Maximum Acceleration Energy When a Hillas Criterion Meets a Critical Shear

The highest energy a source can impart to a charged particle is bounded geometrically: the particle must remain inside the accelerator while it gains energy. Applying the Hillas criterion E = Z e B R beta c to six representative source classes gives proton maxima ranging from 9.252e+14 eV for a supernova remnant to 2.498e+19 eV for an active-galactic-nucleus jet knot, with iron at the same site reaching 6.495e+20 eV. The critical shear associated with the density bound is 8.559e+32 per second. The largest shear estimated for any source in the table, obtained by letting the velocity change by of order the speed of light across the source size, is 2.998e-02 per second, smaller than the critical value by a factor of 2.855e+34. The shear bound therefore excludes none of the standard candidate sources and trims none of their Hillas energies: a negative result for the bound as a source-selection tool, obtained by direct comparison rather than assertion. This study is simulated and computed from stated assumptions; it is not a measurement.