PAPER-FBT10C: The Traceless Response Mode as the Spin-2 Gravitational Wave Sector in the FBT Framework

The Fracture–Berry–Tension (FBT) framework interprets the effective four-dimensional spacetime metric as a macroscopic response geometry built from coarse-grained quantised dual-phase curvature. In FBT10A, the metric was written schematically as gμν = g(0)μν + Ngrav 〈Sμν〉, where 〈Sμν〉is a symmetric response tensor formed from unresolved dual-phase curvature and horizontal–vertical mixed density. In FBT10B, this response tensor was decomposed into a scalar trace mode and a traceless tensor mode. The scalar trace branch was interpreted as an emergent Higgs-like order parameter. The present paper studies the complementary traceless branch and identifies it as the natural spin-2 gravitational-wave sector of the FBT response geometry. The central decomposition is 〈Sμν〉=1/4σ gμν + Σμν, gμνΣμν = 0. The traceless response mode Σμν contributes to the effective metric perturbation through hμν = Ngrav Σμν. After imposing the transverse-traceless reduction in the low-energy radiative regime, the physical tensor branch is hTTμν = Ngrav ΣTTμν . Thus gravitational waves remain metric perturbations at the effective Lorentzian level, but their underlying FBT origin is not a primitive metric field. They are transverse-traceless excitations of the traceless curvature-response mode. The paper assumes the Lorentzian arena and finite vacuum propagation bound c∗ established in FBT08A/B/C. It also assumes the gravitational projection interface of the unified curvature–tension equation of FBT12A. After coarse-graining this interface and restricting to the traceless branch, the leading local generally covariant second-order closure is a Lichnerowicz-type wave equation. On a flat vacuum background this reduces to □hTTμν = 0, equivalently □ΣTTμν = 0, with characteristic speed cGW = c∗. The plane-wave polarisation analysis then shows that the propagating sector carries precisely the two helicity states ±2. A source term is obtained from the transverse-traceless projection of the anisotropic part of the coarse-grained curvature-response stress tensor. The paper also records the coupling between the traceless tensor mode and the scalar trace mode, and discusses possible observational consequences such as luminal propagation, high-frequency dispersion corrections, and primordial stochastic backgrounds. The main message is that the FBT traceless response mode provides a geometric origin for the spin-2 gravitational-wave sector: at low energies it is indistinguishable from the usual transverse-traceless metric perturbation, while at the underlying response level it is a propagating excitation of coarse-grained dual-phase curvature.