We report evidence that the asymptotic low-energy power-law slope ct (
below the spectral break) of BATSE gamma-ray burst (GRB) photon spectr
a evolves with time rather than remaining constant. We find that a hig
h degree of positive correlation exists between the time-resolved spec
tral break energy E(pk) and alpha. In samples of 18 ''hard-to-soft'' a
nd 12 ''tracking'' pulses, evolution of alpha was found to correlate w
ith that of the spectral break energy E(pk) at the 99.7% and 98% confi
dence levels, respectively. We also find that in the flux rise phase o
f hard-to-soft pulses, the mean value of alpha is often positive, and
in some bursts the maximum value of alpha is consistent with a value >
+1. BATSE burst 3B 910927, for example, has an alpha(max) equal to 1.
6 +/- 0.3. These findings challenge GRB spectral models in which alpha
must be negative or remain constant.