  The observation of a gamma-ray burst (GRB) associated with a supernova (SN)
coincides remarkably with the energy output from a binary system comprising a
very massive carbon-oxygen (CO) core and an associated binary neutron star (NS)
by the Binary-Driven Hypernova (BdHN) model. The dragging effect in the late
evolution of such systems leads to co-rotation, with binary periods on the
order of minutes, resulting in a very fast rotating core and a binary NS
companion at a distance of $\sim 10^5$ km. Such a fast-rotating CO core,
stripped of its hydrogen and helium, undergoes gravitational collapse and,
within a fraction of seconds, leads to a supernova (SN) and a newly born,
fast-spinning neutron star ($\nu$NS), we name the emergence of the SN and the
$\nu$NS as the SN-rise and $\nu$NS-rise. Typically, the SN energies range from
$10^{51}$ to $10^{53}$ erg. We address this issue by examining 10 cases of
Type-I BdHNe, the most energetic ones, in which SN accretion onto the companion
NS leads to the formation of a black hole (BH). In all ten cases, the
energetics of the SN events are estimated, ranging between $0.18$ and $12
\times 10^{52}$ erg. Additionally, in all 8 sources at redshift $z$ closer than
$4.61$, a clear thermal blackbody component has been identified, with
temperatures between $6.2$ and $39.99$ keV, as a possible signature of
pair-driven SN. The triggering of the X-ray afterglow induced by the
$\nu$NS-rise are identified in three cases at high redshift where early X-ray
observations are achievable, benefits from the interplay of cosmological
effects.
