Sampling the Graviton Pole and Deprojecting the Swampland

We develop a primal bootstrap framework for effective field theories in the presence of a graviton pole, based on finite-resolution sampling rather than smearing, while also allowing direct control over the number of subtractions. We show that this approach reproduces the known projective bounds obtained from smearing in $D\ge 6$, while yielding slightly stronger bounds in $D=5$. This method also makes it straightforward to impose linearized unitarity directly and provides an access to the extremal spectra. Applying the method to crossing-symmetric dispersion relations, we derive new non-projective bounds that fix the overall scale of the EFT couplings. In $D=5$, for example, we find that $\frac{M}{M_{\mathrm{P}}}\lesssim 7.8$, showing that the EFT cutoff cannot be taken parametrically larger than the Planck scale. At the extremal values of the couplings, the spectra exhibit a surprising structure: for projective bounds, they exhibit peaks around quadratic Regge-like trajectories, while for the non-projective bounds they form sharp quadratic bands. In the latter case, the leading coefficients further display an inverse-quadratic dependence on the band number.