  We set up a numerical S-matrix bootstrap problem to rigorously constrain
bound state couplings given by the residues of poles in elastic amplitudes. We
extract upper bounds on these couplings that follow purely from unitarity,
crossing symmetry, and the Roy equations within their proven domain of
validity. First we consider amplitudes with a single spin 0 or spin 2 bound
state, both with or without a self-coupling. Subsequently we investigate
amplitudes with the spectrum of bound states corresponding to the estimated
glueball masses of pure SU(3) Yang-Mills. In the latter case the 'glue-hedron',
the space of allowed couplings, provides a first-principles constraint for
future lattice estimates.
