Can a small probabilistic neural network represent every distribution on four yes-or-no variables? Four binary variables have sixteen possible patterns. A restricted Boltzmann machine with four visible units and three hidden units has nineteen adjustable parameters. That sounds plentiful, but counting parameters does not settle what the model can represent. This unrefereed candidate gives a negative answer. Choose eight particular patterns and assign equal probability to them, with zero probability elsewhere. The proof shows that the model cannot get arbitrarily close to this distribution, even when its parameters grow without bound. The key is a leakage inequality. If every chosen pattern receives appreciable probability, some probability must remain outside the chosen set. Forty-two exact integer identities, expanded through cube symmetries, cover every relevant hidden-state selection. A short written argument turns that finite certificate into a statement about the entire model closure. The paper also gives a strictly positive excluded distribution and explicit lower bounds on approximation error. Those bounds are tiny and not claimed to be sharp; they do not measure a practical training penalty. In contrast, every distribution on a specified nine-pattern parity-plus-one support is attainable. Geometry matters, not just the number of patterns. At least four hidden units are therefore necessary for universal approximation, but this work does not show that four are sufficient. This is an Evidence Press unrefereed candidate release dated sixth September twenty twenty-six. The full paper, exact evidence and limitations are linked. Producer replay and internal AI review are not independent mathematical validation. This AI-generated OpenAI synthetic voice is a communication aid, not additional evidence.