  New resonances with masses of order a few ${\rm TeV}$ might be discovered at
the LHC. We show that no resonance that couples to electrons only through
Standard Model interactions can decay to both $e^+e^-$and $\gamma \gamma$ with
significant branching ratios. This means that finding both electron-positron
and two-photon final states is evidence that electrons couple directly to the
new physics associated with the resonance and furthermore that the resonance is
not spin-1. The least fine-tuned such examples involve electron compositeness.
One such example, Kaluza Klein excitations of the graviton in the version of
the Randall Sundrum Model where Standard Model matter is located on the ${\rm
TeV}$ brane, can be distinguished from other possibilities by its predicted
branching fractions into the two modes.
