SP3 Hydrodynamic Nesting of Atomic Space: A First Calculation of Orbital Geometry, Boundary Conditions, and the Limits of a Simple Wave Model

SP3 proposes that a participating physical space medium extends through both inner atomic space andouter space, and that a nucleus organizes local configurations in which electron-associated states canoccur. This paper reports a first, deliberately minimal calculation testing whether hydrodynamic wavegeometry and boundary conditions can illuminate the familiar s, p, d, and f orbital hierarchy. Separatinga scalar wave equation in spherical coordinates gives angular modes indexed by ℓ = 0, 1, 2, 3 anddegeneracies 2ℓ + 1 = 1, 3, 5, 7, matching the spatial-orbital counts. A spherical cavity also generatesdiscrete radial modes, but its Bessel-root frequencies do not reproduce the hydrogen 1/n² bindingenergy spectrum. Introducing the Coulomb potential reproduces the spectrum and the restriction ℓ ≤ n −1, but only by importing established quantum mechanics. Vortex-street pattern selection, vortex-ringpersistence, and spherical standing-wave organization offer complementary physical analogies; noneindependently derives electron charge, spin, Pauli exclusion, or stable atomic binding. The result is abounded advance: a geometrical correspondence, a documented failed energy model, and a sharplydefined next calculation seeking a nuclear-medium coupling that yields an effective inverse-distancebinding interaction without assuming it