Signature of a multidimensional space lattice: higher derivative gravity and evidence from galactic dynamics

Abstract The breakdown of Newtonian dynamics at galactic scales poses a fundamental question regarding the macroscopic behavior of gravity. In this work, we approach this anomaly as an inverse problem, performing a rigorous numerical deconvolution of the gravitational field using high-fidelity kinematic data from the SPARC database. Our analysis reveals that the effective gravitational interaction is characterized by a non-local, damped Helmholtz-like propagator of the form $$G\left( r \right) {\propto }e^{-{\upalpha }r}\cos \left( kr+c \right) /r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mfenced> <mml:mi>r</mml:mi> </mml:mfenced> <mml:mo>∝</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:mo>cos</mml:mo> <mml:mfenced> <mml:mi>k</mml:mi> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> </mml:mfenced> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> . This analytical kernel consistently yields correlation coefficients exceeding 0.99 across a morphologically diverse sample of galaxies. To establish a first-principles foundation, we mathematically derive this propagator as an emergent geometric consequence of embedding the observable 3-brane into a multidimensional space-lattice. Notably, this factorized framework provides a natural ultraviolet (UV) regularization of the gravitational potential at the origin, effectively eliminating the classical 1/ r singularity. Furthermore, we demonstrate that a density-driven geometric screening suppresses extra-dimensional excitations in high-density environments, rigorously ensuring consistency with stringent Solar System constraints. Ultimately, this data-driven, non-local extension of the gravitational propagator offers a robust theoretical alternative to the dark matter hypothesis. It suggests that galactic kinematics are deterministically governed by macroscopic spatial resonances and higher-order geometric degrees of freedom inherent to the cosmic fabric.