FULCRUM: Pair-Dependent Preferences Beyond Transitive Rankings

Assigning each option one context-independent scalar score, as reward models and fixed priority orderings do, makes every preference transitive. Judgements on several criteria need not be: when the criterion that decides depends on the pair compared, the relation can cycle and no single ordering represents it. We study FULCRUM, a calculus for structured argumentation in which the priority over comparison criteria is chosen from the pair itself, with the resulting relation converted into defeats under standard semantics. Its endogenous MAX-GAP rule induces strict cycles of every length that no preorder realises, and we characterise exactly, through pairwise disagreement and a minimum weighted feedback arc set, the best accuracy any linear order can attain on a given set of comparisons. For a reasonable ordering in the sense of Modgil and Prakken, closure and consistency hold without transitivity, and the aggregation used to score conjunctions decides whether the relevant reasonableness clause holds. A soft-gated comparator makes the structure learnable with criterion-level attribution. Empirically, in a controlled world whose true rule is gap-driven it gains up to +7.3 accuracy points over a linear Bradley–Terry model, above the transitive ceiling; it is neutral on transitive preferences, and more flexible pairwise comparators perform better when the dependence has a different form.