Predictive-distribution entropy makes a strong selection rule in retrieval-augmented question answering: across five QA benchmarks, keeping the candidate answer that a frozen respondent LLM produces with the lowest answer-token entropy lifts mean answer $F_1$ from 0.4769 to 0.5148 over the retriever's top-ranked passage, with no gold answers. Yet this lowest-entropy rule, which prior entropy-based selectors adopt, fails in a specific and consequential way: a misleading passage makes the respondent confidently wrong, driving its entropy down precisely where the signal looks most trustworthy. We show that the failure comes from the passage the respondent reads -- and the context that passage is read in is an input we can intervene on. We introduce LODESTAR, to our knowledge the first method to score a text intervention by the uncertainty it induces in a third-party frozen respondent, compared across one question's candidates. LODESTAR uses reinforcement learning to train, once and offline, a polarizer -- a short fixed natural-language string inserted into the respondent's prompt and never into its weights; its training labels are built offline from gold answers and two LLM judges, and inference reads neither. Evaluating every competing selector under the same frozen respondent and the same candidate pools on 5,008 questions, LODESTAR attains the highest mean $F_1$ of any inference-ready selector (0.5148 to 0.5339), the highest exact match (0.4136), and the highest GPT-4o judge score of the frozen-respondent configurations judged (0.6435); its three-seed mean wins all 70 method-by-dataset $F_1$ cells against fourteen published configurations while remaining paired-significant against every one. The gain holds both in-domain and out-of-domain, and ablating the polarizer shows it is what makes the respondent read a misleading passage less often (26.0% against 30.3%).
Chain-of-thought (CoT) reasoning improves large language model (LLM) performance while also providing an observable interface to the model's reasoning process. Existing approaches that leverage verbalized CoTs to monitor reasoning correctness, however, largely evaluate the semantic correctness or consistency of individual intermediate steps, rather than how the reasoning process evolves across the trace. As a result, failures distributed across the reasoning trajectory, rather than those localized to a single incorrect step, remain comparatively underexplored. Furthermore, verbalized CoTs need not faithfully reflect the model's internal reasoning, motivating analyses that do not treat individual statements as literal accounts of internal computation. In this work, we therefore ask whether the dynamics of visible CoT can be leveraged to systematically distinguish successful from failed reasoning without assuming such semantic faithfulness. We study a range of LLMs on verifiable Boolean satisfiability tasks with variable complexity, enabling controlled comparisons near each model's capability frontier. Tagging CoT sentences by reasoning function reveals premature verification collapse on SAT problems: incorrect traces enter clause checking earlier, repeat similar operations, and finalize sooner. On UNSAT problems, models presumptuously move towards incorrect SAT conclusions, checking candidate assignments rather than deriving contradictions across constructed cases. Subsequently, a targeted proof-search prompt intervention raises Llama3-70B accuracy from 13.3% to 85%, correcting 84.6% of these errors. These results show that capability failures can manifest as distributed, task-dependent changes in the structure of visible reasoning, and that CoT dynamics agnostic to whether the verbalized trace reflects the model's internal computations can help diagnose and correct failures.