O2 and O3 have been long considered the most robust individual biosignature gases in a planetary atmosphere, yet multiple mechanisms that may produce them in the absence of life have been described. However, these abiotic planetary mechanisms modify the environment in potentially identifiable ways. Here we briefly discuss two of the most detectable spectral discriminants for abiotic O2/O3: CO and O4. We produce the first explicit self-consistent simulations of these spectral discriminants as they may be seen by James Webb Space Telescope (JWST). If JWST-NIRISS and/or NIRSpec observe CO (2.35, 4.6 μm) in conjunction with CO2 (1.6, 2.0, 4.3 μm) in the transmission spectrum of a terrestrial planet it could indicate robust CO2 photolysis and suggest that a future detection of O2 or O3 might not be biogenic. Strong O4 bands seen in transmission at 1.06 and 1.27 μm could be diagnostic of a post-runaway O2-dominated atmosphere from massive H-escape. We find that for these false positive scenarios, CO at 2.35 μm, CO2 at 2.0 and 4.3 μm, and O4 at 1.27 μm are all stronger features in transmission than O2/O3 and could be detected with S/Ns ≳ 3 for an Earth-size planet orbiting a nearby M dwarf star with as few as 10 transits, assuming photon-limited noise. O4 bands could also be sought in UV/VIS/NIR reflected light (at 0.345, 0.36, 0.38, 0.445, 0.475, 0.53, 0.57, 0.63, 1.06, and 1.27 μm) by a next generation direct-imaging telescope such as LUVOIR/HDST or HabEx and would indicate an oxygen atmosphere too massive to be biologically produced.