Beyond Labels: A Composable Ordinal Interface for Language Models

An ordinal label can be correct yet insufficient for subsequent reasoning. We study this gap for language-model predictions that admit an underlying scalar quantity, using nested proportional composition as a concrete setting. We introduce the Differentiable Fuzzy Inference Layer (DFIL), a dual-path head that retains a standard classifier while exposing a supervised scalar and Gaussian memberships with learnable ordered centers. Rather than composing intermediate labels, DFIL performs proportional operations in scalar space and discretizes only the final result. For nearest-center decoding, we characterize exactly when intermediate labels suffice for further composition, construct minimal exact refinements for fixed finite proportional multiplier sets, and derive optimal state budgets for margin-separated threshold queries. For a fixed composition structure and a supplied Cartesian uncertainty box that preserves center ordering, we also derive the exact set of attainable nearest-center labels while accounting for parameters shared across composition and decoding. For the analyzed ordered-center families, we further reduce Gaussian label reachability to a two-variable feasibility problem without requiring globally monotone decoding. Experiments on quantifier reasoning and related ordinal tasks, including human-judgment data and matched controls, separately assess predictive accuracy, structural consistency, and the contribution of the learned membership bank. This work frames ordinal prediction as the design of an inspectable, composable interface rather than a terminal category decision.