Correct Labels Can Destroy Uniform Learnability: A Sharp Honest-Insertion Threshold

Can adding correctly labeled examples make a learnable problem unlearnable? For each $\kappa \ge 3$, we construct one fixed class of total binary functions and one fixed known marginal. A learning rule achieves uniformly vanishing risk against all honest insertion channels with fewer than $n$ additions to $n$ IID observations, whereas $n$ additions can force minimax risk toward $1/2$. The adversary retains every original example, uses only target-correct labels, and acts before the test query, but hides which observations were IID. The proof swaps the roles of clean and inserted samples between two target mixtures. Their observed data have a common probability mass approaching one, but their predictions disagree on most queries. An exact transport characterization gives the largest overlap that honest additions can create between two target mixtures' observed data at each budget. Our example is a countable class of binary-digit formulas under the uniform law on $[0,1]$, not a geometrically regular low-dimensional class. The failure is uniform over targets; each fixed target remains consistently learnable. For this class, marking one original IID observation, chosen independently of its value, restores uniformly vanishing risk at every budget without adding data. Thus correct labels and reliable sampling information play different roles in uniform learning.