The Geometry of the Hollow Answer: Representational Divergence and the Crisis of Humanistic Knowing

The large language model did not create a crisis in the humanities. It revealed one already present. When a competent answer can be produced from a hollow prompt, what is exposed is not the tool's capability but the degree to which the academy had already been certifying retrieval as learning. The central formal category is the unknown known: certainty resting on surface expression with no verified knowledge network beneath it. Drawing on representational divergence as a heuristic formal apparatus, the paper formalizes humanistic knowledge as a directed graph GK whose edges require explicit validation, and the asserted knowledge state as a surface subgraph whose divergence from GK is measurable via Hausdorff distance. This structural variance, the Blaeu Rent, compounds across the credentialing system, the civil service, corporate governance, and intergenerational knowledge transmission. A distributional analysis demonstrates that the gap falls hardest on communities of color, first-generation learners, and economically marginalized populations, whose differential exposure to high-stakes AI-mediated decisions amplifies the harm from biased outputs. The paper develops a public resource allocation model, a minimax decision framework for institutional reform, and a five-component social cost pricing function. The liberal arts corrective is framed not as a moral argument but as a strict optimization strategy: in a world where surface fluency has a marginal cost of zero, the capacity to verify GK is the only high-value signal the labor market will price. GPU-accelerated computation makes the cost of the knowledge gap formally expressible at population scale for the first time.