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We study the space of all kinematically allowed four photon and four gravitonS-matrices, polynomial in scattering momenta. We demonstrate that this space isthe permutation invariant sector of a module over the ring of polynomials ofthe Mandelstam invariants $s$, $t$ and $u$. We construct these modules forevery value of the spacetime dimension $D$, and so explicitly count andparameterize the most general four photon and four graviton S-matrix at anygiven derivative order. We also explicitly list the local Lagrangians that giverise to these S-matrices. We then conjecture that the Regge growth ofS-matrices in all physically acceptable classical theories is bounded by $s^2$at fixed $t$. A four parameter subset of the polynomial photon S-matricesconstructed above satisfies this Regge criterion. For gravitons, on the otherhand, no polynomial addition to the Einstein S-matrix obeys this bound for $D\leq 6$. For $D \geq 7$ there is a single six derivative polynomial Lagrangianconsistent with our conjectured Regge growth bound. Our conjecture thus impliesthat the Einstein four graviton S-matrix does not admit any physicallyacceptable polynomial modifications for $D\leq 6$. A preliminary analysis alsosuggests that every finite sum of pole exchange contributions to four gravitonscattering also such violates our conjectured Regge growth bound, at least when$D\leq 6$, even when the exchanged particles have low spin.

Classifying and constraining local four photon and four graviton S-matrices