Topological Forcing of Zeros on the Critical Line via Inversion Symmetry

This paper formulates a cohomological obstruction mechanism where inversion symmetry forces a zero on the critical line by making "zero-avoidance" topologically impossible. The author establishes that if a de-symmetrized zeta-type function avoids the origin on the critical line, its image carries a winding class in the first cohomology group H¹(C^×; Z) Z. Because the inversion map ι(z) = z⁻¹ reverses this class (sending n to $-n$), a requirement for inversion self-consistency ([γ] = [ι ∘ γ]) forces the winding number to be zero. Consequently, if a curve possesses a nonzero winding number, it cannot satisfy this symmetry restoration while remaining in the punctured plane; the resulting topological contradiction can only be resolved if the curve crosses the origin, thereby necessitating a critical-line zero.