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The NIST/SEMATECH e-Handbook of Statistical Methods explains score-based binomial proportion confidence intervals and gives the Wilson score interval endpoints.

Official reference: www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm

Suppose $\sqrt{n}(\hat{\theta }-\theta )\stackrel{d}{\to }N(0,v(\theta ))$ with $v(\theta )>0$, and let $\hat{v}$ consistently estimate $v(\theta )$. A two-sided Wald interval isThe Slutsky theorem makes the standardized error asymptotically standard normal, giving coverage tending to $1-\alpha$ at each regular parameter value. The estimated standard error is $\sqrt{\hat{v}/n}$, not $\sqrt{\hat{v}}$.

This is an asymptotic construction. For a binomial proportion, the usual substitution gives $\hat{p}\pm z_{1-\alpha /2}\sqrt{\hat{p}(1-\hat{p})/n}$, which can extend beyond $[0,1]$ and collapses when $\hat{p}$ is zero or one. Those features explain its poor finite-sample behaviour near boundaries; binomial proportion confidence intervals use other constructions when appropriate.