A Weyl-Curvature-Constrained Effective Field Theory of Gravity: Exact Background Stability and a Falsifiable, Parametrized Quasinormal-Mode Signature

Penrose’s Weyl Curvature Hypothesis (WCH) proposes that the vanishing of the Weyl tensor at the initial cosmological singularity encodes the very low gravitational entropy of the early universe. We explore a specific, falsifiable dynamical consequence of promoting this idea from a boundary condition to a continuous effective-field-theory (EFT) coupling in the gravitational action, of the schematic form ξ(R) CµνρσCµνρσ. We give the exact field equations obtained by variation of this action, and we prove – both analytically and by an independent, from-scratch symbolic (computer-algebra) computation – that the Schwarzschild metric solves the modified vacuum field equations exactly and without any fine-tuning of ξ(R), because the Bach tensor vanishes identically on any Ricci-flat four-dimensional background. This corrects a stronger and less transparent claim (fine-tuning of ξ′(0)) that appears in an earlier, non-peer-reviewed draft of this line of reasoning and that we explicitly disown here. Treating the quartic-derivative term perturbatively, in the sense of Simon’s method of perturbative constraints, avoids the Ostrogradsky ghost associated with na¨ıve fourth-order equations of motion. We then parametrize, via dimensional power counting anchored to the exactly computed Weyl invariant C2 = 48M²/r⁶, the leading-order correction such a term induces in the axial Regge–Wheeler potential governing black-hole ringdown, introducing a single dimensionless coupling ˆϵ = ξ₀/M². Using a literature-validated third-order WKB solver (accurate to ≲1.5% against Leaver-method values in the pure-GR limit), we compute how the fundamental quasinormal-mode damping time depends on ˆϵ, perform a multi-multipole sensitivity analysis, and construct a toy Bayesian inference pipeline that recovers an injected coupling from a synthetic noisy ringdown measurement.The central, falsifiable prediction of the model is a monotonic, sign-definite relationship between ˆϵ and the fractional shift in ringdown damping time, Δτ/τ_GR, that can be constrained directly by gravitational-wave ringdown spectroscopy. We are explicit throughout about which results are exact (the action, the field equations, and the Bach-flatness theorem) and which are leading-order EFT estimates awaiting a full linearized Bach-tensor perturbation calculation (the precise radial profile of the potential correction). All code, data, and unit tests supporting this paper are included and are fully reproducible (fixed random seed).