Mathematical Foundations of Structural Ethics: The Geometry of Moral Obligation Under Resource Constraints

This paper presents a new mathematical framework for grounding moral obligations in the structural constraints of finite resources. Agents are modeled as multi-dimensional systems whose continued existence requires satisfying fundamental needs—each with its own decay rate and finite buffer. This yields three qualitatively distinct regimes: Sufficiency, where resources exceed combined needs; the Zero-Instantiation Constraint (ZIC), where resources exactly match needs; and Scarcity, where total needs exceed what the environment can supply. Across these regimes, the dynamics of resource flows determines both the limits and the possibilities of moral action. In Sufficiency, obligations are real but bounded, and a non-empty region of supererogatory action becomes available. At ZIC, obligations are uniquely fixed and leave no slack. Below ZIC, dual preservation is impossible, making classical moral theorizing—obligation, fairness, maximizing the good—structurally undefined. The framework distinguishes duty from generosity, sustainable sacrifice from collapse, and informed self-giving from hope under uncertainty. It also introduces non-fungible, person-specific need dimensions that render another’s existence constitutive of one’s own—explaining phenomena such as parental sacrifice, grief-induced functional decline, and rational ultimate self-sacrifice. In shock-prone worlds, the analysis further shows that two-agent systems are structurally brittle: no dyad can maintain redundancy under finite buffers. By contrast, three-agent configurations admit genuine shock-tolerant resilience, establishing a triadic minimum for robust interdependence. The resulting theory does not rely on utility, preference, or idealized norms. Instead, it shows how moral demands emerge from the existential structure of embodied agents operating under real constraints. This offers a new foundation for structural ethics and clarifies where moral reasoning succeeds—and where the world itself makes moral solutions impossible.