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A constant is not a variable, and it isn't actually stored anywhere. As it isn't stored, it's not an instance member, and it's not static.
The constant is just a name for a value. When the code is compiled the value is inserted where the constant is used. (This has implications if you use a constant declared in a different assembly. Changing the declared value of the constant doesn't change the value used until you recompile the code that uses the constant.)
So, a constant declared locally works exactly as a constant declared anywhere else, it's only the scope that differs.