Your concept of burden of proof is so twisted I am not sure what the point is.

Burden of Proof
A fallacy is when someone makes an argument based on unsound reasoning. Burden of proof is one type of fallacy in which someone makes a claim, but puts the burden of proof onto the other side. For example, a person makes a claim. Another person refutes the claim, and the first person asks them to prove that the claim is not true. In a logical argument, if someone states a claim, it is up to that person to prove the truth of his or her claim.

A negative claim is a colloquialism for an affirmative claim that asserts the non-existence or exclusion of something. The difference with a positive claim is that it takes only a single example to demonstrate such a positive assertion ("there is a chair in this room," requires pointing to a single chair), while the inability to give examples demonstrates that the speaker has not yet found or noticed examples rather than demonstrates that no examples exist (the negative claim that a species is extinct may be disproved by a single surviving example or proven with omniscience). The argument from ignorance is a logical fallacy. There can be multiple claims within a debate. Nevertheless, it has been said whoever makes a claim carries the burden of proof regardless of positive or negative content in the claim.