TITLE:
Galactic Rotation Curves in the 4DEU Framework: Explicit Derivation of the Extra-Velocity Law from 3D Spatial Curvature Induced by the Gravitational Constraint
AUTHORS:
Domenico Maglione
KEYWORDS:
Four-Dimensional Electromagnetic Universe (4DEU), Galactic Rotation Curves, Curvature-Induced Extra Velocity, Effective Density, 3D Spatial Curvature, Gravitational Constraint
JOURNAL NAME:
Journal of High Energy Physics, Gravitation and Cosmology,
Vol.12 No.3,
July
13,
2026
ABSTRACT: The present work serves as a mathematical addendum to the previous analysis of galactic rotation curves within the Four-Dimensional Electromagnetic Universe (4DEU) framework by Maglione, D. (Journal of High Energy Physics, Gravitation and Cosmology, 12, 431-470, 2026), in which the extra velocity component was fitted through the law
v
cve
(
r
)=
V
0
(
r/
R
ref
)
ε/2
(Equation (12) therein). In 4DEU, this extra component is interpreted not as dark matter, but as the effect of purely three-dimensional spatial curvature induced by the gravitational constraint that locally blocks the 3D-only cosmic expansion in gravitationally bound systems. Although the kinematic law, its physical interpretation, and the corresponding effective-density/curvature profiles were already presented in the above-mentioned 4DEU rotation-curve paper by Maglione, the reverse analytical connection—namely, how the same law can be recovered starting from the outer-curvature behavior—was not written out there in full mathematical detail, in order to avoid excessively burdening the paper. In the present addendum, that derivation is given explicitly. Starting from the local relation between the three-dimensional Ricci scalar (3)R and the effective density and combining it with the standard expression linking effective density to the circular velocity profile, it is shown step by step how the curvature-induced term leads to the same kinematic law previously adopted in the fits. The derivation therefore makes explicit that, within the 4DEU framework, the law used to reproduce the extra-velocity component of galactic rotation curves follows from the 3D spatial curvature generated by the gravitational constraint.