Ignorance Is Shared, Knowledge Is Private: Efficiently Mapping Knowledge Boundaries

Large language models hallucinate most on facts that are weakly represented in their training data, and standard remedies, such as sampling more answers or generating longer reasoning traces, spend compute without first asking whether the model knows the answer at all. We study this question on 791,060 evidence-grounded Arabic/English long-tail facts, yielding 1.58 million behavioral probe points per model across five open LLMs (8B-120B). The long tail is closed to everyone: every model leaves 67-93% of the bank unknown per language, and unknown sets overlap across model families with Jaccard similarity 0.75-0.91, above the overlap their sizes alone would produce in every pair. As a result, one model's unknown map flags another's unknowns at 0.76-0.99 precision without probing the second model. Known sets, in contrast, overlap far less, and scaling within a family leaves the tail largely unknown. This structure makes boundary detection cheap exactly where it matters: a single sampled answer flags unknown facts at 0.91-0.97 precision, a linear probe over hidden states separates known from unknown without generation at AUC 0.89-0.95, and only the rarer known-vs-weak distinction needs $k\ge4$ samples. A tiered budget therefore cuts labeling cost 2-3$\times$ versus uniform sampling. Reasoning, in contrast, buys little knowledge at high cost: models think up to twice as hard on facts they do not know, the effort dial changes how long a model spends before producing a wrong answer rather than whether the answer is wrong, and at least 98.8% of reasoning tokens spent on unknown facts precede a wrong answer in every model. Before spending tokens thinking, a system should first estimate whether it is inside its knowledge boundary.