Title:It's the Problem, Not the Path: Budget and Difficulty Confounds in LLM Reasoning Trajectories

Abstract:Reasoning traces of large language models are widely read as containing "breakthrough" moments and early-legible fates. Both readings rest on measurements missing a counterfactual control at the level of the claim; we supply both controls. First, a restart-controlled truncation probe separates when a solution fits the continuation budget from when a prefix carries value that fresh computation cannot buy, comparing per-anchor continuation solve rates against from-scratch restart curves at matched total generated-token budget. Applied to 178 problem-model cells (89 MATH problems x two small open models, an outcome-blind but difficulty-targeted cohort), exactly 1 of 178 cells survives as prefix-limited; restart dose-response separates a compute-starved model from a capability-limited one; and wherever the matched budget lies inside the restart grid, continuing the model's own prefix beats restarting (9 of 9) -- predominantly compute compression rather than expanded reachability. Second, a pre-registered, difficulty-controlled test finds no detectable outcome information in early-window internal signals beyond a problem-difficulty baseline, and two generation-free analyses of public corpora show why this control is needed: a trace-blind difficulty proxy reaches AUROC 0.873 on 192K DeepSeek-R1 generations -- inside the published probe range -- and a closely matched reconstruction of the closest published early-window positive recovers a comparable pooled result (0.849) while within problem it is statistically indistinguishable from chance at all ten anchors (0.496 at t=4); a post-hoc within-targeted probe finds only a small average residual, concentrated in three low-failure problems. High pooled probe AUROCs cannot by themselves establish within-attempt information; a question-only baseline or within-problem evaluation is required.

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