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We present the massless six-point one-loop amplitudes in the open and closedsuperstring using BRST cohomology arguments from the pure spinor formalism. Thehexagon gauge anomaly is traced back to a class of kinematic factors in purespinor superspace which were recently introduced as BRST pseudo-invariants.This complements previous work where BRST invariance arguments were used toderive the non-anomalous part of the amplitude. The associated worldsheetfunctions are non-singular and demonstrated to yield total derivatives onmoduli space upon gauge variation. These cohomology considerations yield anefficient organizing principle for closed-string amplitudes that matchexpectations from S-duality in the low-energy limit.

One-loop superstring six-point amplitudes and anomalies in pure spinor superspace