Geometric Comparison of the 0-Sphere and 1-Sphere (TikZ Source: zenodo_18356895)

Summary: This figure (Fig. 3 in the main paper) provides a topological contrast between the 0-sphere (S0) and the 1-sphere (S1) to justify the necessity of an external embedding domain for internal transport.Key Visual Components (TikZ Logic):(a) S0: 0-Sphere: Visualized as two disconnected points (blue/olive) at x=±a. The dashed circle indicates a geometric constraint, but S0 itself lacks a continuous path between the points.(b) S1: 1-Sphere: A solid circle representing a continuous manifold where intrinsic angular motion is possible.Theoretical Implication: The contrast emphasizes that for the 0-sphere model, any energy transport between kernels must occur via an external medium (Region D), as the sphere itself is topologically disconnected.Scientific Context: Essential for explaining the transition from discrete localization to the requirement of radiative energy transport in the 0-sphere electron model.