Natario's Zero-Expansion Warp Drive: Does Removing Expansion Remove the Negative Energy?

Natario's warp construction sets the expansion of the Eulerian flow to zero identically, so the bubble transports its contents without the volume expansion and contraction that define Alcubierre's original. It is natural to hope the negative energy was tied to the expansion and left with it. Evaluating the Eulerian energy density for the zero-expansion field, where the density is set by the curl of the shift vector, gives 2.68e+21 kilograms per cubic metre in magnitude for a bubble of one hundred metres radius at light speed, and 2.68e+25 kilograms per cubic metre at one metre, against a density ceiling of 9.72e+57 kilograms per cubic metre that neither approaches. The density is negative at the wall, with a ratio to the ceiling of 2.76e-37 at the headline radius. The verdict is plain: zero expansion is a genuine geometric improvement and no energetic rescue. The negativity lives in the shift's shear and curl, not in its expansion, and it survives the redesign at full order of magnitude. Computed, not measured.