User blog comment:Upquark11111/An Explanation of Loader's Number/@comment-11227630-20171209072636/@comment-26454151-20171210023112

Remember, however, that there are only two ways that a type can enter the context. They are through "start" and "weakening." In both cases, it is required that the type must be judged to be $$\ast$$ or $$\square$$ before entering the context. By not specifying the condition $$\Gamma \vdash A:s_1$$, all that does is imply that $$s_1$$ could be either $$\ast$$ or $$\square$$. The fact that the type is in the context in the first place means that we have already proved that it is of type $$\ast$$ or $$\square$$. Hence, by omitting that condition, that is how Loader makes this to be λPω instead of λP.