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Pi is not a physical constant but a mathematical one; I.e. it can be derived a priori, without a physical universe. A physical constant would be Planck's, or the gravitational constant, or c. Those may well be "arbitrary" for all I know.

c, G, and h need predefined measures of meter, second, and kg to be defined, whether or not they're "arbitrary". There true measure is 1, 1, and 1.

Depends. You can choose your measuring system to make some physical constants 1, but not all. And the choice is not obvious for all constants.

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