Beam search keeps only a fixed number of promising partial solutions at each depth. It controls cost but can permanently discard the route to the best solution.

Beam Search Trades Coverage for Bounded Width

Time, tokens, money, tool calls, and risk tolerance are finite resources. Carrying remaining budgets in state allows routers and loops to degrade or stop predictably.

Budgets Are First-Class Graph State

A reasoning graph can branch, merge, revise, and reuse intermediate hypotheses rather than form one chain. It needs state equivalence, evidence labels, pruning, and a final verification criterion.

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