From Energy to Structure: The Nested Atom

Earlier papers derived mass (From Energy to Mass, DOI 10.5281/zenodo.22042755), motion from center of mass (COM) shifts when energy overlaps (From Energy to Motion, DOI 10.5281/zenodo.23441924), and angular momentum from the R/c lag (From Energy to Angular Momentum, DOI 10.5281/zenodo.24125261). Here those two mechanisms alone, COM shifts and R/c lag, build an atom. Three organizational levels share one density function, ρ(r)/ρ₀ = 1 + cos(kπr³), with k = 1, 2, and 3 at correspondingly scaled radii. In a computational engine with nothing else in it, no conservation rules and no stability aids, eleven spheres self-assemble into a nested, breathing structure that holds through a 30,000-step run. Electrons emerge as high-density nodes where k=3 spheres overlap, not as particles: six for four spheres in a tetrahedral arrangement, moving as the spheres breathe, and fading and reforming as the overlaps shift. The k=2 nuclear group presents the opposite character, with low-density lacunae where its spheres' low-density bands coincide. The k=1 quarks sit in the densest junction of the whole group, in elongated triangles that crab-walk, and they can lead the atom's changes of direction. Group sizes are limited by the mechanism itself: density piles up as spheres join, until bursts eject the excess. Testing values showed groups hold at 3 or 4 for k=1, 6 (occasionally 7) for k=2, and 12 (rarely 13) for k=3. Grouping, exclusion, and hierarchical nesting all emerge without fine-tuning of absolute parameter values. The engine and analysis tools are available on request.