The Compactness Bound on Jet Lorentz Factors in Gamma-Ray Bursts

High-energy photons escape a gamma-ray burst source only if relativistic motion dilutes the pair-creation optical depth. The compactness argument converts luminosity and variability time into a lower bound on the bulk Lorentz factor. This paper evaluates that bound across representative bursts and compares it with Lorentz factors inferred from afterglow onset. Using the calculation described in the Methods, the headline results are: Gamma_typical = 110, Gamma_bright_fast = 246, Gamma_090510like = 215. Typical long bursts require Lorentz factors of order one hundred, and bright rapidly varying bursts require several hundred. Afterglow-inferred values sit above these floors in the cases where both exist, so the compactness bound is satisfied but not saturated. Any emission model placing the gamma-ray site at much lower Lorentz factors must move the emission radius outward rather than dispute the bound. This study is simulated and computed from stated assumptions; it is not a measurement. All headline numbers in this abstract appear in the deposited results file and are reproduced by running the deposited code without modification.