Proving that an equal partition does not exist
The current expectation is that most likely there is no efficient (polynomial-size, polynomially-verifiable) way to do that.

The partition problem is NP-complete.  For a NP-complete problem, we have an efficient way to certify that the answer to the question is YES, but not necessarily any efficient way to certify that the answer is NO.  If you find an efficient way to verify that the answer is NO (i.e., a polynomial-length proof that can be verified in polynomial time), then you have proven that the problem is in co-NP.  If you prove that any NP-complete problem is in co-NP, then you have proven NP = co-NP.  Currently, it is widely conjectured that NP $\ne$ co-NP.  Assuming that conjecture is correct, there is no polynomial-length polynomially-verifiable proof that an equal partition does not exist.