Modular Entropic Gravity: An Axioms-First Derivation of the Gravitational Sector

The Modular Entropic Gravity (MEG) programme has developed a theory of gravity through a phenomenology-driven derivation recorded across a multi-paper Zenodo archive and an extensive monograph. The historical route began with the observed phenomenology of disc-galaxy rotation curves, identified the entropy field S(x) as the operational gravitational potential at galactic scales, and proceeded outward to the variational principle, the coherence length ℓ₀, environmental screening, structure formation, cosmological evolution, and the CMB acoustic structure. The four MEG axioms and the underlying vacuum entanglement kernel were identified during this derivation rather than imposed at its start. This paper undertakes the inverse construction. Starting from the four axioms acting on the vacuum entanglement kernel, the scalar sector of MEG gravity is derived forward: the scalar entropy field as the dominant mode of the kernel's vertical projection through a two-channel construction (topological selection by the Z₃ vacuum and vacuum-geometric supply of the coherence scale); the Principle of Least Entropic Stress as the kernel's local quasi-static form; the covariant scalar Lagrangian as the Lorentz-invariant extension of the kernel's quadratic form; the Einstein-scalar field equations by variational extremisation; the matter coupling through the saddle-point completion supplied by Axiom 4; the environmental screening through the kernel's density-dependent coherence length; and the recovery of the Newtonian and post-Newtonian limits, the equivalence principle, and the conservation laws. The construction culminates in an operational Gauss law for entropy flux, ∮∇S_b·dA = -(4πG/c²)M_b(V), identifying gravitational charge with the surface flux of baryonic entropy gradient and fixing the coupling normalisation uniquely by the simultaneous consistency of Axiom 4, horizon thermodynamics, and Gauss's law on nested surfaces. The paper positions itself within the MEG programme's foundational claim: that the four axioms acting on the vacuum entanglement kernel constitute the structure from which physical laws emerge. The programme has substantively addressed the deepest theories of modern physics — the axiomatic structure of quantum mechanics through the dissolution of the Born rule and the DHR superselection construction; the gauge structure of the Standard Model through the lateral channel of the kernel; the Einstein-Hilbert action of general relativity through the quantum Fisher information route developed in the companion dynamical paper (doi:10.5281/zenodo.20271117); and the conservation laws of dynamics through the kernel's symmetry structure. These are the upstream theories from which the body of modern physics is built; the programme's strategy is foundational in the sense of addressing them specifically, with the rest of physics downstream by inheritance. Some elements of the programme — the explicit derivation of the SO(8) vacuum gauge group, the full derivation of the Lorentzian manifold substrate from abstract kernel data, the microscopic determination of ℓ₀ — remain research targets and are identified explicitly. Establishing the foundational claim with full certainty is necessarily a long-term enterprise. The present paper is one step of this programme: the axioms-first derivation of the scalar gravitational sector. The companion dynamical paper carries the axiomatic route to the tensor sector through the quantum Fisher information identification of the Einstein-Hilbert action from the kernel's shear correlator. The companion strong-field paper (doi:10.5281/zenodo.20270963) develops compact-object phenomenology, binary-pulsar timing compatibility, and the parametrised post-Einsteinian framework. The variational origin of the matter coupling and the saddle-point completion of Axiom 4 are developed in detail in the companion paper on the variational and geometric origin of Axiom 4 (doi:10.5281/zenodo.20272143). The historical phenomenology-driven route recorded in the monograph and the present axiomatic route converge on the same gravitational theory. The convergence is evidence that the axioms are doing structural work: the monograph found the right gravitational theory because it was implicitly tracking what the axioms encode; the present paper makes that implicit tracking explicit. The honest-reporting standard is maintained throughout: structural derivations are stated with the standing they earn, open items are recorded explicitly, and the bounded scope of specific claims (the regulator's ghost-free analysis, the operator-level closure of the JLMS matching, the full degeneracy analysis on general curved backgrounds) is acknowledged.