The claims didn't exist because even USA acted as if they were not valid, and even if the southern states had claims from sea to sea before the Treaty of Paris, they renounced to them in that Treaty, which fixed the borders And that all disputes which might arise in future on the subject of the boundaries of the said United States may be prevented, it is hereby agreed and declared, that the following are and shall be their boundaries, viz.; from the northwest angle of Nova Scotia, viz., that nagle which is formed by a line drawn due north from the source of St. Croix River to the highlands...
After that, the states would limit their claims to the Mississipi River, and that was the land they surrendered to the Federal Governement.
The treaty was a scam, cleverly negotiated by Ben Franklin to screw the Europeans, which, of course, it ultimately did.
Mexico didn't sign the treaty.