We present Jaime's Theorem: the Mean Reciprocal Rank of any Retrieval-Augmented Generation system over a document D is bounded above by Φ (D) = min (g (DDI), (1−NI) ^γ, QS), where DDI is the Content Density Index, NI is the Noise Index, and QS is the Query Space Richness. The bound is measurable before any retrieval occurs and is independent of the retrieval architecture, embedding model, or language model. The bound is tight: for each active term, a corresponding ideal retriever achieves MRR arbitrarily close to Φ. The theorem is grounded in the Semantic Discontinuity Principle: every communication medium carries observable signals of semantic discontinuity — typographic whitespace in text, acoustic silence in audio — whose density determines DDI. The theorem implies a Density-Adaptive Chunking algorithm that measures DDI, NI, QS before indexing and selects the optimal chunking strategy. Validated on a production audio corpus (30 sessions, 100 speakers) and a document corpus (7, 500 production documents, four types): observed MRR is at or below Φ in all cases, within 0. 02–0. 04 of the bound. As Codd's relational model established the conditions for queryable data, Jaime's Theorem establishes the conditions for retrievable documents.
Alejandro Jaime (Sun,) studied this question.