PhasePilot: Auditing and steering phase boundaries in budgeted in-context reasoning

Abstract Reasoning performance in in-context learning can change discontinuously as inference-time resources vary, even at fixed model size. We characterize this behavior with an operational framework that treats the prompt as a budgeted state space defined by context cap k, demonstration count N, and task depth D. We define an order parameter m (exact-match accuracy) and a sensitivity χ=∂m/∂logN; the peak of χ yields a critical example count Nc, complemented by transition width and plateau gap to make claims falsifiable. We validate this estimator against alternative threshold-detection methods and report stability and uncertainty via bootstrap. We introduce PhasePilot, a prompting protocol that steers phase boundaries under a strict 4096-token budget cap enforced by cap-based allocation (prompt cap + generation cap per call × calls). PhasePilot shifts Nc leftward by 2.6–4.1× on controlled tasks (median 3.4×) by reallocating budget into an explicit state scaffold (5–6 task-specific variables within a 4-block I/O schema: Input/State/Reasoning/Answer) and by suppressing distractor entropy. On 11 external benchmarks, PhasePilot achieves a median +25.9-point gain over CoT (IQR [22.3, 29.2]) across GSM8K, MATH L3–5, GPQA Diamond, BBH, AIME 2023–24, ARC-Challenge, HotpotQA, DROP, ProofWriter, LogiQA, and LongCtx-DQA; within BBH, per-configuration gains span +9.0 to +46.2, exposing strong task-structure heterogeneity. Per-instance gain distributions, realized-token diagnostics, and output audits support that improvements reflect reasoning quality rather than extraction artifacts. Under the same cap, PhasePilot lies on the empirical compute frontier on the controlled suite (94.8% accuracy with one call versus 75.1% for ToT(b=4) with 12 calls). Mechanistic probing with pre-registered head selection and randomized controls provides supporting evidence linking the boundary shift to a sparse mid-layer head set, with patching effects 5.8× larger than random-head baselines.