Functional Symmetry and Cohomological Obstructions to Critical-Line Zeros

This paper does not attempt to prove the Riemann Hypothesis. Instead, it explains how zeros on the critical line may be forced under specific structural conditions for zeta-type functions. Without symmetry, the restriction of a function to the critical line is generally complex-valued. A zero on the line then requires both the real and imaginary parts to vanish at the same point. Since there is only one real variable, this is generally a difficult condition. With functional symmetry, the situation changes. The functional equation relates the value of the function on the critical line to its complex conjugate. This imposes a square-root condition on the phase of the function. If one assumes that there are no zeros on the critical line, this phase must be globally well-defined. On a closed phase path, one can measure the winding number or degree of this phase. If the square-root condition imposed by the functional equation conflicts with the parity of this degree, then the assumption that there are no zeros becomes impossible. In that case, a zero on the critical line is forced. The central idea is that symmetry does not automatically prove that all zeros lie on the critical line. However, functional symmetry can impose a topological constraint on the phase. When this constraint conflicts with a cohomological degree obstruction, the function must cross zero on the critical line. Hecke structure provides arithmetic phase data, the Atkin–Lehner involution supplies the functional-equation symmetry, and Ramanujan-type bounds support local phase stability. The essential mechanism is the square-root phase condition produced by the functional equation and the resulting cohomological degree obstruction.