Nonlocal correlations created in networks with multiple independent sources enable surprising phenomena in quantum information and quantum foundations. The presence of independent sources, however, makes the analysis of network nonlocality challenging, and even in the simplest nontrivial scenarios a complete characterization is lacking. In this Letter we study one of the simplest of these scenarios, namely that of distributions invariant under permutations of parties in the minimal triangle network, which features no inputs and binary outcomes. We perform an exhaustive search for triangle-local models, and from it we infer analytic expressions for the boundaries of the set of distributions that admit such models, which we conjecture to be all the tight Bell inequalities for the scenario. Armed with them and with improved outer approximations of the set, we provide insights on the existence of a classical-quantum gap in the triangle network with binary outcomes.

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• Quantum communication
• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism
• Quantum foundations