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The Guadalupe Hidalgo Treaty did not give the US Gov title to any lands that were in private use.
The only lands that were not in private use were the high ridges. All arable and grazable lands are rightly privately owned.
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I don’t believe that the Federal Government should hold it either.. Give it to the people.. They used to give homesteaders grant or they could give it to the state. I’m just saying that the left thinks since it;s the Federal Governments then they own it and will stop everybody from using it period..