  Gravitational models of self-tuning are those in which vacuum energy has no
observable effect on spacetime curvature, even though it is a priori
unsuppressed below the cut-off. We complement Weinberg's no go theorem by
studying field theoretic completions of self-adjustment allowing for broken
translations as well as other generalisations, and identify new obstructions.
Our analysis uses a very general Kallen-Lehmann spectral representation of the
exchange amplitude for conserved sources of energy-momentum and exploits
unitarity and Lorentz invariance to show that a transition from self-tuning of
long wavelength sources to near General Relativity on shorter scales is
generically not possible. We search for novel ways around our obstructions and
highlight two interesting possibilities. The first is an example of a unitary
field configuration on anti-de Sitter space with the desired transition from
self-tuning to GR. A second example is motivated by vacuum energy sequestering.
