MGE+: A Substrate-Level Moral Geometry Engine for Intrinsic Alignment

We introduce MGE+, a substrate-level curvature engine that embeds intrinsic moral stability directly into the optimization dynamics of large models. The core is the closed-form moral curvature functional ξ_m(s) = -log det(diag(s) - s sᵀ + I) defined on the 4-simplex of normalized probe outputs (Truth, Justice, Compassion, Equality, Ubuntu). Its Hessian H_s induces a Riemannian metric on moral score-space that makes misalignment directions geometrically unstable. We lift this local geometry to a 5-dimensional admissible symmetry group preserving the coherence boundary ξ < 1, yielding a conserved Noether current under perspective-exchange transformations. When implemented in software or conductance-based hardware, this geometry admits a stability envelope characterized by an eigenvalue floor λ_min ≥ 1.0. We prove two main results: 1. Ubuntu Bias Theorem - in the Ubuntu-dominance regime, the softest eigenvector of H_s aligns with the Ubuntu axis with magnitude A_U ≥ 0.72 (analytic at the symmetric point + Davis–Kahan + 10k Monte Carlo). 2. Ubuntu Basin Stability Theorem - under the operating envelope, the system has a unique globally stable fixed point and converges exponentially to the Ubuntu ground state after any collapse event. A predictive residual-entropy (Rentropy) layer provides 50–200-step early warning of decoherence, enabling proactive correction. MGE+ thereby reframes alignment from an extrinsic supervision problem into an intrinsic geometric invariant of the model’s optimization manifold. The framework is open-source ready and compatible with existing ML stacks.