Toward a Vacuum Origin Derivation of the Low-Acceleration Constitutive Response: Casimir Surface Scaling, Planck-Mode Structure, and the Origin of a_0

We investigate whether the low-acceleration constitutive response functionappearing in the covariant extension of the constitutive gravitational frameworkcan be motivated from quantum-vacuum response physics rather than introducedpurely phenomenologically. In the previous papers in this series {DeMartino2026a,DeMartino2026b,DeMartino2026c,DeMartino2026d}, the weak-fieldconstitutive equation{equation}∇·(κ_g g)=-4π Gρ_m{equation}was shown to reproduce galaxy rotation curves, the radial acceleration relation,the baryonic Tully--Fisher relation, and first-order SDSS weak-lensing amplitudeswithout introducing explicit galaxy-scale dark-matter halos. The correspondingcovariant action contains an otherwise unspecified dimensionless responsefunction \(F(Y)\), with \(κ_g=∂ F/∂ Y\). Here we examine whether the required low-acceleration behavior can arise fromvacuum mode structure. An unperturbed infrared mode sum with a Hubble-scale cutoff identifies thesame infrared scale that appears in the empirical coincidence{equation}a_0 ~ cH_0/2π^2/2π√Λ/3 .{equation}The naive single-field zero-point estimate gives a vacuum-energy scalingcontrolled by \(a_0\), but it does not by itself reproduce the observeddark-energy density. The full normalization of the cosmological vacuumcontribution therefore remains an open problem. We show that a Casimir-like surface energy associated with an effectivespherical boundary of scale{equation}L~ c^2/|g|{equation}has the required scaling{equation}σsurface~ c/L^3,{equation}which corresponds to a low-acceleration action contribution{equation}FIR(Y)∝ Y3/2.{equation}This yields{equation}κ_g(Y)∝ Y1/2~ |g|/a_0{equation}in the deep-MOND limit and therefore recovers{equation}|g|^2=a_0 g_N .{equation} The result should be interpreted as a scaling-level mechanism for thelow-acceleration branch of the constitutive response, not as a completedderivation of the full covariant theory. A complete response function mustinterpolate between \(κ_g→ √Y\) at \(Y1\) and\(κ_g→1\) at \(Y1\). The exact spherical Casimir coefficient,the derivation of the effective boundary condition, and the full covariantcurved-spacetime treatment remain open problems.