The Structure Gap: When More Data Cannot Replace Structured Reasoning

Abstract

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 cor