Keep a Branch: Completion-Aware Width Control for Verifier-Guided Reasoning

A branch kept alive by test-time search can survive a layer without advancing, advance without finishing, and finish without winning the final vote. We study how verifier-guided reasoning should choose its width, the number of branches retained per layer, under a fixed token budget, and find that each link of this chain fails separately. On 195 MATH problems, widening a frozen 7B beam from width 1 to 5 under a fixed ceiling halves accuracy, from 82% to 42%. The wide beam leaves 112 runs with no completed answer: at width 5, only 7,116 of the 65,536 budget tokens are newly generated text. Exact-distance labels on 1.22 million planning candidates isolate the earlier links, separating retention of an improving branch from sustained progress toward the goal. The diagnosis yields two requirements for useful exploration: an extra branch must justify its cost, and enough budget must remain to finish one retained path. SCA+ turns them into a per-layer width rule built from an estimated marginal gain and an estimated completion reserve. Under total-token accounting, with verifier calls charged like generation, SCA+ exceeds its strongest budget-aware baseline on MATH by 4.7 points with 16% fewer tokens while both discover correct answers equally often. The margin comes from completion quality: narrower width concentrates tokens per candidate, raises verifier scores at completion and shifts a fixed final vote.