  This work extends a contemporaneous effort (Panaitescu & Vestrand 2022) to
study the properties of the lower-energy counterpart synchrotron emission
produced by the cooling of relativistic Gamma-Ray Burst (GRB) electrons through
radiation (synchrotron and self-Compton) emission and adiabatic losses. We
derive the major characteristics (pulse duration, lag-time after burst,
brightness relative to the burst) of the Prompt Optical Counterpart (POC)
accompanying GRBs. Depending on the magnetic field life-time, duration of
electron injection, and electron transit-time Dto from hard X-ray (GRB) to
optical emitting energies, a (true) POC may appear during the GRB pulse (of
duration dtg) or after (delayed OC). The signature of counterparts arising from
the cooling of GRB electrons is that true POC pulses (Dto < dtg) last as long
as the corresponding GRB pulse (dto ~ dtg) while delayed OC pulses (Dto > dtg)
last as long as the transit-time (dto ~ Dto). If OC variability can be
measured, then another signature for this OC mechanism is that the GRB
variability is "passed" only to POCs but is lost for delayed OCs. Within the
GRB electron cooling model for counterparts, POCs should be on average dimmer
than delayed one (which is found to be consistent with the data), and harder
GRB low-energy slopes bLE should be associated more often with the dimmer POCs
The range of low-energy slopes bLE in [-1/2,1/3] produced by electron cooling
and the average burst brightness of 1 mJy (with 1 dex dispersion) imply that
POCs of hard GRBs can be dimmer than R=20 and difficult to detect by robotic
telescopes (unless there is another mechanism that overshines the emission from
cooling electrons) and that the POCs of soft GRBs can be brighter than R=10,
i.e. as bright as the Optical Flashes (OFs) seen for several bursts.
