We study the centimeter- to millimeter-wavelength synchrotron spectrum of the core of the radio galaxy M 87 at ≲0.8 mas ~ 110R_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 α of ≳−0.37 (S ∝ ν^+α) between 22 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 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 ≳ 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.