Search spends a finite budget on candidates whose evaluations may be wrong

A full b-ary tree grows exponentially with depth when every retained node expands into b children. Count the root, internal nodes, and leaves according to which operations the implementation performs. A beam policy retains only selected states between levels, but it can still generate more candidates than it retains. Conflating retained width with total generation underestimates cost. Prompt length, evaluator repetitions, tool calls, and state copying can make node costs unequal, so a fixed per-node model is only a transparent starting point.

A value label such as promising or impossible can be a heuristic judgment rather than a proof. If a sound task-specific verifier certifies a branch infeasible, pruning has a different guarantee from pruning because the same language model dislikes it. Test false rejection of useful branches and false acceptance of invalid ones. Larger candidate sets can increase diversity, but correlated generation can repeat the same error. An independent-success formula for many samples is not justified merely by changing random seeds or asking the same model several times.

A practical controller can spend more work where uncertainty or a task-specific check could change the decision. It can stop expanding duplicate states, bound depth, or preserve a few diverse candidates rather than only the highest self-score. These are policies to evaluate. They do not become optimal by being called adaptive. Reserve budget for a final verifier, state reconciliation, and reporting rather than spending every token on branch generation. If the evaluator itself is costly or poorly calibrated, allocating more evaluation can add expense without improving the chosen plan.

Keep speculative search local unless effects are explicitly authorized and coordinated. Two candidate plans should not both execute a purchase, publication, or shared write merely to see which is better. Use a simulator, immutable read-only observation, or separated draft artifact where appropriate, and treat real side effects through the task's operation protocol. Search-state rollback cannot undo an external effect by deleting the branch text. Record what was generated, what was evaluated, what was actually executed, and how correctness was checked.

Evaluate breadth, depth, diversity, and verifier choice while holding task sets and resource limits clear. Include unsolvable tasks, misleading high-scored candidates, duplicate branches, and cases where a cheap external check rejects a fluent answer. Report success with actual consumed work, not only an oracle best-of-many score unavailable to the serving policy. Distinguish an ablation of search from a change in model, prompt, or tool access. A benchmark improvement is evidence about those conditions and should not be generalized into guaranteed reasoning correctness on arbitrary tasks.

The root-finding lab demonstrates a stronger guarantee in a much narrower problem. Reset, set both c=2 and upper bound U=4, predict midpoint error bound 2 initially,1 after one Step, and 0.5 after two, then inspect. It bisects only x²−c on a known bracket, with compensated sign checks and a fifty-iteration cap. Step advances one refinement per 0.5 seconds. This does not generate language branches or score plans. Its certified bracket reduction depends on the explicit monotone-function assumptions absent from an arbitrary model evaluator.