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Proof requires axioms, and the universe hints towards what they are but isn’t explicit.

Per thermododynamic axioms, incompleteness is implied by the arrow of time and information atrophy. This would imply an unknowable beginning.

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You got the direction backwards. Increase in entropy means that the farther back you go in time, the less information the universe has in it in some sense. The initial state of the universe would therefore be very simple, having low entropy. If the state were simple enough, then in principle, we could figure it out in exact detail. The thing that thermodynamics says is unknowable is the distant future of the universe. We can predict that it will be dark and cold and empty and sparsely filled with ever-more red-shifted radiation, but it's impossible, even in principle, to know the details of exactly where each individual particle in going to end up.

Maybe we are thinking of S=k ln(0 or 1) then? Undefined or zero is not a great place to start:)

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