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 |
|---|---|---|
| (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 – | 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) |
| (Obligation Vector) | What an agent must do, with direction and magnitude | Tangent vector on (rank-1 tensor, Ch. 6) |
| (Interest Covector) | What a patient needs, measuring obligations | Cotangent vector on (rank-1 tensor, Ch. 6) |
| (Satisfaction) | How well obligations meet interests | Fundamental contraction: covector vector scalar (Ch. 6) |
| (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: for admissible re-descriptions (Ch. 5, 11) |
| Contraction + Residue | Reducing a tensor to a summary scalar, and what is lost | via iterated contraction; captures discarded structure (Ch. 14) |
| Bond Index (Bd) | Quantitative alignment score for AI systems | Scalar measure of structural invariance 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, 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.