KV cache compression is critical for long-context LLM inference, yet existing training-free methods fall into a dilemma: attention-based eviction clusters retained tokens around semantically “hot” regions, creating context holes that degrade long-range reasoning, while diversity-based selection retains directionally broad but semantically irrelevant tokens. We identify the root cause: at prefill time, future decoding queries are unknown, making query-conditioned signals unreliable as a sole selection criterion. The retained KV set must instead provide broad directional coverage of the context—so that arbitrary future queries can find supporting keys—while filtering out tokens that expand the coverage span without contributing to generation. We propose AnchorKV, which realizes this principle through block-wise partitioning with uniform per-block budgets and anchor-driven cosine dissimilarity scoring for linear-time diversity estimation. A compression-rate adaptive hybrid score dynamically reweights diversity and importance. Experiments on LongBench across three model families show AnchorKV consistently outperforms existing training-free methods, with the largest margins at low retention ratios. A query-subsampled variant reduces prefill latency by up to 2.7× with minimal accuracy loss. We provide theoretical guarantees on approximation error and variance reduction.

Large Language Models (LLMs) often suffer from “Reasoning Collapse” on challenging mathematical reasoning tasks, where stochastic sampling produces lexical variations of the same erroneous logic rather than genuine semantic exploration. We observe that failed reasoning traces are often associated with a low-rank bias manifold in the model’s hidden-state geometry, which reduces exploration toward corrective solution directions. To address this, we propose Spectral Orthogonal Exploration (SOE), a geometric inference framework under a “Student Guides Teacher” paradigm. Instead of using a weak auxiliary agent for imitation, SOE uses it as an orthogonal probe to introduce semantically heterogeneous reasoning signals into the teacher’s orthogonal complement of its dominant subspace. This intervention steers the teacher toward more diverse reasoning trajectories and improves exploration beyond standard sampling. Experiments on mathematical benchmarks show that SOE improves average accuracy by 62.4% and average sampling efficiency by 113.7% over baseline methods, suggesting that geometric interventions can be effective for mitigating reasoning collapse in mathematical reasoning. We further provide preliminary evidence that SOE is also effective on logic and code generation benchmarks. Code is available at https://github.com/dayuwang401/spectral-orthogonal-exploration.