The fundamental plane of black hole activity is a relation between X-ray luminosity, radio luminosity, and black hole mass for hard state Galactic black holes and their supermassive analogs. The fundamental plane suggests that, at low-accretion rates, the physical processes regulating the conversion of an accretion flow into radiative energy could be universal across the entire black hole mass scale. However, there is still a need to further refine the fundamental plane in order to better discern the radiative processes and their geometry very close to the black hole, in particular the source of hard X-rays. Further refinement is necessary because error bars on the best-fit slopes of the fundamental plane are generally large, and also the inferred coefficients can be sensitive to the adopted sample of black holes. In this work, we regress the fundamental plane with a Bayesian technique. Our approach shows that sub-Eddington black holes emit X-ray emission that is predominantly optically thin synchrotron radiation from the jet, provided that their radio spectra are flat or inverted. X-ray emission dominated by very radiatively inefficient accretion flows are excluded at the >3sigma level. We also show that it is difficult to place FR I galaxies onto the fundamental plane because their X-ray jet emission is highly affected by synchrotron cooling. On the other hand, BL Lac objects fit onto the fundamental plane. Including a uniform subset of high-energy peaked BL Lac objects from the SDSS, we find sub-Eddington black holes with flat/inverted radio spectra follow log L_x=(1.45pm0.04)log L_R-(0.88pm0.06)logM_{BH}-6.07pm1.10, with sigma_{int}=0.07pm0.05 dex. Finally, we discuss how the effects of synchrotron cooling of jet emission from the highest black hole masses can bias fundamental plane regressions, perhaps leading to incorrect inferences on X-ray radiation mechanisms.
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