Nuclear Magic Numbers from Wave Geometry: A Zero-Parameter Derivation from Z₃ Möbius Topology

The nuclear magic numbers — 2, 8, 20, 28, 50, 82, and 126 — mark nucleon counts of exceptional nuclear stability. Since 1949 they have been explained by the nuclear shell model, which requires adding a spin-orbit coupling term with fitted constants to a harmonic oscillator potential. The sign and strength of that coupling are chosen to reproduce the known magic numbers, not derived from a deeper principle. This paper presents a geometric derivation that reproduces all seven magic numbers with zero free parameters. The single input is a biconical horn with Z₃ Möbius topology — the resonant cavity formed by a standing wave on a one-third-twist strip in Dimensional Oscillation Theory. The 30° cone angle is derived directly from the topology: the Z₃ strip accumulates 120° of twist per complete circuit; the equator-to-pole path is one quarter of a cycle, giving 30° of polar twist; self-consistency requires the cone face to match this angle. The derivation proceeds in two stages corresponding to two physically distinct kinds of magic number. For the first three (2, 8, 20): standard Bessel-function waveguide theory shows that exactly three angular modes — the point, triangle, and pentagon — propagate freely through the horn tips, with all higher modes trapped. The threshold is set by the 60° full opening angle of the 30° cone. The first three magic numbers follow from simple tetrahedral counting with no correction. For the remaining four (28, 50, 82, 126): the Möbius chirality of the horn determines which half of each trapped mode has lower energy, replacing the fitted spin-orbit constant with a topological property. The co-rotating circular polarization penetrates deeper into the narrowing horn and sits lower in the attractive nuclear potential. Parity exclusion forces intruder levels into lower shells. The closed formula Magic(n) = n(n² + 5)/3 reproduces all four values exactly and predicts 184 as the next magic number. A connection to Gabriel's Horn shows that the first three odd-polygon shells (triangle, pentagon, heptagon) fill exactly 15/16 of the total horn volume, with remaining capacity decreasing as 1/(N+1)² per additional shell — an exact inverse-square law with no free parameters. The derivation also gives a geometric account of nuclear deformation: spherical nuclei at magic numbers reflect equal chirality loading at both horn tips; quadrupole deformation between magic numbers reflects incomplete chirality balance; and pear-shaped nuclei just above magic numbers reflect intruder filling across a parity boundary.