We establish the exact reconstruction thresholds for a class of broadcasting models on hypertrees over a symmetric binary alphabet. As a consequence, we show that the condensation threshold coincides with the Kesten-Stigum threshold for random NAE-SAT and random hypergraph bicoloring with arity at most four at any temperature, confirming a prediction of Ricci-Tersenghi et al. '19. We also determine the exact weak recovery threshold for the two-community hypergraph stochastic block model in certain parameter regimes. The proof combines an improved robust non-reconstruction analysis with a computer-assisted rigorous population dynamics algorithm. It relies heavily on information-theoretic tools, in particular the theory of binary memoryless symmetric channels and channel comparison methods.

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