Core Objects at a Glance
The framework introduces a number of mathematical objects, each with a specific role. The table below provides a compact reference. Readers may wish to return here as new objects are introduced in the text.
| Object / Symbol | Informal Meaning | Formal Character & Chapter |
|---|---|---|
| \mathcal{M} (Moral Manifold) | The “space” of morally relevant situations | -dimensional Whitney-stratified space with smooth strata; nine canonical dimensions serve as a base chart, with domain-specific applications potentially adding further coordinates (Ch. 5) |
| Nine Dimensions D_{1}–D_{9} | The axes of moral variation | Derived from 3×3 scope/mode grid: D₁ Consequences/Welfare, D₂ Rights/Duties, D₃ Justice/Fairness, D₄ Autonomy/Agency, D₅ Privacy/Data, D₆ Societal/Environmental, D₇ Virtue/Care, D₈ Procedural Legitimacy, D₉ Epistemic Status (Ch. 5, §5.3) |
| O^{\mu} (Obligation Vector) | What an agent must do, with direction and magnitude | Tangent vector on \mathcal{M} (rank-1 tensor, Ch. 6) |
| I_{\mu} (Interest Covector) | What a patient needs, measuring obligations | Cotangent vector on \mathcal{M} (rank-1 tensor, Ch. 6) |
| S = I_{\mu}\, O^{\mu} (Satisfaction) | How well obligations meet interests | Fundamental contraction: covector \times vector \rightarrow scalar (Ch. 6) |
| g_{\mu\nu} (Moral Metric) | The “exchange rate” between moral dimensions | Symmetric rank-2 tensor encoding trade-off structure; context-dependent (Ch. 6, 9) |
| Stratification | Discrete jumps in moral structure (e.g., consent thresholds) | Whitney stratification with semantic gates at stratum boundaries (Ch. 5, 8) |
| BIP (Bond Invariance Principle) | Moral evaluations must not depend on mere labeling | Gauge symmetry: E(d) = E(d') for admissible re-descriptions (Ch. 5, 11) |
| Contraction + Residue \mathcal{R} | Reducing a tensor to a summary scalar, and what is lost | T^{(n)} \rightarrow S via iterated contraction; \mathcal{R} captures discarded structure (Ch. 14) |
| Bond Index (Bd) | Quantitative alignment score for AI systems | Scalar measure of structural \times invariance \times residue compliance (Ch. 16, 18) |
| No Escape Theorem | Structural containment blocks cognitive escape routes | Under 4 requirements (canonicalization, grounded evaluation, audit, verification), re-description cannot evade constraint (Ch. 17) |
| f(n) = g(n) + h(n) | Computational engine of moral reasoning: accumulated cost + heuristic estimate | A* evaluation; g = BF (Ch 5–6), h = obligation heuristic, Oᵐ = −gᵐᵛ∂ₙh (Ch 11) |
The objects above map roughly onto the book’s five parts. Part I motivates the geometric approach. Part II (The Framework) introduces\mathcal{M}, the nine dimensions, tensors, the metric, stratification, and the BIP. Part III (Dynamics and Symmetry) develops curvature, connection, and the Noether conservation law. Part IV (Agents and Collectives) builds the contraction pipeline and the residue. Part V (Implementation) translates these objects into the Bond Index, the No Escape Theorem, and the DEME architecture.
Key Results at a Glance
The framework produces seven central theorems. Each is conditional on stated assumptions; Appendix F catalogues the full dependency chain.
| Result | Statement (informal) | Assumptions |
|---|---|---|
| Thm 9.2 Structured Pluralism |
The space of admissible moral metrics admits a partial order that is not total: genuine moral disagreement is geometrically irreducible. | BIP, Def 4.5 |
| Thm 12.1 Moral Noether |
If the moral Lagrangian is invariant under re-description (BIP), then harm is a conserved Noether charge. | BIP, C² Lagrangian |
| Thm 12.3 Gauge Group |
The maximal gauge group consistent with Hohfeldian structure and bounded harm is D₄ × U(1)_H. | Axioms A1–A5 |
| Thm 18.1 No Escape |
Under canonicalization, grounded evaluation, structural audit, and verification integrity, an AI system cannot circumvent moral constraints by re-description. | Reqs 1–4, BIP |
| Prop 15.3 Non-Commutativity |
The order in which moral dimensions are contracted affects the outcome: contraction paths are not interchangeable. | Def 15.1 (pure math) |
| Thm 11.1 Admissibility | Core moral heuristics never overestimate true cost to equilibrium | BF model, Ch 8 stratification |
| Thm 11.2 Intractability | Exact moral geodesic planning is intractable in manifold dimension | Whitney stratification, 9D manifold |