Axioms of Diagnostic Urgency: A Characterization Theorem.

Diagnostic prioritization, deciding which diagnoses in a differential should be evaluated first, is fundamental to emergency medicine but lacks a formal foundation. We model diagnostic urgency using four parts of a diagnostic scenario: outcome distribution, irreversibility, preventable treatment benefit, and time lost as treatment effectiveness decays. We propose six axioms: four monotonicity conditions, coordinate independence, and continuity. We then prove a representation theorem: any urgency function satisfying these axioms and the stated regularity conditions can be represented by an additive score with one component for each part. We also give one axiom-valid construction using expected severity, an irreversibility index, directional preventable expected severity, and hyperbolic temporal decay. The theorem-aligned score is the primary urgency measure throughout; we report a severity-weighted entropy score separately as an exploratory enrichment, characterizing exactly the simplex sub-region on which it respects the severity-dominance axiom. The theorem-aligned score correlates with DALY-derived condition weights across 42 emergency conditions (r=0.934) and, in a retrospective MIMIC-IV v2.2 condition-level criterion-validity analysis, with in-hospital mortality (r=0.684, 95% CI [0.494, 0.837]) and ICU admission (r=0.637); the entropy enrichment correlates with mortality at r=0.747, but its incremental value over the theorem-aligned score is not statistically significant (ΔR2=0.04, p=0.11). The score separates higher- from lower-acuity conditions (AUC = 0.96 across 42 conditions). The framework gives a formal basis for studying the clinical principle of "ruling out the dangerous diagnoses first"; it is not yet a prospectively validated clinical decision rule.