When Adaptive Self-Consistency "Wins" Dissolve: Matched Mechanics, Paired Confidence Intervals, and Controls for Small-Model Stopping Rules

Comparisons of adaptive self-consistency stopping rules (ASC, ESC, confidence- and calibration-based stopping, learned gates) routinely report one rule beating another. We argue such "wins" are often methodological artifacts, and give a cheap protocol that dissolves them. Self-consistency, which samples many reasoning chains and majority-votes, is a strong test-time-compute lever but its cost grows linearly in the sample budget; adaptive rules cut that cost by stopping early per question, and are each evaluated in-distribution on mid-to-large models. We re-run a unified head-to-head (fixed-budget, ESC, confidence-threshold, and a trained calibrated gate) on open small models (Qwen2.5-0.5B/1.5B/7B; GSM8K, MATH-500, BBH) under three guards prior comparisons omit: (a) comparing signals only under matched stopping mechanics, since incremental and 2-stage schedules have different saving ceilings and otherwise dominate any ranking; (b) a paired bootstrap with a TOST equivalence test and a reported minimum-detectable-effect, not marginal CI overlap; and (c) oracle and random-stop controls. Under these guards the reported advantages vanish. This is, by design, a negative-methodology paper. (1) Scale: saving is flat across 0.5B-1.5B and jumps at 7B (confidence 0.81), a jump robust to problem- and arrival-order resampling but confounded by 4-bit vs. fp16 precision, so we report it as suggestive. (2) Which rule: a paired bootstrap and TOST make every signal comparison inconclusive (minimum-detectable-effect 0.11-0.29, often larger than the ~0.15 savings compared), an underpowered null. (3) Signal: the agreement share is a reliable ranking but not a probability (its raw ECE is essentially the aggregate bias |mean_share - accuracy|, removed by one monotone recalibration). (4) External validation: applying the audit to the released rollouts of Adaptive-Consistency, the most-cited early-stopping baseline, reproduces the same two axes on its data, a signal-free random-stop control flipping from 0.95 (weak model) to 0.14 (competent model), and it surfaces a data defect in that release. We claim statistical significance for no cross-method or transfer comparison; the contribution is the reusable audit and the graded nulls it produces.