I would first set up shop with a random number generator based off distant cosmic rays and a sufficiently powerful telescope in a secure, but verifiable by everyone else, enviroment.
Then I would let it pick a random star, any star, out in the cosmos.
Then I would decide all proposals on this planet based on how accurately, precisely, and completely they predict future observations of said star. If this policy is ever deviated from, everyone also equipped with eyes and measuring instruments will also know and can thus pick someone else to listen to.
With all disputable parameters, such as the interval between each observation, also picked randomly by cosmic rays and announced at a predetermined time.
There will be no upper or lower limit to the number of logical steps acceptable for said predictions. Nor will there be any limit to the scope or content of claims. Nor any other restriction that could not be resolved through better observational predictions.
If there is no credible way to link the claim with stellar observations, or if current technology does not render the precision and accuracy feasible, I would delegate the decision to subordinate decision making authorities with a provisional status, until technology and reason and so on advance to a sufficient stage in the future.
The results, with all intermediary information, will also be published and disseminated to everyone, at predetermined intervals, with every disputable factor decided by the random influx of cosmic rays. And again if there is a majority consensus in whatever direction contrary, they can then select who is most trustworthy from amongst themselves.
If the limits for a single star are exhausted then another one will be selected at random and so on.
Therefore, all possible considerations, all possible interest groups, all speculations in general, will be treated as equitably as possible, at least until we run out of stars to look at. And at that future age I’m confident some even greater methods will yet be devised!