha}$) 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.
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