Good news for evidentialists! Still doesn't help functionalists or causalists, though.

Sometimes there is an analogy made between Newcomb's problem and the choice between a life of faith versus a life of sin given Calvinism (a particular sort of predestinarian Christian theology). The argument goes that one should live a life of sin under Calvinism if and only if one should two-box in Newcomb's problem; similarly, faith corresponds to one-boxing. See the end of Scott's post here, or Arif Ahmed's comments near the start of Evidence, Decision and Causality.

I think this misses an interesting feature of Calvinism, though, which makes this analogy break down when you consider functional decision theory in addition to the standard causal and evidential decision theories. Calvinism holds not only that your choice of how to live has no causal effect on your salvation or damnation, but also that when God predestines you for salvation or damnation, this is done by God's pleasure alone, and not by predicting what you would do (the technical phrase is that it's done ante praevisa merita).

So while the correlational, evidential connection still remains between your choice of what life to lead and your outcome, there's not the same kind of prediction-based functional dependence there is in Newcomb's problem. Thus, functional decision theorists should join the causal decision theorists in living a life of debauchery if Calvinism is true.

So, I might be getting something wrong, but why doesn't Löb's theorem imply that statement? A semi-formal argument, skipping some steps:

$1.\forall p(PA⊢(Prov\left(p\right)\to p\left)\right)\to (PA⊢p))$ (Löb's theorem)

$2.\forall p\left(\neg \right(PA⊢p)\to \neg (PA⊢\left(Prov\right(p)\to p)\left)\right)$ (Contraposition)

$3.\exists p\neg (PA⊢p)$ (e.g., 1 + 1 = 3)

$4.\exists p\neg (PA⊢(Prov\left(p\right)\to p\left)\right)$ (from 2)

$5.\neg \forall p(PA⊢(Prov\left(p\right)\to p\left)\right)$ (by change of quantifiers)