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 \citep{DeMartino2026a,DeMartino2026b,DeMartino2026c,DeMartino2026d}, the weak-fieldconstitutive equation\begin{equation}\nabla\cdot(\kappa_g \mathbf{g})=-4\pi G\rho_m\end{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 \(\kappa_g=\partial F/\partial Y\). 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\begin{equation}a_0 \sim \frac{cH_0}{2\pi}\sim\frac{c^2}{2\pi}\sqrt{\frac{\Lambda}{3}} .\end{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\begin{equation}L\sim \frac{c^2}{|\mathbf{g}|}\end{equation}has the required scaling\begin{equation}\sigma_{\rm surface}\sim \frac{\hbar c}{L^3},\end{equation}which corresponds to a low-acceleration action contribution\begin{equation}F_{\rm IR}(Y)\propto Y^{3/2}.\end{equation}This yields\begin{equation}\kappa_g(Y)\propto Y^{1/2}\sim \frac{|\mathbf{g}|}{a_0}\end{equation}in the deep-MOND limit and therefore recovers\begin{equation}|\mathbf{g}|^2=a_0 g_N .\end{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 \(\kappa_g\to \sqrt{Y}\) at \(Y\ll1\) and\(\kappa_g\to1\) at \(Y\gg1\). The exact spherical Casimir coefficient,the derivation of the effective boundary condition, and the full covariantcurved-spacetime treatment remain open problems.