When Do Transformer Components Compose? Validating a Log-Pool Decomposition Criterion

A recent log-pool framework introduces a compositional utility gap as a criterion for decomposing a probability distribution into interacting contributions. We ask whether this gap, applied to architectural decompositions of trained transformers, has empirical content independent of any subagent interpretation. We measure the gap at three resolutions: atomic per-component gaps, random partitions, and partitions found by a genetic algorithm. A partition is feasible when the smallest group gap is non-negative. Held-out recomputation is stable; cross-corpus transfer depends on resolution and architecture. Across Llama 3.1 8B, Pythia 1.4B, and Gemma 2 2B, random-partition feasibility differs sharply under three nulls (uniform, size-matched, residual-matched): Llama is permissive, Pythia rare, Gemma zero in every cell; the genetic algorithm finds a feasible partition in all but one trained cell. Under per-tensor-$\sigma$-matched parameter-scrambled controls, both random and search feasibility collapse in every cell. Under these stated conventions, the gap is a substantive measurement tool for learned transformer structure, while leaving open whether the measured property reflects subagent structure, distributional coherence, or broader decomposition coherence.