User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-11227630-20160601045613

In definition of $$C_{n+1}(\alpha,\beta)$$, one can use $$\chi(\gamma,\delta)$$ even if $$\gamma\ge\alpha$$, whose definition rely on $$\chi(\alpha,\beta)$$, and it goes wrong for "loop" definition.

Maybe $$\chi(\eta,\delta)$$ where $$\eta<\alpha$$ is better, which is used in this notation. However, in that notation, it seems that $$\chi_\alpha(\beta)=\theta(\alpha,\beta)$$.