combinatorics

The Square Law of Parallel Agent Collisions: Pair Growth, Zone Size, and Merge Cost

Potential collision pairs grow as n-squared; bounded zone size is what restores near-linear conflict growth

When many agents operate in the same mutable workspace, the number of potential collision pairs grows quadratically. That combinatorial fact does not by itself tell operators how to partition the team. This article keeps the square-law insight, then replaces an incorrect partition formula with a clearer tradeoff: within-zone collisions fall as zones get smaller, while cross-zone merge cost rises as zones get smaller. The optimal design usually comes from choosing a bounded zone size, not from a universal square-root law in the number of zones.