Uniqueness of a possible exceptional real Dirichlet zero ($\#\{\beta \in [1-c/\log q,1]:L(\beta ,\chi )=0\}\le 1$)

For primitive real nonprincipal Dirichlet characters, there is an absolute $c>0$ with at most one zero in this interval, counted with multiplicity. Positivity of $-{\zeta }^{\prime }/\zeta -L^{\prime }/L$ and the local partial fractions give $0\le 1/(\sigma -1)+C\log q-\sum (\sigma -\beta {)}^{-1}$. Testing $\sigma =1+a/\log q$ with small fixed $a$ rules out two nearby real zeros. Any possible zero is consequently simple; existence is not asserted.