Typical gamma-ray burst (GRB) spectra are characterized by a spectral break E-p, which for bright BATSE bursts is found to be narrowly clustered around 300 keV. Recently identified X-ray ashes, which may account for a significant portion of the whole GRB population, seem to extend the E-p distribution to a broader range below 40 keV. Positive correlations among E-p and some other observed parameters have been noticed. On the other hand, within the cosmological fireball model, the issues concerning the dominant energy ingredient of the fireball as well as the location of the GRB emission site are still unsettled, leading to several variants of the fireball model. Here we analyze these models within a unified framework and critically reexamine the E-p predictions in the various model variants. Attention is focused on the predictions of the narrowness of the E-p distribution in different models and the correlations among E-p and other measurable observables. These model properties can be tested against the current and upcoming GRB data, through which the nature of the fireball as well as the mechanism and site of the GRB emission will be identified. In view of the current data, various models are appraised through a simple Monte Carlo simulation, and a tentative discussion about the possible nature of X-ray ashes is presented.
