Space-Charge Density Limits in High-Intensity Proton Rings Against the Density Ceiling

This paper computes the characteristic densities of a high-intensity proton synchrotron beam from published operating parameters and compares them, quantity by quantity, with the Aldred density ceiling of 9.72e+57 kilograms per cubic metre, the neutron-branch bound at prescription coefficient one eighth. The headline comparison gives a margin of 67.7 orders of magnitude between the operating point and the ceiling: the bunch in a megawatt-class proton driver has a mass density 67.7 orders of magnitude below the ceiling, yet its intensity is firmly capped by space-charge tune spread and loss control; collective self-fields, not any absolute density bound, govern high-intensity rings. Every number is obtained from the deposited calculation, which uses only published device parameters, internationally tabulated constants, and algebra stated in full. The result is negative in the useful sense: it shows that the ceiling imposes no constraint on the system as designed or operated, identifies the empirical and engineering limits that act instead, and states precisely what change in operating parameters would be required before the ceiling could matter. The study is a computation from stated assumptions; nothing in it is a measurement.