The appearance at the anatomical level of a canonical laminar microcircuit suggests that each six-layer column of granular cortex may mediate a canonical computation, but it remains unknown what that computation is and which aspects of neuronal activity mediate it. Predictive processing theorists have suggested that gradient-driven Bayesian inference forms this canonical computation, and have put forward circuit models to show how neuronal activity encodes the necessary predictions and errors, such as predictive coding (spike rates) and predictive routing (neuronal oscillations). By combining electrophysiology an optogenetics in a visual oddball task in mice and nonhuman primates, we tested the predictive coding and predictive routing models. Spiking data refuted predictive coding: highly predictable stimuli were never explained away, and highly unpredictable oddballs did not evoke omnipresent prediction errors. Passing to the local field potentials in the spectral domain, ɣ-band local-field potential (LFP) oscillations conveyed feedforward prediction errors in lower sensory areas of cortex; ⍺/β-band oscillations weakened in unpredictable conditions compared to predictable ones; and θ-band oscillations additionally signalled slower, longer-scale temporal prediction errors. In combination with the previous findings, predictive routing explains these experiments where neither ubiquitous predictive coding nor feedforward adaptation can.