Causal Propagation and Gravitational Waves from Projective Spectral Dynamics

We construct the Lorentzian completion of the spectral entropy framework in which the renormalized variation of a Laplace-type operator generates an infrared Einstein response. While the original formulation was elliptic and sufficient to derive the effective field equations and their thermodynamic interpretation, it did not address causal propagation. We define a Lorentzian spectral action via the real part of the hyperbolic determinant, equivalently through a Schwinger–Keldysh prescription, ensuring a real and retarded quadratic kernel. Linearization about Minkowski spacetime shows that, in the regime $R\ell_\chi^2 \ll 1$ and $\omega \ll \ell_\chi^{-1}$, the theory propagates exactly two transverse-traceless tensor modes of helicity $\pm2$. Scalar and vector sectors remain non-dynamical in the infrared, and the additional pole generated by higher-derivative corrections lies at $k^2 \sim \ell_\chi^{-2}$, beyond current observational reach. The polarization content therefore coincides with that of general relativity in the accessible domain. When the vector sector is sourced by a projected momentum density it obeys an elliptic gravitomagnetic constraint, fixing the frame-dragging potential $g_{0i}$ without an additional propagating degree of freedom, conditionally on the projected matter current. Beyond the leading Einstein sector, the ratio of Seeley–DeWitt coefficients fixes a universal $k^4$ correction to the dispersion relation, \[ \omega^2 = c^2 k^2 - \gamma \ell_\chi^2 k^4 + \mathcal{O}(k^6), \] with $\gamma = 1/(180\zeta)$ and $\zeta=\mathcal{O}(1)$ encoding scheme-dependent normalization factors. In the natural spectral scheme one finds $\gamma=1/30$. This structure maps directly onto the parametrizations used in gravitational-wave catalogues and yields a bound on the pre-geometric scale $\ell_\chi$ from interferometric data. The Lorentzian completion thus embeds the spectral entropy dynamics in a fully causal framework and establishes gravitational waves as infrared probes of the underlying pre-geometric scale.