Reading through Geyer's PDF it seems that there may be some technical issues which a mathematician should investigate in my construct here. For example, it may be possible that for n nonstandard and less than N the construct here doesn't fully constrain the tail behavior. In other words, way out in the tail where the density is infinitesimal you might have non-uniqueness (you could have different infinitesimal distributions with the same nonstandard entropy), for example higher moments might not be fully constrained. so maybe you need an additional criterion to constrain the tail behavior in order to get the standard normal distribution out of this construct.
It doesn't keep me up at night, because I suspect that everywhere that the density is appreciable it will converge to the normal distribution via the usual arguments about calculus of variations, and we're only really interested in areas where there is appreciable density, (if the distribution we pick is basically zero at x=300 but not quite the same kind of zero as exp(-300^2) it will have no effect on actual statistics but it could be an interesting problem for a math grad student or something.

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