Crossover criterion for learning-style matching

Let ${\mu }_{p,f}$ be the expected score for learner-preference group $p$ under randomly assigned format $f$. A two-style visual/auditory matching claim requires ${\mu }_{V,V}>{\mu }_{V,A}$ and ${\mu }_{A,A}>{\mu }_{A,V}$. Thus the within-group format contrast changes sign. An interaction contrast $I=({\mu }_{V,V}-{\mu }_{V,A})-({\mu }_{A,V}-{\mu }_{A,A})$ is positive under this pattern, but $I>0$ alone is insufficient: one format might remain better in both groups. With preference group on the horizontal axis and format as the plotted line, the matching prediction is a crossover. This is a criterion for the stated hypothesis, not evidence that fixed styles exist.