The Tensor Sector of the Conformal Einstein-Gap: Gravitational-Wave Propagation, Standard-Siren Distance, and Black-Hole Ringdown in the Geometric Relay Programme

This deposit presents the tensor sector of the conformal Einstein-gap in the Geometric Relay Programme (GRP), built on the Entangled Relativity framework of O. Minazzoli. The paper derives the gravitational-wave propagation equation on a FLRW background for the conformal Einstein-gap: Q_munu = G_munu(g) - G_munu(kappa g), with phi = ln(kappa). The main tensor-sector results are: c_GW = c, m_eff^2 = 0, and only the two general-relativity tensor polarizations are present. The only formal homogeneous imprint is the Planck-mass/friction term: alpha_M = phidot / H = d ln(kappa) / d ln(a), which gives the standard-siren relation: dL_GW / dL_EM = sqrt(kappa_0 / kappa(z)). A background-consistency diagnostic then shows that the dark matter cannot be a rolling conformal Einstein-gap at the homogeneous level. On FLRW, the background counter-curvature density is: rho_Q = -3 H phidot - (3/4) phidot^2. A frozen relay gives rho_Q = 0, while requiring the rolling gap to supply the dark matter pins: alpha_M approximately -1.2. This would induce an excluded variation of G_eff across the matter era. Therefore the viable GRP dark sector must be the perturbative, memory-clustered counter-curvature with a frozen background conformal factor. The resulting standard-siren prediction is consequently small: alpha_M_bg approximately 0, dL_GW approximately dL_EM, up to perturbative lensing/Shapiro-type residuals. The paper also proves a background-independent principal-symbol theorem: since Q_munu contains at most first derivatives of the metric, it does not alter the graviton principal symbol. The vacuum tensor quasi-normal modes of a black hole are therefore exactly those of GR. Possible deviations can only arise in matter-sourced sectors, where L_m is not equal to T, and are independent of the cosmological lambda bracket. The central open lock is explicitly identified: the coarse-grained perturbative counter-curvature must produce a CDM-like density, <Q_00_pert> proportional to a^-3, without rolling kappa. The decisive future test is the cold law, w_eff approximately 0, against generic Green-Wald-type backreaction.