Geometric Localization of Electromagnetic Duality Symmetry Based on Noether's Theorem and the Inevitable Derivation of Magnetic Charge Conservation inside ℵ-Trijindaduonion Spacetime

This study extends Noether’s theorem to a 32-dimensional non-alternative and non-associative ℵ-trijindaduonion (Tℵ) Artinian local ring structure, providing a purely algebraic proof for the necessity of electric and magnetic charge polarity configurations in macroscopic spacetime. The simultaneous localization of the first law of thermodynamics—as demanded by time-translation symmetry—and the strong-weak electromagnetic duality symmetry (S-duality) is realized within a spatiotemporal dimensional reduction pyramid. Under these constraints, the asymmetric manifestation of the electron as a negative charge (e−) and the macroscopically stable monopole as a South pole (gS) is demonstrated to be a unique, mathematically inevitable algebraic solution to maintain the global Noetherian invariants of the universe, rather than a stochastic consequence of spontaneous symmetry breaking. It is rigorously proven that the complete parity symmetry violation of space and the negative sign in Faraday’s law of induction constitute a coupled mechanism of symmetries and conservation laws that prevents the spatiotemporal causal matrix from collapsing via nilpotent maximal ideals. This framework completes the mathematical paradigm of grand unification.