Euclid in the Retrieval Engine: An Anatomical Review of an AI Assessment Thread and a Fresh Appraisal of Euclidean Geometry in Visual Retrieval-Augmented Generation

This paper has two parts. The first is an anatomical review of a two-turn conversation with a general-purpose AI assistant. In the first turn the assistant assessed a short interdisciplinary note claiming that Euclid's Elements, Book II, Proposition 5 and the reflection coefficient of the Smith chart share one partition ratio; in the second it was asked to find the present author's own papers on Euclidean geometry. The review dissects each turn for accuracy, evidential basis, calibration and completeness. It confirms the mathematical core of the first assessment with a short derivation, showing that the Euclidean identity, divided through by the square on the half-sum, is exactly the power-balance relation 1 − |Γ|² + |Γ|² = 1 for a real load. It also finds that the second turn under-retrieved: at least one further record with Euclidean framing in its title, and a substantial body of related geometric work, were missed. That retrieval failure motivates the second part, a fresh assessment of Euclidean geometry in visual retrieval-augmented generation (visual RAG), where page images and photographs are embedded as vectors and retrieved by similarity. The paper shows that the same identity, in the form of the polarisation identity, underlies the equivalence of cosine and Euclidean ranking after normalisation. It then examines what Euclidean geometry does well and where it fails: distance concentration in high dimensions, anisotropy and hubness, the modality gap between image and text embeddings, the non-metric geometry of ColPali-style multi-vector late interaction, compression, and hierarchical content that fits hyperbolic space better. It closes with failure modes, evaluation practice and a calibrated verdict, supported by original figures, a computed experiment and tables.