Under what demented theory of law did you pull that crap out of your butt?
That "demented theory" comes from discussions about FRE 408 (which PA seems to follow). If there is no claim, or no dispute about the claim or the amount due, or if the statement of fault occurs outside the context of compromise negotiations, the furnishing of valuable consideration will be admissible to prove liability.
OP:
"I went outside and was talking with my wife and a couple friends that came over to help us search. The vet comes out with a check for 350, the amount of the visit the day before, hands it to me and says THERE we're even. I laughed again, took the check and told her we're NOT EVEN CLOSE to even. She then told me to get off of her property that I'm not a customer and if I dont leave ASAP she will call the police on me. I left......"
It doesn't sound like the check was given during compromise negotiations, or in response to any claim made by the OP. According to his original post, he was still in the process of searching for his dog when the vet came out, handed him a check, and claimed "we're even" (which also may be used against her in court, by the way).
See
Big O Tire Dealers, Inc. v. Goodyear Tire & Rubber Co., 561 F.2d 1365 ("The discussions had not crystallized to the point of threatened litigation") if you're still confused.
EDIT: After writing the above, I started to think: "Maybe he's not questioning whether the check would be admissible because of Rule 408. Instead, maybe he's clueless as to the logical inferences that can be made, and of the check's relevancy." Given your vast misunderstandings of the law in past threads (as well as in this one), I now suspect that is the case.
Here's something that should clear things up for you (I dug this up out of an evidence casebook): "One plausible inference to draw from offers of compromise is that those making the offer...believe they were at fault in the incident giving rise to a claim against them. If they have this belief, one can then make the further inference that they were in fact at fault. In short, one way in which compromising or offering to compromise a claim is relevant is as a tacit admission of fault or liability."