Dear colleague! Look back on the atomic physics and the sum rules known for atoms. You will answer your question by yourself!
Any strict (i.e. without additional ad'hoc assumptions) sum rule plays a role of a "conservation law". If it is violated, then some of assumptions (or all of them) are wrong. If it is fulfilled - then either you found a new "conservation law" (provided you did not introduce ad'hoc assumptions), or some of them (assumptions) may be correct. An excellent example in particle physics is related with so-called conception of "duality" (see papers in '70-'90 issues of Phys. Reports, for example).