We study passive global optimisation: from twenty noisy samples of an unknown one-dimensional function, with no further queries, estimate where its global minimum lies. We build a benchmark and a self-checking harness and use them for a controlled comparison. The Neural Function Minimizer (NFM), which we introduce, moves a position across the domain for fifteen steps and re-reads the samples from it at every step; a same-size Set Transformer answers in one pass. In a 2x2 design each has one output head or an added mixture head, with the same data and inputs and three seeds each. Within the training families the two are tied on location error, and the mixture loss lowers it by about 0.45 points of the domain in both. The NFM's answers have lower regret, by 2.90 and 3.12 points of the range with one and two heads (significant after correction, and again on a block of fresh cases), mainly because its right-basin answers lie closer to the floor; a Set Transformer looped to the NFM's computation does not reproduce this within the training families, though it has the lower error on unseen functions (exploratory). The NFM also differs from the Set Transformers in encoder, local read, activation, losses and training time. At near-tied minima the form of the answer decides: point answers hedge in about a quarter of exact ties, mixture modes almost never, mixture means in more than 80%. Gaussian-process estimators, with no training, have the lowest regret and are the most accurate on unseen functions. We will release the benchmark, the models and the harness.

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