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Nisan comments on Formulas of arithmetic that behave like decision agents - Less Wrong
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I only looked at this for a bit so I could be totally mistaken, but I'll look at it closely soon, it's a nice write up!
My thoughts:
A change of variables/values in your proof of proposition 3 definitely doesn't yield conjecture 4? At first glance it looks like you could just change the variables and flip the indicies for the projections (use pi1 instead of pi0) and in the functions A[U,i]. If you look at the U() defined for conjecture 4, it's exactly the one in proposition 3 with the indices i flipped and C and D flipped, so it's surprising to me if this doesn't work or if there isn't some other minor transformation of the the first proof that yields a proof of conjecture 4.
One can try to take the proof of Proposition 3 and switch C and D around, but one quickly runs into difficulty: The second line of that proof contains
"cooperating provably results in a better payoff than defecting". This is enough to convince the agent to cooperate. But if you switch C and D, you end up with a proposition that means "cooperating provably results in a worse payoff than defecting". This is not enough to convince the agent to defect. The agent will defect if there is no proof that cooperating results in a better payoff than defecting. So at least one would have to say a bit more here in the proof.