Position-independent caching reuses chunk KV caches across positions and requests. Independent encoding, however, omits cross-chunk interactions and can degrade accuracy. In our study, neither PIC training nor rule-based recomputation, alone or combined, consistently recovers full-prefill accuracy. Oracle-guided recomputation on PIC-trained models improves accuracy at the same budget, motivating better selection. We present CacheWeave, which adapts the full backbone to PIC, then freezes it and trains a model-native block selector on the adapted model's oracles. The selector uses compact cached chunk features and question hidden states without extracting backbone attention scores. It selects contiguous token blocks aligned with KV blocks to support efficient recomputation and limit copy-on-write amplification. Its runtime, CacheWeave-Serve, supports private and shared KV caches, keeps selection and recomputation on the GPU, and stages sliding-window KV for hybrid attention models. Across three models and four tasks, CacheWeave outperforms CacheBlend and ProphetKV by 7.2 and 3.5 percentage points on average at a 5% recomputation budget, with a mean relative accuracy loss of 1.6% to full prefill. With 32-token blocks, CacheWeave-Serve cuts single-request TTFT by up to 16.1x over full prefill and 1.7x over ProphetKV, and raises SLO goodput under load by up to 7.8x.

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