Collimation and the Causal Bound: Opening Angles, Lorentz Factors and the Shape of the M87 Jet

This paper addresses a single accounting question in the physics of relativistic jets: whether the measured opening angles of relativistic jets are narrow enough, given their Lorentz factors, for the flow to remain in causal contact across its own width. Using published Lorentz factors and opening angles for five jets, and the measured parabolic width profile of the M87 jet, the calculation derives new bounds and budgets from standard synchrotron and special-relativistic relations, with every number produced by the deposited compute script. The headline results are: all connected = True, width at 100pc = 1.45. M87 inner: 0.436, M87 outer: 0.314, 3C 273: 0.349, Cygnus A: 0.262, BL Lac: 0.393. All five cases satisfy the causal condition, most comfortably, with products well below unity for the wide slow base of M87 and approaching a few tenths for the fastest narrow jets. The study is a computation from stated assumptions and published inputs; it is simulated, not measured, and its conclusions are stated at the precision those inputs support.