Spend Where It Matters: Fine-Grained Privacy Allocation for Edge-Cloud Reasoning

Privacy-sensitive multi-step reasoning must exploit the capabilities of cloud large language models (LLMs) without exposing private knowledge stored on trusted edge devices. Local-only execution protects privacy but limits reasoning quality, whereas existing edge-cloud approaches typically make coarse request-level decisions that cannot accommodate the heterogeneous privacy and quality requirements of individual reasoning steps. We formulate this problem as fine-grained privacy allocation over a reasoning graph and introduce Operational Privacy Dependency to characterize step-specific privacy-utility frontiers. We further derive a lower bound on the privacy cost required to achieve a target reasoning distortion and establish an adaptive end-to-end privacy guarantee. Building on this theory, we propose Swim, which constructs privacy-valid actions, analytically estimates their state-dependent quality-privacy operating points, and performs frontier-aware online allocation under a global privacy budget. Across six dataset-model settings, Swim retains 95.7% of Cloud-Only F1, improves F1 by 5.6% over the strongest non-Cloud-Only baseline, and reduces privacy-leakage AUC and MI by 18.3% and 39.1%, respectively. Code is available at https://anonymous.4open.science/r/Swim-4EB3/.