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Spectral Geometry of LoRA Adapters Encodes Training Objective and Predicts Harmful Compliance

We study whether low-rank spectral summaries of LoRA weight deltas can identify which fine-tuning objective was applied to a language model, and whether that geometric signal predicts downstream behavioral harm. In a pre-registered experiment on \texttt{Llama-3.2-3B-Instruct}, we manufacture 38 LoRA adapters across four categories: healthy SFT baselines, DPO on inverted harmlessness preferences, DPO on inverted helpfulness preferences, and activation-steering-derived adapters, and extract per-layer spectral features (norms, stable rank, singular-value entropy, effective rank, and singular-vector cosine alignment to a healthy centroid). Within a single training method (DPO), a logistic regression classifier achieves AUC~1.00 on binary drift detection, all six pairwise objective comparisons, and near-perfect ordinal severity ranking ($ρ\geq 0.956$). Principal component analysis on flattened weight deltas reveals that training objective is PC1 (AUC~1.00 for objective separation), orthogonal to training duration on PC2. Query-projection weights detect that drift occurred; value-projection weights identify which objective. Cross-method generalization fails completely: a DPO-trained classifier assigns every steering adapter a lower drift score than every DPO adapter (AUC~0.00). In a behavioral evaluation phase, DPO-inverted-harmlessness adapters show elevated harmful compliance on HEx-PHI prompts (mean ASR 0.266 vs.\ healthy 0.112, $Δ= +0.154$), with near-perfect dose--response ($ρ= 0.986$). The geometry-to-behavior rank correlation is $ρ= 0.72$ across 24 non-steered adapters. These results establish that within a controlled manufacturing regime, LoRA weight-space geometry carries objective identity, intensity ordering, and a coarse link to harmful compliance, and that cross-method monitoring requires per-method calibration.

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