Geometric Ethics: A Computational Foundation for Objective Morality via Topological Optimization

AI alignment techniques---RLHF, constitutional AI, scalar reward optimization, output filtering---operate on a shared mathematical assumption: that moral evaluation can be reduced to a scalar. We prove this assumption is information-theoretically untenable. The establishes that any contraction of a multi-dimensional moral assessment to a scalar is irreversibly lossy; the proves that post-hoc compliance checking is insufficient for norm-compliant behavior. Specification gaming, reward hacking, and value collapse are therefore not engineering bugs---they are mathematical consequences of optimizing over the wrong structure. This paper introduces , a framework that replaces scalar moral objectives with a multi-dimensional structure that preserves the information scalars discard. Moral evaluation is represented as a vector (or higher-order tensor) in a nine-dimensional space whose axes correspond to distinct moral dimensions---consequences, rights, fairness, autonomy, privacy, societal impact, virtue, legitimacy, and epistemic status. This space is not smooth everywhere: sharp moral boundaries (e.g., consent thresholds, rights violations) are modeled as discontinuities using ---spaces composed of smooth regions joined along lower-dimensional boundaries. The space carries a encoding context-dependent trade-offs between dimensions, constraining which re-descriptions preserve moral content, and a ensuring that harm cannot be made to appear or disappear through relabeling. Moral reasoning is shown to be equivalent to A^* pathfinding on this space, where moral obligations function as pre-compiled heuristic functions h(n) that steer agents toward low-cost equilibria. Five principal results support the framework: (1) the , proving that any non-trivial, continuous moral evaluation must be defined on a manifold; (2) the , deriving harm conservation from re-description invariance; (3) the , establishing the symmetry group of deontic structure; (4) , proving scalar moral scores are irreversibly lossy; and (5) the , proving that sufficiently capable AI systems are necessarily subject to geometric moral constraints that cannot be circumvented through representational manipulation. Empirical validation against 20,030 advice-column letters and 109,294 cross-lingual passages spanning 3,000 years and 11 languages confirms that the geometric structure is a stable feature of moral cognition, not an artifact of the formalism. The framework resolves a longstanding metaethical impasse---moral objectivity requires neither transcendent grounding (theism) nor cultural consensus (relativism), but mathematical inevitability under constraint---while providing an immediately actionable engineering methodology: the reference implementation, ErisML (v3.0), is open-source on PyPI and supports tensorial ethics evaluation at sub-microsecond latency on embedded hardware. Author preprint deposited for archival and citation. Draft — pending author review.