Measuring Black-Box Confidence via Reasoning Trajectories: Geometry, Coverage, and Verbalization

Reliable confidence estimation gates safe deployment of chain-of-thought (CoT) reasoning through text-only APIs, yet the dominant black-box baseline, self-consistency over K samples, is linearly expensive and ignores the geometry of the trace. We introduce a black-box trajectory-confidence score that embeds a CoT as a sliding-window trajectory and measures its convergence toward external answer anchors with a one-parameter softmax, requiring no logits, hidden states, or supervised calibrators. On six (benchmark,reasoner) settings over MedQA-USMLE, GPQA Diamond, and MMLU-Pro × {Gemini 3.1 Pro, Claude Sonnet 4.6}, fusing this score with coverage and verbalized-confidence channels at K=4 Pareto-improves self-consistency at K=8 in 6/6 settings (median AUC 0.78 vs. 0.71, ΔAUC=+0.075); a fixed-pick control (+0.060) and an E5 cross-embedder replication rule out answer-switching and single-vendor artifacts. Mechanistically, the geometry signal peaks in the penultimate reasoning window across all benchmarks and reasoners and inverts at the terminal window on GPQA Diamond, exposing answer commitment before literal verbalization. Three increasingly unscaffolded regimes decompose black-box confidence into a judge-mediated Coverage prior (C), within-trace Geometry (G), and a conditional Verbalization channel (V); across 18 benchmark × reasoner × proposer settings, C and G carry independent signal in 18/18 and 16/18, while V contributes residual signal in only 6/18. A judge-family swap (GPT-5-mini → Claude Sonnet 4.6) leaves G-only AUC unchanged (∣Δ∣≤0.013) and shifts C-only AUC by at most ±0.02 (κ=0.82), and fusion beats the best single channel in 17/18 settings (median AUC 0.78, max 0.92). Together, these results show that black-box CoT confidence can be read more reliably from a trace's geometric convergence in embedding space than from sample-vote agreement, at lower sampling cost and without access to logits or hidden states.