The Structure Gap: When More Data Cannot Replace Structured Reasoning

A persistent belief in modern AI is that sufficiently large learners trained on sufficiently much data can make explicit domain knowledge optional. This paper examines a limitation of that belief. We define structure-encoded problems: tasks whose input-output map is determined, or sharply constrained, by known structure such as a solver, symmetry group, graph, physical law, measurement operator, grammar, combinatorial search space, or optimization objective. For such tasks, the central question is not expressivity--a dense neural network may contain the desired map--but finite-data identification, extrapolation, and certification. We introduce the structure gap: the finite-resource gap between a method that receives the governing structure and a learner that receives only examples. The strongest useful form of the claim is asymptotic rather than absolute: structure-agnostic learning may approach a structured solver as coverage grows, but every finite dataset can leave residual uncertainty that the structure resolves by construction. We prove this exactly for a Gaussian inverse problem: the known-operator Wiener solver attains Bayes risk, while the dense empirical learner in the correct linear class has persistent expected excess risk. We also prove a cell-coverage lower bound and an unobserved-orbit non-identifiability result. Controlled experiments in known-operator inversion, sparse recovery, and graph reachability with size extrapolation instantiate the mechanisms. When validated structure is available, it can constrain the hypothesis space before learning begins, leaving learning to address components the structure does not specify.