Appendix F: Mathematical Ledger — Status of Formal Claims
This appendix catalogues every numbered formal statement in the manuscript — definitions, axioms, theorems, propositions, lemmas, corollaries, and conventions — together with its epistemic status and dependencies. The purpose is transparency: a reader should be able to see at a glance what is assumed (as a modeling choice mapping ethical phenomena to mathematical structures), what is proved (given those assumptions), and what is conjectured or proposed for future investigation.
Five epistemic categories are used. Formal Definition: a standard mathematical construction carrying no ethical modeling claim (e.g., “topological manifold”). Modeling Axiom: a mathematical structure posited as a model of an ethical phenomenon; these are the bridge assumptions that connect the mathematics to moral reasoning, and they are the primary target of empirical and philosophical scrutiny. Conditional Theorem: a result that is mathematically proved given stated assumptions (listed in the Dependencies column); the theorem is as strong as its premises. Proved: a result that follows from standard mathematics alone, without ethical modeling assumptions. Conjecture/Proposal: a formally stated claim that is either unproved, depends on open problems, or represents a proposed extension not yet implemented. The table is organized by chapter. “TA” denotes the chapter’s Technical Appendix.
| Ref | Name | Status | Dependencies | § |
|---|---|---|---|---|
| Def 4.1 | Topological Manifold | Formal Definition | None | §4.1 |
| Def 4.2 | Smooth Manifold | Formal Definition | Def 4.1 | §4.1 |
| Def 4.3 | Tangent Bundle | Formal Definition | Def 4.2 | §4.2 |
| Def 4.4 | Affine Connection | Formal Definition | Def 4.3 | §4.6 |
| Def 4.5 | Riemann Curvature Tensor | Formal Definition | Def 4.4 | §4.6 |
| Def 4.6 | Fiber Bundle | Formal Definition | Def 4.2 | §4.7 |
| Def 4.7 | Whitney Stratification | Formal Definition | Def 4.1 | §4.8 |
| Def 4.8 | Gauge Invariance | Formal Definition | Def 4.6 | §4.9 |
| Def 5.1 | Moral Manifold (Informal) | Modeling Axiom | Def 4.2 | §5.2 |
| Def 5.2 | Dimension of the Moral Manifold (k = 9) | Modeling Axiom | Def 5.1 | §5.2 |
| Def 5.3 | Agent Bundle | Modeling Axiom | Def 4.6, Def 5.1 | §5.4 |
| Axiom 5.1 | Bond Invariance Principle (BIP) | Modeling Axiom | Def 5.1, Def 4.8 | §5.3 |
| Def 5.4 | Moral Stratification | Modeling Axiom | Def 4.7, Def 5.1 | §5.6 |
| Def 5.5 | Constraint Set | Modeling Axiom | Def 5.4 | §5.6 |
| Def 5.6 | Absorbing Stratum | Modeling Axiom | Def 5.4 | §5.6 |
| Def 5.7 | Moral Singularity | Modeling Axiom | Def 5.4 | §5.6 |
| Def A.1–A.6 | Moral Manifold (Formal Defs) | Formal Definition | Defs 4.1–4.7, Def 5.1 | §5 TA |
| Def 6.1 | Obligation Vector Oµ | Modeling Axiom | Def 5.1, Def 4.3 | §6.2 |
| Def 6.2 | Interest Covector Iµ | Modeling Axiom | Def 5.1 | §6.3 |
| Def 6.3 | Satisfaction S = IµOµ | Modeling Axiom | Defs 6.1–6.2 | §6.4 |
| Def 6.4 | Evaluation Tensor Eµᵥ | Modeling Axiom | Defs 6.1–6.3 | §6.6 |
| Prop 6.1 | Invariance of Satisfaction | Proved | Defs 6.1–6.3, BIP | §6 TA |
| Prop 6.2 | Decomposition of Disagreement | Proved | Def 6.4 | §6 TA |
| Prop 6.3 | Metric Dependence of Gradient | Proved | Def 4.4 | §6 TA |
| Def 8.1 | Stratified Moral Space | Formal Definition | Defs 4.7, 5.4 | §8.2 |
| Defs 8.2–8.3 | Whitney Conditions A and B | Formal Definition | Def 4.7 | §8.2 |
