(Full) Seeley--DeWitt Reconstruction of Einstein Gravity: NJL Origin, Source-Only State Dependence, and Operational Test

This work develops a conservative but conceptually sharp framework for emergent gravity that stays strictly within the Einstein–Hilbert regime while introducing a controlled, testable state dependence on the source side only. The starting point is not a fundamental spacetime metric but the dynamics of open quantum systems. The fundamental variable is the density matrix ρρ, evolving under a GKSL semigroup. Under detailed balance, the dissipative part of this evolution can be reformulated as a gradient flow of relative entropy in a quantum optimal-transport (OTq) geometry. Decoherence and pointer states then provide the classicalized reference frames (rods, clocks, records) needed to give operational meaning to proper time, distances and causal structure. On the “macro” side, the effective gravitational action is organized via the Seeley–DeWitt (SDW) heat-kernel expansion for a Dirac-type operator. In four dimensions, the a2 coefficient locks the Einstein–Hilbert term with a universal Newton constant G0 (Einstein-strict regime), while higher coefficients (a4 and beyond) encode state-dependent matter/source contributions. A central principle of the framework is a strict No-μR rule: no state-dependent factor ever multiplies the Ricci scalar. All admissible state dependence is pushed into the matter/source sector. In this setting, the active gravitational charge of a source is written as q=μ (pκ) M, where μ (pκ) =1/Φ (pκ) >0 and pκ∈0, 1 is a bounded state invariant (essentially a normalized purity/coherence proxy) constructed from ρ. Test masses still follow geodesics of the emergent metric (WEP preserved) ; any non-universality appears only through μ (pκ) on the source side. The construction is compatible with Bianchi identities and keeps the principal symbol of the Einstein–Hilbert sector untouched (no modified graviton speed, no superluminal propagation). A key outcome is that the framework is explicitly testable at low energy. In the weak-field regime and for a fixed geometry and inertial-mass distribution, one can modulate the coherence of a single source and look for a narrow-band “lock-in” signature Δg/g≃Δμ/μ correlated with that modulation. This places the emphasis on measuring (or bounding) the local susceptibility of the constitutive law ΦΦ, instead of postulating a global functional form. The paper outlines how such a signal can be sought with atom interferometers, quantum gravimetry and related cold-atom protocols, under a strict regime of null tests and systematic vetoes. In short, the proposal does not attempt to solve dark matter, dark energy or Planck-scale physics. Instead, it isolates a minimal, Einstein-strict, source-only extension of gravity, rooted in standard SDW/EFT and open quantum systems, and designed from the ground up to be falsifiable in realistic low-energy experiments.