We show that the two-point function of a quantum field theory with de Sitter momentum space (herein called DSR) can be expressed as the product of a standard delta function and an energy-dependent factor. The proportionality to a standard delta function is a nontrivial result valid in any theory without a preferred frame and relies crucially on the on-shellness condition. Different theories are distinguished by the specific form of the energy-dependent proportionality factor. Applied to models exhibiting running of the dimensionality of space, this result is essential in proving that vacuum fluctuations are generally scale invariant at high energies whenever there is running to two dimensions. This is equally true for theories with and without a preferred frame, with differences arising only as we consider higher order correlators. Specifically, the three-point function of DSR has a unique structure of “open triangles,” as shown here.