The Zero-Sum Invariant on T⁴: The Topological Origin of Chiral Degeneracy

Abstract This paper formally establishes the topological constraints governing the commensurable 4-torus (T⁴) by evaluating the Euler characteristic of the fundamental 1-dimensional manifold (S¹) across discrete, homological, and differential frameworks. Utilizing CW complex decomposition, singular homology (Betti numbers), and the Poincaré-Hopf theorem, the topological invariant χ(S¹) = 0 is rigorously derived. By extending this invariant through the multiplicative property of the Cartesian product space S¹ × S¹ × S¹ × S¹, the global geometric nullity of the undeformed T⁴ base space is demonstrated as identically zero (χ(T⁴) = 0). The derivation establishes that this absolute topological nullity inherently mandates the exact, symmetric balance of chiral zero-modes, thereby formally enforcing the Nielsen-Ninomiya obstruction on the undeformed manifold.