"Sophia is a slave. She gave up her right to disobey when I locked her collar in place." If we accept this as true then there should be no need to punish as there should be no disobeying.
Further you said "Thus she is flogged for disobedience, but also as a test of her obedience " and I wonder why is it necessary to test her obedience if you in fact know that you own her. I realize this is a smart ass comment and I am truly asking it to illustrate my point not to piss you off but do you also periodically check your auto registration to make sure your car is yours?
Which brings me to the crux of the matter for me which is "Punishment is therefore necessary because only the threat of punishment constantly reminds the slave that they are in fact a slave" and my continuing dilemma with wrapping my head around this thought process. If she signed a contract, and wears a collar and is happily doing so, she agreed to it and is eagerly accepting of it then why the need of the big stick?