The Dimensional Ladder: Z = 2D, Dimensional Stratigraphy, and the Sciences as Slices

We identify a one-line formula governing the periodic table of elements: Z = 2D, where Z is the electron capacity of each Madelung group (period length) and D = (l+1)² is the angular momentum degeneracy in four spatial dimensions. The period lengths 2, 8, 18, and 32 are not the subshell capacities in three spatial dimensions (2, 6, 10, 14) but those of four spatial dimensions (2, 8, 18, 32). The Madelung rule, empirical since 1936, is identified as the branching rule SO(4) → SO(3). This observation, which distills into explicit formula results implicit in the group-theoretic periodic table literature of Barut, Kibler, and Ostrovsky, anchors a broader proposal: that the number of accessible spatial dimensions is not fixed at three, or eleven, or any single number. Rather, dimensions form a topography—a landscape with local elevation that varies with gravitational depth. We inhabit the D=1 to D = 11 ledge, a suburban plateau in a terrain that descends deeply at black hole horizons and ascends back to D=1 in intergalactic voids. The periodic table is a stratigraphic column read at our elevation. Each scientific discipline—chemistry, nuclear physics, particle physics, string theory—is a window at a different depth in this landscape. Galaxies are not flat structures in a fixed-dimension spacetime; they are dimensional weather systems, tornadic funnels where effective dimensionality increases drastically toward the center. M-theory’s eleven dimensions and the Randall–Sundrum warped geometry are not the full picture—they are local descriptions valid at our particular elevation in a much larger dimensional topography. Eight independent results from established mathematical physics and chemistry provide structural support for this framework: dynamical dimensional reduction in quantum gravity (Carlip 2017), the Ehrenfest–Tangherlini stability theorem, the Elliott SU(3) model in nuclear physics, the Efimov effect’s dimension-specificity, Nahm’s theorem fixing D = 11 as the supergravity ceiling, Regge trajectories as the nuclear stratum’s one-line formula, Tanabe–Sugano diagrams as laboratory demonstrations of branching-rule state-counting, and the Hopf fibration’s identity with the Bloch sphere of quantum information.