The Shadow Rule: Dark Matter as the Dimensional Shadow of Ordinary Matter in the Extended Randall-Sundrum Framework

This paper proposes that dark matter is not a separate particle species but the gravitational shadow of ordinary matter, arising from a dimensional mismatch between the gravitational interaction and the matter it reads. In the Extended Randall-Sundrum framework 1,2, the gravitational interaction operates as a one-dimensional bulk coupling, while matter exists at higher-dimensional addresses determined by generation. Each first-generation constituent (quark or lepton) at 2D casts a 1× shadow of its mass. The shadow operates per-constituent, not per-composite: a proton (three quarks) carries a 3× shadow; a hydrogen atom (three quarks plus one electron) carries a 4× shadow. A general formula follows: for any nucleus with Z protons and N neutrons, the constituent count is 4Z + 3N and the dark fraction is (4Z + 3N)/(4Z + 3N + 1). Applied to cosmological abundances (75% hydrogen, 24% helium, 1% metals by mass), the predicted dark matter fraction is 83.4% — within one percentage point of the observed 84.3%. The remaining gap is consistent with trace contributions from stellar nucleosynthesis and higher-generation particle production. The framework predicts generation-dependent dark matter halo profiles, offering a specific resolution of the core-cusp problem.