PRISM: A Geometric Risk Bound that Decomposes Drift into Scale, Shape, and Head

Abstract

Comparing post-training LLM variants, such as quantized, LoRA-adapted, and distilled models, requires a diagnostic that identifies how a variant has drifted, not only whether it has degraded. Existing similarity scores such as CKA and SVCCA can flag degradation, but they do not directly link representation drift to risk or mechanism. We propose PRISM, Proxy Risk Inference via Structural Mapping, which exploits the linear output head of LLMs and the empirically near-isometric structure of their backbones to derive a closed-form upper bound on the cross-entropy risk gap between a target model and a post-training variant. The bound is calibrated for variant ranking and decomposes drift into three independently measurable axes: scale mismatch, shape mismatch, and head divergence. Each axis corresponds to a distinct failure mode, including shape distortion under low-bit quantization, scale separability under LoRA forgetting, and head divergence under GGUF k-quantization. As a result, the dominant axis suggests a remediation direction rather than merely raising a degradation flag. Because the shape term is differentiable, the same geometry can also serve as a training-time regularizer against catastrophic forgetting. Across two model families and five benchmarks, PRISM ranks variants with mean Spearman correlations of 0.820 for post-training quantization and 0.831 for LoRA forgetting, and its axis-guided shape regularizer outperforms experience replay in aggregate at mitigating downstream forgetting.