The Geometric Schwinger Effect III: Toroidal Vacuum Structure of the Bosonic Vacuum and Electron–Positron Pair Emergence

This work is the third part of the “Geometric Schwinger Effect” programme and focuses on the internal structure of the bosonic vacuum. Building on the modular foundations developed in Parts I and II, it models the vacuum as a toroidal medium whose compact phase supports domain-wall (kink–antikink) solitons. An intrinsic bias Feff, constructed from geometric and topological sources, can drive a transition in which a doubly helical ground state splits into a kink–antikink pair, interpreted as an effective electron–positron excitation of the vacuum. The paper analyses the geometry, topology and qualitative stability properties of these toroidal solitons, and relates them to known structures in soliton physics, superfluids, electromagnetism and modular spectral theory.