Tensor Algebra of the Plexor Field: Mathematical Formalism of the Topological-Plexor Theory

A rigorous mathematical formalism of the Topological-Plexor Theory (TPT) is presented—a gauge field theory in which strong electron correlations and superconductivity are derived from the geometry of discrete spacetime. The physical interpretation and experimental predictions are detailed in the main article [7]. The fundamental object is the 600-cell with H₄ symmetry, whose projection onto a two-dimensional plane generates the fundamental frustration angle θ₀ ≈ 7.356^◦. It is shown that this projection reduces the spinor group SU(2) to an effective abelian group U(1), and the two-valuedness of the projection generates an effective pseudospin. The Plexor field tensor P_µν = ∂_µΓ_ν − ∂_νΓ_µ is defined, where Γ_µ is the geometric connection. It is proven that the noncommutativity of coordinates [x,ˆ yˆ] = ia² provides a built-in ultraviolet regularization, limiting the maximum field strength to a finite value P_max ∼ θ₀/(πa²). The Plexor Lagrangian is constructed, unifying the free field energy, chiral current, and order parameter. It is analytically proven that Cooper pairing of electrons with opposite chiralities is a direct consequence of minimizing the plexor field energy, and the binding energy is estimated as  with exact dimensional correspondence. The formalism is free of phenomenological parameters and is fully derived from the first principles of noncommutative geometry.