Are there any derivations of the born rule from MWI that are not circular and are mathematically consistent? I ask this because Florin Moldoveanu insists that such a derivation from MWI doesn't exist. So I'm wondering if anyone can defend the MWI
The usual ab initio derivations suffer from circular reasoning. Sometimes game theoretical arguments are invoked to argue why an observer should bet on the Born rule, but these arguments all make hidden assumptions. There is, however, a derivation of the Born rule from the weaker statement that a system in an eigenstate of an observable will be found in that eigenstate upon measuring that observable with certainty. This can be derived by considering an ensemble of $N$ copies of a system, each in the same quantum state. The ensemble is a system in its own right, we can consider measuring the relative frequencies of obtaining the different possible outcomes and comparing that to Born's rule.
If the Born rule probability of finding result $\lambda_i$ is given by $p_i$ but in the ensemble of identical systems we find a fraction of $q_i$, then the quantity:
$$X = \sum_i \left(p_i - q_i\right)^2$$
corresponds to an observable for the ensemble of $N$ identical systems prepared in the same state. One can show that in the limit of $N\to\infty$, all the possible states of the ensemble are in the null-space of the observable, so Born's rule is always satisfied in the limit of an infinite number of measurements.