The Kaufmann--Clote question on end extensions of models of arithmetic and the weak regularity principle

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in\mathbb{N}$ and any countable model of $\mathrm{B}Σ_{n+2}$, we construct a proper $Σ_{n+2}$-elementary end extension satisfying $\mathrm{B}Σ_{n+1}$, which answers a question by Clote positively. We also give a characterization of countable models of $\mathrm{I}Σ_{n+2}$ in terms of their end extendibility similar to the case of $\mathrm{B}Σ_{n+2}$. Along the proof, we will introduce a new type of regularity principles in arithmetic called the weak regularity principle, which serves as a bridge between the model's end extendibility and the amount of induction or collection it satisfies.