The von Neumann brain: Observer production through self-replicating constructors arising from Boltzmann fluctuations

A Boltzmann fluctuation could produce a self-replicating von Neumann constructor whose reproductive lineage builds functioning observers. I refer to these constructed observers as von Neumann brains and derive their count distribution under reproductive failure, inherited information and a shared risk of environmental termination. An exact necessary and sufficient criterion determines when the number matching a specified subjective state has the law of independent binomial selection from the completed observer population. A correlated-sibling construction shows that two genealogies can have the same total observer-count distribution, resource stock, terminal depths and mean matching count, yet different probabilities of producing any matching observer. The resulting law identifies how shared ancestry and environmental termination induce correlations in completed observer counts, while its state-resolved extension determines which of those observers enter the chosen class. Reproduction can allow a rarer seed to contribute more observers in expectation while remaining less likely to produce any observer in a given opportunity. Inherited state loss can also change the optimal amount of reproduction. Under an explicit thermal preparation law, the resource stock required for further reproduction enters the formation penalty and gives an exact stopping criterion. A separately funded extension permits a constructor to supply a qualifying observation during copying, counting repeated instants of the continuing machine without treating them as additional individuals. Together with a persistent-factory benchmark for serial production, these results determine how reproductive organisation, transmitted information and resource availability affect the contribution of a fluctuation-initiated lineage. Particular cosmological comparisons require the physical state-matching rule, initial formation laws and observer-sampling convention to be supplied consistently.