Feynman diagrams are the standard tool for computing scattering amplitudes in quantum field theory. Each diagram represents a term in a perturbation series, with coefficients that must be computed and often renormalised. This paper shows that scattering amplitudes in Pentagon Physics are traces over the 2I character table. The 81-entry character table of the binary icosahedral group 2I encodes all allowed transitions. Feynman diagrams are character table entries. The amplitude for a process is the trace of the corresponding representation matrix. No perturbation theory, no renormalisation — just finite group algebra.