Paraphrase neighborhoods are fit far better by a quadric than an affine subspace

Bedratyuk generates controlled families of near-paraphrase sentences from three template domains and embeds them with all-mpnet-base-v2 and all-MiniLM-L6-v2 [bedratyuk-2026-controlled-paraphrase-geometry] Fitting implicit affine/quadric/cubic surfaces via total least squares to each local PCA neighborhood shows large systematic residual drops with polynomial degree (mpnet RMSE 0.2296 affine to 0.0165 quadric to 0.0082 cubic) [bedratyuk-2026-controlled-paraphrase-geometry] The paper does not classify the fitted quadric by eigenvalue signature and explicitly states its second-order descriptors are norm-based diagnostics, not full differential-geometric curvature invariants, so the "one-sheeted hyperboloid" typing is not established by the source [bedratyuk-2026-controlled-paraphrase-geometry] A downstream 18-class few-shot ablation finds synthetic points generated on the fitted surface give no consistent accuracy or macro-F1 benefit, a measured null causal result [bedratyuk-2026-controlled-paraphrase-geometry]