The Goldbach exceptional set: an exponent 7/12 and conditional refinement

Let $E(X)$ count the even integers between $4$ and $X$ that are not sums of two primes. We combine Pintz’s explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwiłł to prove $E(X){\ll }_{\epsilon }{X}^{7\mathrm{/}12+\epsilon }$ whenever all Dirichlet $L$-functions have a common fixed zero-free half-plane $\text{Rea}s>{\theta }_0$, with ${\theta }_0<1$. The uniform $7\mathrm{/}8$ theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most $\theta$, where $1\mathrm{/}2\le \theta \le 2\mathrm{/}3$, we prove $$E(X){\ll }_{\theta ,\epsilon }{X}^{g(\theta )+\epsilon },g(\theta )=\frac{\theta (5-4\theta )}{2(2-\theta )}\mathrm{.}$$ Thus every fixed $\theta <2\mathrm{/}3$ gives an exponent strictly below $7\mathrm{/}12$. The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.