Probing Discrete Vacuum Geometry: Predictions for a High-Order Azimuthal Weak-Lensing Modulation in Galactic Halos

The standard cosmological model ($\Lambda$CDM) successfully accounts for a broad range of cosmological observations, while galactic rotation curves and weak-lensing profiles remain key phenomenological probes of non-baryonic dark matter. In this paper, we investigate an exploratory non-canonical scalar effective-field construction in which additional gravitational support is represented by an effective vacuum stress contribution in the weak-field regime. Within an adopted non-linear boundary-value formulation, the resulting field equations admit outer-field configurations whose asymptotic scaling is compatible with observed galactic rotation plateaus. Extending the effective construction to weak gravitational lensing, we consider a discretized perturbative representation of projected halo geometry and examine its consequences for azimuthal shear structure. If such effective discretization contributes to halo shear, local metric perturbations may induce higher-order angular modulations of the background lensing field. Within the adopted symmetry assignment, a leading high-order azimuthal mode emerges together with a corresponding benchmark normalization scale in the asymptotic regime. Future high-resolution weak-lensing surveys, including data from the Euclid Consortium, may test this framework through stacked multipole analyses searching for isolated high-order azimuthal excess relative to adjacent spectral modes.