Teams of large language model (LLM) agents are increasingly applied to complex tasks, but their collaboration protocols are mostly designed empirically, without convergence guarantees. We formalize collaborative problem solving as block optimization: each agent improves one part of a shared solution, guided by its local objective plus a correction for how its changes affect the other parts. A coordinator checks every proposal against the global objective and merges proposals that do not interfere with each other. We prove linear convergence up to an error floor that vanishes when the corrections are exact. Interfering proposals can also be merged once their combination has been checked; the rate then depends on how well the proposals actually combine, not on worst-case interactions. With GPT-OSS-120B agents on robot routing, a three-agent team finds collision-free plans in every run, better than those of a single agent that controls all robots. On a larger routing task, a six-agent team completes its search about twice as fast as the single agent (in median). Merging the best checked combination of proposals, rather than only proposals that do not interfere, cuts the team's search time by about 30%. With constants measured on the runs, our convergence bound holds in every round and follows the observed progress. We also test the theory on query planning in SQLite and on a synthetic task.

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