Horizontal Injectivity of the Dirichlet Eta Function and the Riemann Hypothesis.

The manuscript asserts that the Dirichlet eta function eta(s) is horizontally injective within the open critical strip 0 < Re(s) < 1. It demonstrates that for a given height b != 0, the condition eta(a_1+bi) = eta(a_2+bi) requires a_1 = a_2.  

Because the zeros of the Riemann zeta function zeta(s) mirror the zeros of eta(s) in the open strip, a zero off the critical line (Re(s_0) != 1/2) would imply eta(s_0) = 0. Applying the Riemann functional equation and the Schwarz reflection principle, this would force the reflection point s^* to also evaluate to zero, yielding eta(s_0) = eta(s^*). Proving horizontal injectivity eliminates this possibility, restricting all non-trivial zeros to Re(s) = 1/2.