A prime plus a square: A conditional 11/20 upper bound

Let $E_{\mathrm{\square }}(N)$ count the positive integers at most $N$ that cannot be expressed as a prime plus a nonnegative integer square. Assuming that every nontrivial zero of every primitive Dirichlet $L$-function has real part at most $7\mathrm{/}8$, we prove $E_{\mathrm{\square }}(N){\ll }_{\text{eps}}{N}^{11\mathrm{/}20+\text{eps}}$ for every $\text{eps}>0$. The proof combines a conductor-weighted mean-square estimate with a uniform estimate for certain families having a prescribed quadratic component. For these families, the large sieve depends on the varying part of the conductor, while an approximate functional equation controls the fourth moment using the full conductor. A zero-detection argument then absorbs all bands of zeros with real part at least $129\mathrm{/}167$. The remaining bands and the minor arcs give the exponent $11\mathrm{/}20$. The proof includes even denominators, completion and positivity of the singular series, and removal of proper prime powers.