Bunch Density Scaling from the High-Luminosity LHC to a 100 TeV Hadron Collider Against the Density Ceiling

This paper computes the characteristic densities of the upgrade path from the High-Luminosity LHC to a 100 TeV hadron collider 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 63.5 orders of magnitude between the operating point and the ceiling: moving from the High-Luminosity LHC to the 100 TeV design raises peak bunch mass density only by a factor of 0.482, leaving a margin of 63.5 orders of magnitude to the ceiling; luminosity-driven focusing, not any fundamental density bound, sets the practical density in both machines. 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.