Galaxies and Stellar Systems as Vortex Opercula: Gastropod Shell Geometry, Dark Matter as Mother-Wave Body, and the Universal Apex Relation in the PEW Framework (Article 7)

The Primordial Energy Wave (PEW) framework models all physical entities as topological vortex configurations of a single complex scalar field Ψ = ÷eiθ. Previous articles established the gastropod shell topology — a flared upper gate and a quasi-closed apex — as the fundamental geometry of all PEW vortices Art. 1–6. The present article extends this geometry to galactic and stellar scales with a key reversal of perspective: galaxies and stellar systems are not the vortices themselves, but their opercula — the thin baryonic condensation membranes sealing the apertures of much larger Mother-Wave fluctuations whose three-dimensional bodies constitute dark matter. From the logarithmic spiral geometry of the shell, we derive a universal, parameter-free relation: the distance from the operculum centre to the apex is Dapex = Roperc / ln(φ) ≈ 2.078 × Roperc Status A — analytically derived at a fixed elevation angle α = arctan(1/ln φ) ≈ 64.3° above the disc plane. This ratio 1/ln(φ) ≈ 2.0781 is the universal constant of the Mother Wave, derived solely from the φ-scaling law of the PEW Lagrangian with no free parameters. Applied to the Milky Way (Roperc = 25 kpc): Dapex ≈ 52 kpc, consistent with the observed BHB stellar density peak at 50–60 kpc and the orbital radii of the Magellanic Clouds. Applied to the solar system (Roperc = 120 AU): Dapex ≈ 249 AU, consistent with the Voyager north–south termination shock asymmetry (94 AU vs. 84 AU). Applied to the Earth–Moon system (Roperc = 384 400 km): Dapex ≈ 799 000 km, with Nturns ≈ 4.3 turns. The number of turns grows with structural complexity: ~4.3 (Earth–Moon), ~10.5 (solar system), ~51 (Milky Way). The dark matter halo is identified with the shell body — the distributed amplitude field Ã2(r) of the Mother-Wave fluctuation, invisible because below the baryonic condensation threshold. The exterior face of each operculum opens onto pure Mother-Wave vacuum: the Local Void for the Milky Way (galactic south pole), the Local Bubble for the solar system, and the interplanetary solar wind for the Earth–Moon system — all confirmed observationally. Galactic disc undulations (warp) are interpreted as a superposition of free Mother-Wave oscillations (~60%) driving the slow coherent precession and perturbations of the galactic fluctuation itself (~40%) producing rapid asymmetric features detected by Gaia. The stability analysis of Article 4 remains fully valid as the 2-D in-plane cross-section; the present article provides its 3-D out-of-plane complement. All results are classified by epistemic status: A analytically derived, B formally derived with stated assumptions, D structural correspondence open to derivation. No result beyond Roperc requires a free parameter.MSC: 83F05, 85A15, 83C75 · PACS: 98.20.-d, 98.35.Gi, 95.35.+d