We present (SGE), a mathematical framework providing rigorous foundations for deterministic, verifiable ethical reasoning in autonomous systems. SGE addresses fundamental limitations of prior geometric approaches by modeling the space of ethically relevant configurations as a ---a union of smooth manifolds of varying dimensions connected along boundary strata---rather than a smooth manifold. This structure captures moral discontinuities, incommensurable values, and genuine ethical dilemmas that smooth models cannot represent. We make five principal contributions. First, we show that stratified spaces are a among standard geometric structures for representing ethical phenomena including threshold effects, lexical priorities, and moral dilemmas (Theorem 2.3). Second, we establish a (Theorem 4.3) characterizing all satisfaction functionals satisfying five explicit axioms---including a novel locality axiom and scale-normalization assumption---with a complete proof. Third, we prove (Theorems 3.9--3.11) showing that any decision problem on a compact stratified space reduces to a finite graph problem with explicit error bounds. Fourth, we establish (Theorem 6.4) for the quantifier-free, non-temporal fragment of our ethical specification language via o-minimal structures, with temporal properties handled by standard model checking on finite approximations. Fifth, we derive (Theorems 7.1--7.3) for learning ethical content from data. We introduce the Bond Invariance Principle (BIP), which requires ethical judgments to depend only on morally relevant relationships, not arbitrary representation choices, and prove that our axiom system implies BIP. SGE serves as the theoretical foundation for the DEME architecture, providing mathematical justification for design choices including multi-dimensional moral vector spaces, governance profiles with veto regions and lexical priorities, and layered enforcement architectures. Since initial development, a comprehensive reference implementation (196 commits, 15 development sprints) has realized all core theoretical constructs as rank-6 tensors with hardware-accelerated backends, and the Bond Invariance Principle has been empirically validated across 11 languages (80.0\ \(p = 0.023\)). Author preprint deposited for archival and citation. Draft — pending author review.
Andrew Bond (Sun,) studied this question.