The Dynamical Sector of Modular Entropic Gravity: Propagating Modes, Gravitational-Wave Compatibility, and the Information-Geometric Route to the Einstein–Hilbert Action

We develop the dynamical sector of Modular Entropic Gravity (MEG) by treating the Einstein-Hilbert action as a derived rather than postulated object. The vacuum distinguishability kernel of the broader programme, identified with the Bogoliubov-Kubo-Mori monotone metric on the manifold of vacuum-adjacent states, decomposes into compression, shear, and mixing pieces; we show that the shear sector contains the Einstein-Hilbert action on curved backgrounds as the leading two-derivative diffeomorphism-invariant content of an integrated stress-tensor correlator, with the gravitational coupling G_ind fixed by vacuum structure. The linearised theory on flat space gives a quadratic action whose modular reparametrisation invariance, inherited from the equality case of the data-processing inequality, yields linearised diffeomorphism invariance; the Fierz-Pauli uniqueness theorem then completes the construction to the full Fierz-Pauli operator. The two derivations are mutually locked through standard no-go theorems for self-coupled massless spin-2 fields, providing the strongest internal-consistency check of the construction. The leading-order structural results for the gravitational sector are c_T = c on Lorentz-invariant and cosmological backgrounds with corrections suppressed by (ℓ_β ω / c)², two polarisations for the massless tensor mode, and a running Planck-mass parameter α_M^MEG = 0 at leading QFI/EH order. The leading-order MEG predictions coincide with general relativity and are compatible with GW170817, standard-siren amplitude bounds, and the primordial tensor-to-scalar ratio constraint r < 0.034. MEG-specific gravitational-wave signatures concentrate in identified sub-leading channels: higher-derivative dispersion, scalar-tensor mixing on inhomogeneous backgrounds, and inflationary entropic-sector content collected in an effective damping parameter ν_MEG(z) for late-time propagation. The construction rests on the data-processing inequality of quantum information theory specialised to the modular structure of the vacuum. The relative-entropy choice that underlies the kernel is selected uniquely from candidate divergences by five structural conditions, and is forced by the bottom-up route from phenomenology through the modular Hamiltonian and the Bisognano-Wichmann construction. On this architecture, classical general relativity is the time-symmetric macroscopic projection of MEG's intrinsically directional coarse-graining flow: correct in its tested classical domain, structurally incomplete in the regimes where the coarse-graining direction becomes phenomenologically operative.