Causality, Positivity, and the Cutoff of Higher-Derivative Gravity: Sign-Indefinite Tidal Forces and the Species Scale

The low-energy effective field theory of gravity admits curvature operators that correct the graviton three-point vertex, and these operators are not free parameters. We give a self-contained and symbolically verified analysis of the obstruction. Working with the Aichelburg–Sexl shockwave, we first establish by explicit computer algebra in D=5 and D=6 that the shockwave curvature is algebraically of type N, that every quadratic curvature invariant vanishes identically on it, and that the Lanczos–Lovelock tensor vanishes componentwise — so the shockwave remains an exact solution of Einstein–Gauss–Bonnet gravity and, by the same argument, of any theory built from the Riemann tensor. We then compute in closed form the eigenvalue spectrum of the tidal tensor ∂_i∂_j h that controls the polarization dependence of the Shapiro delay: it is traceless, with a single radial eigenvalue +(D-4)(D-3) h/r^2 and D-3 degenerate angular eigenvalues -(D-4) h/r^2, verified symbolically for 5 ≤ D ≤ 9. The tracelessness is the crux. Because a traceless nonzero tensor necessarily carries eigenvalues of both signs, a Gauss–Bonnet coupling λ of either sign produces a time advance for some graviton polarization once the impact parameter drops below b_* ∼ √(|λ|); no choice of sign evades the problem, only λ=0 does. Asymptotic causality therefore requires new states at M ≲ |λ|^-1/2, and eikonal unitarity requires them to form a Regge tower of unbounded spin. We show that a twice-subtracted dispersion relation reproduces the identical scaling by wholly independent means, and we observe that the causality cutoff coincides parametrically with the species scale Λ_sp = M_Pl/√(N), so that the tower demanded by causality, the tower that lowers the species scale, and the tower of the Swampland Distance Conjecture are one and the same. Tree-level bosonic and heterotic strings saturate the bound, λ M_s^2 = O(1), while maximally supersymmetric type II strings evade it by having no correction to the cubic vertex at all. Finally we quantify how weak the empirical situation is: current gravitational-wave constraints on Einstein-dilaton-Gauss–Bonnet gravity, α_GB ≲ 1.7 km, are weaker than the causality bound by a factor 8.6×10^21 in length, once one grants that there is no gravitationally coupled higher-spin tower below the TeV scale.