User blog comment:B1mb0w/J Function Sandpit J 4/@comment-1605058-20151014132917

Your description of the function in the last section (which I believe was supposed to define $$J_4$$ and not $$J_3$$ as it does now) doesn't make the function well-defined. Apart from the issue with fundamental sequences Deedlit addressed above, there is an issue with that there are many ways to express a number. For example, when solving for $$J_4(3)$$, we must look at $$J_4(2)$$. Now:

$$J_4(2)=f_1(f_1(2))$$, so if we replace $$2$$ with $$J_4(2)$$, we will get $$J_4(3)=f_1(f_1(J_4(2)))=f_1^4(2)$$.

At the same time, $$J_4(2)=f_\omega(2)$$, so $$J_4(3)=f_\omega(J_4(2))>f_1^4(2)$$.

Even worse, we could write $$J_4(2)=f_1(f_1(1+1^2))$$ and then replacing $$2$$ won't do anything, leading to $$J_4(3)=f_1(f_1(2))$$.