Abstract: Observable scalar rankings constructed from observable characteristics are widely used to organize search, matching, and allocation across markets. In practice, such rankings are often treated as portable sufficient statistics: conditional on an individual’s rank, other observable characteristics are presumed irrelevant for equilibrium allocation and downstream outcomes. We test this scalar sufficiency assumption by tracing observable compression from search behavior to equilibrium matching and household outcomes in China’s marriage market. Using a large-scale randomized online-dating experiment that independently manipulates male height and income, we first estimate causal height–income substitution patterns governing directed search. We reject a common linear observable index: substitution rates vary sharply and systematically across female height groups. We then construct behaviorally disciplined group-specific observable rankings and transport them to nationally representative marriage data from the China Family Panel Studies (CFPS). Conditioning on these indices attenuates assortative matching but does not eliminate it, leaving substantial residual sorting within narrow index strata. Finally, we reject additive separability in household-income surfaces: husband-height gradients depend strongly on wife height, implying economically large interaction effects. Observable scalar compression, therefore, breaks down from search to equilibrium to outcomes. Even when externally disciplined by randomized variation, observable rankings are not portable sufficient statistics once two-sided competition and market-clearing constraints bind. These findings highlight fundamental limits to scalar-index representations in matching models and to the use of observable rankings in education, labor markets, credit allocation, and platform design.