Same Answers, Different Targets: The Hidden Joint Law of Structured Supervision

Teaching the same answers together can produce different training targets. Two Python workers must return complementary subsets of an input, but the task does not require them to share an implementation style. We construct datasets with identical local programs, marginals, relations, and legal support, yet different frequencies of complete responses: their joint laws. Across three paired Qwen3-8B runs, this change moves held-out same-style generation from 24.3–25.1% to 72.0–72.6%; a common-update-denominator rerun reproduces the contrast at one existing seed. Changing only the weights on the same 1,024 scored validation responses also reverses likelihood-based checkpoint selection on all 32 held-out specifications. A classical minimum-information projection separates task-required from constructor-added dependence. On released programs, upweighting same-source pairings raises a prespecified voter's wrong acceptance by 4.31 points over 75 tasks with unseen inputs. A separate experiment holds the actual first program fixed: a new second attempt shares more errors and supplies less rescue, reducing at-least-one success by about one point. The joint law is thus a learner-visible part of supervision with consequences for both model selection and what one answer adds to another.