Cardinal–Ordinal Correlations as Bidirectional Discretization: A RED-Based Operator Account

Cardinals (three) and ordinals (third) are routinely treated as closely related members of the numeral system, yet their semantic contributions diverge: cardinals measure quantity, while ordinals identify positions within an order. This article develops an explicit account of the correlation as a bidirectional discretization architecture in which (i) an ordinalization operator refines a coarse quantity into a structured, addressable progression, and (ii) inverse unification operators collapse refined microstructure back into macro-level ordinalized wholes. The proposal is expressed in a diagnostic methodology (RED: Reverse Entailment Diagnostic) adapted to numerals, testing whether a processual representation can be reified into a discrete countable object (an ordinal) and then reverse-entails back to its cardinal basis under controlled collapse. Formally, ordinalization is modeled as a lexicographic-sum construction R(I,{F_i}) = Σ_{i∈I} F_i, yielding a directed system of refinements whose ‘continuity’ is processual (deterministic gluing across steps) rather than analytic density. The reverse direction is captured by projection and aggregation maps (collapse c, Fold, group formation G, and GEN) that unify fibers into single indexed objects. Empirically, the framework predicts when (and why) languages allow numeral expressions to shift between quantificational, adjectival, and nominal uses (e.g., the third; came third; the three; three of them), and it derives diagnostics for distinguishing genuine ordinal objecthood from mere contextual ordering.