Entropic Linear–Response Organisation of Future–Mass Projection Kernels

Future–Mass Projection (FMP) gravity replaces local dark matter sources by a nonlocal bilocal kernel acting on the baryonic energy–momentum tensor. In covariant formulations this kernel is defined on a closed time path (CTP) with a finite horizon ΔT, but its Newtonian limit in realistic, nonlinear galactic configurations remains opaque. In this exploratory work we organise the Newtonian FMP kernel in terms of an “entropic” linear–response ansatz on the space of coarse–grained surface density profiles Σ(R) of axisymmetric stellar discs. Starting from a coarse–grained functional S[Σ] = Sloc + Spair we define a background–dependent linear–response kernel KS(R, R′; ¯Σ ) as the Hessian of S around a chosen background disc ¯Σ(R). Restricting to Gaussian statistics in the fluctuations and to a simple, radially modulated covariance kernel, we obtain an entropic contribution to the FMP source, ΣF(R)=∫0∞\ddR′Kent(R,R′;Σ¯)δΣ(R′), which is linear in the fluctuations δΣ = Σ − ¯Σ but parametrically dependent on the background disc. We model Kent as a superposition of a local term and a finite–width nonlocal term, controlled by a radial weight function and a Gaussian covariance kRernel. For an exponential Milky Way–like disc we explicitly enforce the radial zero–DC condition $\int \Sigma_F(R)\,2\pi R\,\dd R=0$ and show how it fixes the ratio of local to nonlocal amplitudes. With a small fluctuation parameter ε ≪ 1 we find entropic boost factors D(R) of order unity, with D(R) ≃ 1.0 at R ≲ 3 kpc, D(R) ≃ 2.3 at R ≃ 8 kpc, and D(R) ≃ 1.2 at R ≃ 20 kpc, consistent with the range required by previous FMP fits to Milky Way rotation curves. The construction is deliberately phenomenological and does not claim to derive the FMP kernel from microphysics; instead it provides a state–dependent surrogate that organises the Newtonian kernel in terms of coarse–grained disc properties and highlights where a future CTP–based derivation would need to reproduce or replace these ingredients.