  While the width-luminosity relation (WLR) among type Ia supernovae (slower is
brighter) has been extensively studied, its physical basis has not been
convincingly identified. In particular, the 'width' has not been quantitatively
linked yet to a physical time scale. We demonstrate that there are two robust
fundamental time scales that 1. can be calculated based on integral quantities
of the ejecta, with little dependence on radiation transfer modeling and 2. can
be inferred from observations. The first is the gamma-ray escape time $t_0$,
which determines the long-term evolution of the bolometric light curve and is
studied in this Paper I. The second is the recombination time of $^{56}$Fe and
$^{56}$Co, which sets the long-term color evolution of the emitted light and is
studied in Paper II. Here we show that the gamma-ray escape time $t_0$ can be
derived with $\sim 15\%$ accuracy from bolometric observations based on first
principles. When applied to a sample of supernovae, the observed values of
$t_0$ span a narrow range of $30-45$ days for the wide range of observed
$^{56}\rm Ni$ masses $0.1M_{\odot}\lesssim M_{^{56}\rm Ni}\lesssim 1
M_{\odot}$. %This (trivial) bolometric WLR This narrow range of the gamma-ray
escape time across the range of luminosities (a trivial WLR) is consistent with
central detonations and direct collisions of sub-Chandrasekhar mass white
dwarfs (WDs) but not with delayed detonation models for explosions of
Chandrasekhar mass WDs, which are therefore disfavored as the primary channel
for the population of type Ia supernovae. Computer codes for extracting $t_0$
from observations and models and for calculating gamma-ray transfer in 1D-3D
are provided.
