A Higher Bachmann-Howard Principle

We present a higher well-ordering principle which is equivalent (over Simpson’s set theoretic version of ${\text{ATR}}_0$) to the existence of transitive models of Kripke-Platek set theory, and thus to ${\mathrm{\Pi }}_1^1$-comprehension. This is a partial solution to a conjecture of Montalb'an and Rathjen: partial in the sense that our well-ordering principle is less constructive than demanded in the conjecture.