Predictive tasks constitute subprocedures within human--AI interactions (HAIs) involving professionals and AI systems. Complementarity occurs when an HAI outperforms a benchmark defined by its constituent agents. Existing formalizations, however, focus on a single human and a single AI, leaving multi-agent interactions insufficiently characterized. We introduce a tree-based mathematical framework for multi-agent complementarity. Ordered agent-role configurations decorate the leaves of rooted planar binary trees and local aggregation rules generate protocol outputs evaluated against either a best-average or a pointwise-min benchmark. We establish four results. First, under linear pooling, every protocol tree provides a bijective coordinate system for the interior of the simplex of global leaf weights. Arbitrary trees with the same ordered leaves admit output- and complementarity-preserving reparameterizations, realized locally through path-independent Tamari rotations. However, HAI constraints can violate admissibility preservation between equivalent protocols. Second, for regression under squared loss, we characterize complementarity through residual correction and derive optimal linear-pooling weights. We also characterize protocol-indifference loci under identical local parameters. Third, selector-based protocols cannot achieve complementarity relative to the pointwise-min benchmark, independently of task and loss. Fourth, in binary classification, HAIs cannot achieve complementarity under endpoint-monotone losses and the pointwise-min benchmark, with an analogous obstruction for multiclass cross-entropy. Numerical illustrations examine the main claims. We provide a mathematical foundation for analyzing multi-agent predictive subprocedures within HAIs, establishing a first step toward understanding how interaction, aggregation, and real-world constraints affect HAI performance.

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