Proportional Justice in AI: A Mathematical Framework for Geometric Fairness, Stateful Decision-Making and Verifiable Accountability

Many deployed AI decision systems operationalize fairness as arithmetic equality: identical rules, thresholds, or treatment for all individuals. However, classical theories of justice - most notably Aristotle’s concept of geometric proportion - imply that justice is often proportional, requiring outcomes to preserve ratios between relevant merits, responsibilities, or risks. This paper formalizes geometric justice as a set of mathematical axioms over allocation and decision functions, introduces measurable fairness objectives that reduce to ratio-preserving constraints, and proves impossibility and instability results showing that stateless decision policies cannot satisfy proportionality together with temporal consistency. We then model stateful deterministic decision systems as dynamical systems with auditable state transitions and show that deterministic memory is necessary for proportional justice, temporal stability, and post-incident reconstruction. The result is a rigorous foundation establishing proportional fairness as a sequential property and positioning deterministic state as a prerequisite for explainable and accountable AI.