User blog comment:Upquark11111/An Explanation of Loader's Number/@comment-11227630-20171210002438

I doubt your iteration method in "construction of $$f_\omega$$" section is not strong enough. I have tried this kind of things in untyped λ-calculus, and get
 * The λ-expression $$\lambda x.x(\lambda x_n.\lambda x_{n-1}.\cdots\lambda x_2.\lambda x_1.(x_1x_nx_{n-1}\cdots x_2x_1)\underbrace{xx\cdots x}_n)$$ is a function on Church numbers comparable to $$f_{\omega\uparrow\uparrow(n-1)}(x)$$.

Then I can't get any further.

In λPω, similar method also lead to nested λ-abstraction when we try to access $$f_{\varepsilon_0}$$. To go beyond $$f_{\varepsilon_0}$$ (like what David Moews did), we may need different methods.