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The Supreme Court has held that the Constitution contains a right to habeas corpus in Boumedine v. Bush. The Lincoln thing was never fully litigated and was probably unconstitutional.
The Constitution doesn’t empower the court to interpret the constitution. If the executive chose to ignore the court it would be perfectly legal.
Well that’s an even older decision:
Yes, the supreme Court gave itself that power. To that end the other branches could justifiably choose to not find that to be valid.