lambda-calc program which halts on only one input
Well nevermind this has an easy answer.

Let $r_n = \lambda x_1 ... x_n . \lambda p . p \: x_1 ... x_n$

Note that for $m &lt; n$, $r_n\: A_1 ... A_m$ normalizes for any normal $A_1 ... A_m$.

Then for any $f$, let $n$ be larger than the number of subterms in $f$.

$f \:r_n$ normalizes, so therefore no $f$ exists which only normalizes for a specific input.