SAME ERROR, DIFFERENT GEOMETRY: RELATIONAL GENERALIZATION ON COMBINATORIAL LANDSCAPES

Standard predictive error can be insufficient on structured combinatorial spaces because models with similar aggregate error can organize residuals differently over the underlying graph. This matters when predictions are used not only to estimate individual fitness values, but also to recover mutational effects, interactions, landscape structure, and search-relevant relationships. We formalize this distinction as relational generalization, separating pointwise predictive risk from first- and second-order graph-relative fidelity. Across four empirical RNA fitness landscapes and seven model classes, natural matched-risk comparisons showed limited first-order separation, motivating a controlled intervention that exactly matched held-out pointwise error while preserving learned residual organization. Despite identical held-out pointwise risk, learned residual directions retained substantial differences in first- and second-order relational error, with second-order error differing by up to 2.8× within the same system and split. These differences altered reconstructed local-optimum and accessibility structure. Mutation-sign disagreements were concentrated among low- to moderate-margin effects, while high-margin effects were largely stable, consistent with a margin bound linking relational error to sign errors above a specified effect margin. Controlled spectral constructions on Domingo tRNA and Tetrahymena thermophila provided a complementary demonstration that equal aggregate error can coexist with different relational geometry and reconstructed topology. Relation-aware training further reduced relational error in both tested biological systems, although pointwise equivalence was not established. Sensitivity analysis showed that the main joint-preservation result was robust across a broad range of first- versus second-order weightings. Overall, our results show that aggregate predictive error alone does not determine relational geometry or structural fidelity, motivating relational generalization as a complementary criterion for evaluating and training models intended for structured biological inference and sequence search.