Constraint transforms and default priors

Modeling
,
5

However, I think even my above “update” is incorrect. By that reasoning, the following should be equivalent:

parameters {
  real<lower=0> C;
}

transformed parameters {
  real lnC = log(C);
}

model {
  target += lnC; // Is this already included implicitly?
}

and

parameters {
  real lnC;
}

transformed parameters {
  real C = exp(C);
}

model {
  target += lnC; // needed to convert to uniform in C>0.
}

However, this does not appear to be the case; the actual models (which include data and other parameters) agree when I remove the Jacobian addition for the model with parameter real<lower=0> C;.

So then it appears that the default prior is uniform in the real parameter, even with a lower limit — the Jacobian is added in by default no matter what (as @Funko_Unko said in their original response).

Can an expert comment (and/or point to appropriate documentation)?

What am I missing?