Large language models (LLMs) seem to offer an easy path to interpretability: just ask them to explain their answers. Yet the features driving an answer often differ from those emphasized in its explanation, meaning post-hoc rationales can misrepresent what actually shaped the model’s output. We quantify this gap by comparing the feature-importance distributions of answers and their explanations. Prior analyses reveal such discrepancies, but large-scale study has been limited by the high computational cost of attribution methods. To address this, we introduce the Post-hoc Self-Consistency Bank (PSCB), a large-scale benchmark linking model decisions with diverse explanations and attribution vectors across datasets, methods, and model families. Using PSCB, we find that Spearman rank correlation provides a more reliable signal of alignment than cosine similarity. Building on this insight, we apply Direct Preference Optimization (DPO) to attribution-based preference data, improving alignment without degrading task accuracy, and show that standard supervised fine-tuning on the same data fails to achieve comparable gains. These improvements generalize robustly across domains, paving the way toward scalable and faithful alignment between LLM decisions and their natural language explanations.

Wider beam search should improve LLM reasoning, but when should you stop widening? Prior work on beam width selection has focused on inference efficiency (CITATION), without analyzing whether wider search can hurt output quality. We study each search step as scoring n candidates and keeping the top b of them. Best-of-n keeps a single winner (b=1); beam search keeps several. Because the scorer is noisy, a worse candidate can outrank a better one, and that risk grows as more candidates are compared. How far you can usefully widen therefore depends on the scorer: how large the true quality gap is relative to the noise. Keeping several hypotheses is easier than naming one winner, so beam search is more forgiving than best-of-n, but that per-step analysis does not by itself tell you the best width for the full task. We compare perplexity scoring and a process reward model (PRM) across three 7–8B models and all ten MR-BEN subjects (5,737 questions). Perplexity yields no accuracy gain at any tested width; the PRM gains up to 8.9 points and continues improving through beam width 4 (n=16 candidates per step). Across 444K search steps, the spacing of ranked scores agrees with a simple Gaussian ranking model. The same model and algorithm therefore support very different useful widths, depending on how reliable the scorer is.