Exactly Certified Dispersive Functionals from Automated Search: An Application to Gravitational Positivity Bounds

Abstract

Dual dispersive positivity bounds on gravitational effective field theories are normally obtained from hand-designed functionals whose optima are decided by floating-point semidefinite programming up to solver tolerance. We present a pipeline in which the functional design is performed by an evolutionary search (structural mutations proposed by a large language model) whose sole selection pressure is an exact-arithmetic certificate: a candidate counts only if per-spin positivity of its smeared spectral density is proven in exact rational arithmetic on the mass continuum, for every audited spin. A controlled ablation finds no search advantage for the language model over conventional evolutionary or random search on this task, so we treat it as the search's mutation operator. The identical machinery reproduces the known scalar EFT-hedron with exact all-spin certificates.

For the $R^4$ coefficient $g_0$ of $D=10$ maximal supergravity we obtain exact certificates for spin-truncated dual functionals: 3.0136 (at $J\le40$) and 3.0708 (at $J\le120$) on the reference domain of Caron-Huot et al., and 3.0401 (at $J\le120$) in the extended-$u$ domain that reference introduced but left uncomputed. These become bounds on $g_0$ only conditional on tail positivity beyond the audited spins (measured on extensive scans; not proven). An intermediate $J\le40$ truncation certifies 2.96514, numerically below the reference's 3.000; we show by direct measurement that this is truncation slack and does not survive tail hardening.

Systematic attempts to certify below 3.000 along four independent routes all terminate at the same tangency wall near 3.0. We present that convergence, together with the failure modes an exact standard exposes (tolerance-decided constraints at the $10^{-10}$ level; grid scans passing functionals whose exact density is wrong by seven orders of magnitude), as the principal measured content of this methods-and-rigor study.