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June 4, 2026Open Access

Toward a Vacuum Origin Derivation of the Low-Acceleration Constitutive Response: Casimir Surface Scaling, Planck-Mode Structure, and the Origin of a₀

Key Points

This research aims to explore the origin of the low-acceleration constitutive response function from quantum-vacuum physics.
Investigated the covariant gravitational framework and its implications for galaxy rotation curves and dark energy behavior.
Developed a model using light-speed cutoffs and infrared mode sums to analyze the vacuum contributions to gravitational responses.
Analyzed scaling behaviors through Casimir-like surface energy effects at varying accelerations.
Identified a significant scaling relationship for the Casimir-like surface energy corresponding to low-acceleration dynamics.
Demonstrated that the response function can be represented as a function of the scaling parameters, leading to new insights into gravitational behavior.
Revealed that a complete response function remains to be formulated, emphasizing the need for further theoretical work.

Abstract

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 equationequation (g g) =-4 Gₘequationwas 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 \ (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 coincidenceequationa₀ cH₀2²2{3}. equationThe naive single-field zero-point estimate gives a vacuum-energy scalingcontrolled by \ (a₀\), 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 scaleequationL c²|g|equationhas the required scalingequation ₒₔₑ₅₀₂₄ cL³, equationwhich corresponds to a low-acceleration action contributionequationF ₈ₑ (Y) Y^3/2. equationThis yieldsequationg (Y) Y^1/2 |g|a₀equationin the deep-MOND limit and therefore recoversequation|g|²=a₀ gN. 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 \ (g Y\) at \ (Y1\) and\ (g1\) at \ (Y1\). The exact spherical Casimir coefficient, the derivation of the effective boundary condition, and the full covariantcurved-spacetime treatment remain open problems.

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