Differential Geometry for Moral Alignment: The Mathematical Foundations of DEME 3.0

We present the mathematical foundations of DEME 3.0 (Democratically-governed Ethical Modules), a framework for AI governance based on differential geometry and tensor calculus. In DEME 3.0, ethically relevant configurations of an AI system are modeled as points on a smooth ``moral manifold'', stakeholder perspectives are represented as coordinate charts, and ethical quantities are expressed as tensors on this manifold. We define obligations as contravariant vector fields and stakeholder interests as covector fields, and show that a scalar ``ethical satisfaction'' functional can be modeled as their tensor contraction. We first prove that this satisfaction score is invariant under changes of coordinates, providing a frame-independent moral scalar. We then go beyond this basic invariance calculation and establish a representation theorem: under axioms of locality, bilinearity, non-degeneracy and coordinate invariance, any pointwise satisfaction functional must be representable---up to a smooth scalar factor---as the canonical contraction of a vector and a covector. A normalization condition removes this degree of freedom, yielding a uniquely defined scalar field. We further endow the moral manifold with a governance-dependent metric tensor, show that simple axioms on local trade-off weights ensure the existence of a Riemannian metric realizing them, and derive geodesic equations for ``trajectories of least ethical resistance''. Finally, we instantiate these ideas in a DEME 3.0 case study: a triage decision scenario implementing DEMEProfileV03 and Geneva-style base ethics modules. We show how the existing EthicalFacts schema can be interpreted as coordinates on the moral manifold, how obligations and interests act as tensors, and how governance profiles induce different ethical metrics and geodesic policies. We situate this work in the broader landscape of AI governance, alignment and mathematical foundations of AI. Author preprint deposited for archival and citation. Draft — pending author review.