The Strong-Field Sector of Modular Entropic Gravity: Level I Completion and the Level II Programme

This paper consolidates the strong-field content of the Modular Entropic Gravity (MEG) programme and develops it under the Level I / Level II organising principle in which Level I treats MEG as a covariant scalar-tensor theory with the Einstein-Hilbert sector as a structural input, and Level II is the entropy-only completion in which the Einstein-Hilbert action itself is derived from the modular substrate of the vacuum entanglement kernel. The paper's principal results at Level I are five. First, the bulk Einstein-Hilbert action pulled back onto a non-perturbative metric dictionary g_μν(S) is exactly the negative of the PLES Dirichlet functional in the static, zero-shift, conformally-flat, no-slip sector, with the Schwarzschild exterior recovered exactly in spherical symmetry. Second, compact-object scalar charges carry the form-factor structure α_A = λ F(C_A), with F(C) ∈ [0.60, 0.92] established by direct numerical solution of the sourced MEG scalar equation on fixed GR-TOV backgrounds across the stable neutron-star branch. Third, the natural coupling scale λ_eff ~ σ_S = √(ζ_SM/2) ≈ 10⁻³ from Standard Model modular content is established through a perturbativity argument in the controlled band [σ_S, 4.7 σ_S]; the gap from ~ to = is honestly flagged as remaining open. Fourth, the binary radiative dipole structure δα_MEG = λ_eff [F(C_1) − F(C_2)] is derived from MEG's foundational entropic law, giving compatibility with the most stringent binary-pulsar bounds across all tested systems (J1738+0333, J0737−3039, B1913+16, J0348+0432), with a margin of approximately ten at the natural representative value and at least a factor of two across the natural-scale band. Fifth, the translation of MEG's predictions into the parametrised post-Einsteinian framework locates the framework near the GR limit, with the dipole ppE phase coefficient at ~10⁻⁸ for asymmetric neutron-star/white-dwarf systems — approximately five orders of magnitude below the LIGO-Virgo-KAGRA bound. The black-hole anatomy in MEG is presented on the basis of the companion paper, with the universal mass-independent horizon datum S(r_H) = 1/2, the Bekenstein-Hawking entropy with the coefficient 1/4 fixed by the same JLMS normalisation that calibrates the galactic phenomenology, and the area law emerging structurally from the effectively constant interior. The exchange current induced by environmental screening is shown to be dimensionally negligible for compact-binary inspiral while operative in the galactic and cosmological regimes that the broader MEG programme treats elsewhere. The Level II programme is articulated with its computational checkpoints made explicit; the principal Level II deliverables — the leading-order value of the induced Newton constant from integrated stress-tensor correlators, and the Fierz-Pauli reduction of the kernel's shear sector — have not been performed in the present paper and belong to a forthcoming companion. The honest-reporting standard is maintained throughout: where derivations close, the results are stated with the standing they earn; where structural obstructions are encountered, they are recorded as open items; where work is partial, the partial result is reported and the remainder is flagged.