Opposite Direction of Church-Rosser Theorem

Solution 1:

I'm afraid there's no such ideal symmetric confluence property in untyped lambda calculus. Consider $b:=(λyx.x)M_1↠λx.x, c:=(λyx.x)M_2↠λx.x$ where $M_1, M_2$ are distinct terms. Clearly terms $b,c$ here cannot be alternative rewritten terms from a common term since there's no relation between the arbitrary distinct terms $M_1, M_2$.

Another way is to think of Church numerals as representing natural numbers, then obviously any number may have infinitely many ways to express, such as $20=19+1 (Suc)=21-1 (Pred)=...$, thus one cannot find a fixed common ancestor term within any finite sequence.