High-Field Compact Tokamak Density in SPARC Against the Density Ceiling

This paper computes the characteristic densities of the SPARC high-field compact tokamak 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 63.9 orders of magnitude between the operating point and the ceiling: SPARC reaches roughly three times the ITER density by using a field more than twice as strong in a far smaller volume, and still its core mass density sits 63.9 orders of magnitude below the ceiling; compact high-field design buys fusion power density through field and temperature, not through any approach to an absolute density bound. 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.