Prompting methods reorganize category-manifold geometry through distinct mechanisms

Kirsanov et al. apply mean-field manifold capacity theory to residual-stream activations of Llama-3.1-8B and Gemma-2-2B under zero-shot, few-shot, and soft-prompt-tuning adaptation [kirsanov-etal-2025-geometry-of-prompting] Despite similar downstream accuracy, the three methods evolve manifold geometry through mechanistically distinct routes: some shrink effective dimension, others change inter-manifold correlation, with effects concentrated in different layer ranges [kirsanov-etal-2025-geometry-of-prompting] This decouples task performance from internal representational organization, showing behaviorally-equivalent adaptation methods rest on geometrically distinct solutions; observational-diagnostic [kirsanov-etal-2025-geometry-of-prompting]