VerdictRoute: Query-Only Routing from Debate Verdict Dynamics

Selective computation succeeds when a router distinguishes inputs that are merely difficult from inputs that will improve under an additional inference step. We introduce deliberation potential, a supervisory abstraction built from the trajectory of an adjudicator's belief during teacher debate. VerdictRoute extracts verdict instability together with debate risk and pivotal-branch sensitivity, then distills this structured signature into a query-only Llama-$3.1$-$8$B router. Our controlled evaluation fixes the deployed self-critique action, a $18.3$ validation firing-rate target, a bounded $256$-token action, and $1.14\times$ average online cost; Repeated-Gain Direct-V uses 2032 stochastic action executions per item to estimate critique value. VerdictRoute reaches $64.1$ average accuracy across GSM8K, MATH, TruthfulQA, and ToolVerify versus 63.3, a 0.8-point gain. It improves risk ECE from 0.071 to $0.058$ and comes within $0.6$ accuracy points of the matched-action oracle. Across five independent debates, verdict instability has 0.81 pairwise agreement and five-rollout consensus correlates at 0.79 with measured critique value. These results show that verdict-change trajectories can provide a useful query-only supervisory signal for bounded self-critique routing.