Given is the following statement.

Given a Problem $$$P$$$, Let $$$T$$$ be the set of all testcases for this problem,

Let $$$t^* \in T$$$ be a test-case such that for a given program $$$S$$$, if $$$S$$$ passes the verdict of $$$t^*$$$, it's guranteed to pass $$$\forall \; t \in T$$$. ( Hence it's a correct solution.)

In simpler terms, Is it always possible to create a test-case for a problem, such that if a solution passes the verdict for that test-case, then it can claimed that the solution is correct.

Prove the existence/non-existence of $$$t^*$$$.

(i.e Prove that it exists, or prove that it doesn't.)

Note : The Given Problem P could be any problem. i.e Prove/Disprove it for all problems that could exist.