(a) *Compton Drag*

This model envisions the propagation of the relativistic outflow in a dense bath of seed photons with energy *E*seed that provide the drag for the expanding outflow whereby cold electrons in the outflow Compton upscatter soft photons [132,133]. The seed photons can be provided either by radiation from the associated supernova remnant that exploded a time ∆*t* ' few hours before the outflow is launched, or if ∆*t* is negligibly small then by the walls of the funnel that has been cleared in the massive star progenitor's envelope by the jet-driven bow shock post core-collapse. These requirements limit the applicability of this model to only long-soft GRBs and do not explain how such an emission would arise in short-hard GRBs. This scenario presents an entirely non-dissipative flow, which is insensitive to the magnetization but yields a high (. 50%) radiative efficiency. To produce the variability, the flow is required to be unsteady. The required *τ<sup>T</sup>* & 1 in this model may make it difficult to produce prompt high-energy emission due to opacity to pair production.

When the prompt GRB emission originates inside the funnel, it is assumed that the funnel is pervaded by a blackbody radiation field emitted by the funnel walls. The spectral peak of the observed prompt emission is then simply the inverse-Compton scattered peak at energy *EIC* ∼ Γ <sup>2</sup>*E*seed, where Γ is the bulk LF of the outflow. Inhomogeneity in the funnel temperature and bulk-Γ of subsequent shells, which could also collide to produce internal shocks, gives rise to a Band-like broadened spectrum. The local polarization, i.e., from a given point on the outflow surface, can be as high as 100%; however, the net observable polarization, e.g., in a top-hat jet, is reduced to Π . 50% for a jet with (Γ*θj*) <sup>2</sup> > 10 [24]. If the jet is narrower than this with (Γ*θj*) <sup>2</sup> < 10, then the net polarization can be much higher with Π . 95% [134]. However, such high polarization requires highly idealized assumptions that are hard to meet in reality.
