The neutral π0 and η mesons are studied in 197Au−197Au collisions at an incident energy of 800AMeV, substantially below the threshold for η production in N−N collisions. While the gross π0 multiplicity increases almost linearly with the number of participant nucleons, the multiplicities of η and hard π0 mesons show a stronger than linear dependence. The nonlinearity is governed by the average transverse-mass excess 〈mt〉−(s−2mN) of the mesons and is insensitive to their final-state interaction in the nuclear medium.