There's couple of related question before. See Reference:
However, it didn't exactly answer all my doubts, and I have some contradiction information as well.
There's several major insights given in the previous posts:
user103515: Biot Savart Law was experimental observation law.
user26872: Biot-Savart law is a consequence of Maxwell's equations.
Emilio Pisanty: both Ampère's law and the Biot-Savart law always hold.
a. Ján Lalinský and Self-teaching worker: A magnetic field around the capacitor does not obey Ampère law.
b. Lelouch: The law is not incorrect except in capacitor type cases when the second term in maxwell's eqn. needs to be taken into account.
My Instructor: There was a paper some where claim that Ampere's law was more general... not sure...
Recently, I've got in touch with lagragian density and action principle a lot. And, this might be a bit suspicious, but Ampere's law looked very, or say exactly, like the boundary term added to the action. This made it act like a gauge, or things of sort, which seemed to be more convient to be understood than Biot Savart. It's coincide with what peanut_butter had observed, that Ampere's law was hard to use unless there's some symmetry.
Could you help me to clarify the point 4, and make a comparative argument between Ampere's law and Biot Savart Law?
Especially, what's the current status on the view of the subject, and weather there's any connection between action principle?
If there's does, then wasn't Biot Savart a special case under Lorentz symmetry? while Ampere's law was a gauge theory in general?(Well, it's gauge of conserved current, so I suppose that's an underling statement that they indicate to the same symmetry?)