We study the centimeter- to millimeter-wavelength synchrotron spectrum of the core of the radio galaxy M 87 at \lesssim 0.8 { mas } ~ { } \sim 110 R _ { 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 22 GHz and 129 GHz . By combining resolution-matched flux measurements from the Very Long Baseline Array ( VLBA ) at 15 GHz and taking the Event Horizon Telescope ( EHT ) 230 GHz core flux measurements in epochs 2009 and 2012 as lower limits , we find evidence of a nearly flat core spectrum across 15 GHz and 129 GHz , which could naturally connect the 230 GHz 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 \gtrsim 100 GHz . The spectral shape can be qualitatively explained if both the strongly ( compact , optically thick at > 100 GHz ) and the relatively weakly magnetized ( more extended , optically thin at < 100 GHz ) plasma components are colocated in the footprint of the relativistic jet .