Description 
We study the space of all kinematically allowed four photon and four graviton Smatrices, polynomial in scattering momenta. We demonstrate that this space is permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants s, t and u. We construct these modules for every value of the spacetime dimension D, and so explicitly count and parametrize the most general four photon and four graviton Smatrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these Smatrices. We then conjecture that the Regge growth of Smatrices in all physically acceptable classical theories is bounded by s^{2} at fixed t. In the case of photons a four parameter subset of the polynomial Smatrices constructed above satisfies this Regge criterion. In the case of gravity, on the other hand, no polynomial addition to the Einstein Smatrix obeys this bound for D ≤ 6. When D ≥ 7 there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture then implies that the Einstein four graviton Smatrix does not admit any physically acceptable polynomial modifications for $D\leq 6$. A preliminary analysis suggests any finite sum of pole exchange contributions to four graviton scattering also such violates our conjectured Regge growth bound, at least when D ≤ 6$, even when the exchanged particles have low spin.
