Specific-Referenced Indefinite Quantifiers as Ordinal Refiners: A RED-Based Operator Extension of Cardinal–Ordinal Correlation

Cardinals (three) measure quantity and ordinals (third) identify positions in an ordered domain, yet natural language permits systematic shifts between these regimes. Building on a bidirectional discretization architecture in which an ordinalization operator R refines coarse quantity into addressable order positions, and inverse unification operators collapse refined microstructure back into macro-level wholes, this article extends the program to a third layer: specific-referenced indefinite quantifiers (e.g., all, both, either, neither, certain, half, exact, the least, the most). The central claim is that these quantifiers function as higher-order ordinal refiners: given an ordinal spine O, they select recoverable blocks or alternative sets over O, preserving enough ordinal identity to support anaphora, ellipsis, contrast, and controlled reverse entailment. Crucially, because the RED methodology is bidirectional, ordinal refinement is not the end-state: licensed collapse maps allow a quantifier-refined ordinal structure to feed back into cardinal inferences (lower bounds, bounded choice summaries, extremal summaries) in a principled way. We formalize this interaction as a composed pathway, Cardinal → Ordinal → SR-Quantifier refinement → Collapse → Cardinal summary, and we introduce RED-QORD (Reverse Entailment Diagnostic for quantifier–ordinal interaction) to classify uses as REFINING vs COARSENING depending on recoverability of ordinal traceability under paraphrase and ellipsis. Two worked case studies (English and German) illustrate the end-to-end protocol and motivate a compact set of distinguishing predictions against lexical-derivational and rank-only baselines. We conclude with implications for neurolinguistic processing (ordinal traceability as a memory-retrieval constraint) and for computational semantics and information extraction (recovering ordered subset structure from quantificational expressions)