Half-Completeness from Standard Gravitational Dispersion Relations

The completeness hypothesis states that quantum gravity realizes every representation allowed by its gauge group. Previous S-matrix arguments used the graviton pole to constrain exchanged representations, including a proof of completeness by imposing an additional high-energy bound on crossing-odd amplitudes. We investigate what follows without this assumption, using standard twice-subtracted dispersion relations, positivity, and crossing symmetry. For external massless SU(2) isospin $J$, we prove ``half-completeness’': either massive exchanges include every isospin $j\le J$, or at least one $j>J$ is required. At $J=1$, we construct an explicit amplitude with only $j=0,1$ exchanges that satisfies these conditions but violates the crossing-odd bound. This establishes that the additional high-energy assumption does not follow from the standard framework. For $J\ge 2$, however, we show that lower-only spectra cannot be realized within a class of string-inspired Euler amplitudes, leaving their existence an open question. More generally, for scalar matter transforming under any compact internal symmetry group, we prove that massive singlets of unbounded Lorentz spin must be exchanged.