Manifold curvature predicts sparse-autoencoder reconstruction-loss scaling floors

Zaher et al. estimate per-layer intrinsic dimension (TwoNN) and multi-scale curvature (local-PCA tangent-variation and heterogeneity) of activation manifolds in Gemma-2 2B and 9B [zaher-etal-2026-geometric-wall] Regressed against fitted SAE scaling-law parameters from 844 Gemma Scope JumpReLU checkpoints, multi-scale curvature kappa_ms is the single dominant predictor of the scaling-law exponent, more than intrinsic dimension alone [zaher-etal-2026-geometric-wall] The full geometric model reaches leave-one-out R^2=0.869 (9B) and 0.976 (2B), and features fit on one model predict the other's scaling law at cross-model R^2>0.92 [zaher-etal-2026-geometric-wall] The analysis is correlational, with no causal manifold-deformation intervention [zaher-etal-2026-geometric-wall]