| Def 8.4 | Threshold Boundary | Modeling Axiom | Def 8.1 | §8.3 |
| Def 8.5 | Phase Transition Boundary | Modeling Axiom | Def 8.1 | §8.3 |
| Def 8.6 | Absorbing Stratum | Modeling Axiom | Def 8.1 | §8.3 |
| Def 8.7 | Constraint Surface | Modeling Axiom | Def 8.1 | §8.3 |
| Def 8.8 | Semantic Gate | Modeling Axiom | Def 8.1 | §8.4 |
| Def 8.9 | Penumbral Zone | Modeling Axiom | Defs 8.4–8.5 | §8.5 |
| Def 8.10 | Moral Phase Transition | Modeling Axiom | Defs 8.1, 8.5 | §8.6 |
| Def 8.11 | Boundary Crossing Data | Formal Definition | Def 8.1 | §8.8 |
| Conv 8.1 | DG Operations at Stratum Boundaries | Formal Definition | Defs 8.1, 8.11, 4.4 | §8.8 |
| Prop 8.1 | Existence of Moral Whitney Stratification | Conditional Theorem | Defs 5.4, 4.7 | §8 TA |
| Prop 8.2 | Absorbing Strata Are Unique Attractors | Conditional Theorem | Def 8.6 | §8 TA |
| Prop 8.3 | D₄ Structure of Hohfeldian Transitions | Conditional Theorem | Def 8.8, Axiom A1–A3 | §8 TA |
| Def 9.1 | Partial Order on Metrics | Formal Definition | Def 4.5 | §9.4 |
| Def A.1 | Admissible Metric | Formal Definition | Def 4.5, Axiom 5.1 | §9 TA |
| Prop 9.1 | Non-Uniqueness of Admissible Metrics | Conditional Theorem | Def A.1 | §9 TA |
| Prop 9.2 | Structural Invariants Constrain the Metric | Conditional Theorem | Def A.1, BIP | §9 TA |
| Lemma 9.1 | D₄ Constraint on Metric Components | Conditional Theorem | Def A.1, Axiom A1–A3 | §9 TA |
| Lemma 9.2 | U(1)ₕ Constraint on Harm Dimension | Conditional Theorem | Def A.1, Axiom A4 | §9 TA |
| Thm 9.1 | Admissible Metric Space Structure | Conditional Theorem | Def A.1, BIP | §9 TA |
| Thm 9.2 | Structured Pluralism | Conditional Theorem | Thm 9.1 | §9 TA |
| Def 10.1 | Moral Connection | Formal Definition | Def 4.4, Def 5.1 | §10.3 |
| Def 10.2 | Parallel Transport | Formal Definition | Def 10.1 | §10.4 |
| Def 10.3 | Holonomy | Formal Definition | Def 10.2 | §10.5 |
| Prop 10.1 | Parallel Transport Preserves Satisfaction | Conditional Theorem | Defs 10.2, 6.3 | §10 TA |
| Prop 10.2 | Holonomy Detects Curvature | Conditional Theorem | Defs 10.3, 4.5 | §10 TA |
| Prop 10.3 | Geodesic Deviation Equation | Proved | Def 4.5 | §10 TA |
| Def 11.1 | Moral Search Problem | Modeling Axiom | Def 5.1, Def 6.4 (BF) | §11.2 |
| Def 11.2 | A* Moral Search | Modeling Axiom | Def 11.1 | §11.2 |
| Prop 11.1 | Obligation-Heuristic Correspondence | Modeling Axiom | Def 6.1, Def 11.2 | §11.3 |
| Def 11.3 | Admissibility | Formal Definition | Standard | §11.4 |
| Def 11.4 | Consistency | Formal Definition | Standard | §11.4 |
| Thm 11.1 | Admissibility of Core Moral Heuristics | Conditional Theorem | Def 11.3, Ch 8 stratification | §11.4 |
| Prop 11.2 | Consistency from Metric Compatibility | Proved | Def 4.5, Def 11.4 | §11.4 |
| Thm 11.2 | Intractability of Exact Moral Geodesic Planning | Conditional Theorem | Def 5.1, Def 11.1 | §11.5 |
| Cor 11.1 | Evolutionary Necessity of Moral Heuristics | Conditional Theorem | Thm 11.2 | §11.5 |
| Cor 11.2 | Heuristic Imperfection Is Structural | Conditional Theorem | Thm 11.2 | §11.5 |
| Prop 11.3 | Satisfaction as Directional Derivative | Proved | Def 6.1, Prop 11.1 | §11.6 |
| Prop 11.4 | A* Optimality on Stratified Spaces | Proved | Def 11.2, Def 8.1 (Whitney) | §11 TA |
| Axiom A1 | Four Hohfeldian Positions | Modeling Axiom | None | §12.3 |
| Axiom A2 | Two Independent Involutions | Modeling Axiom | Axiom A1 | §12.3 |
| Axiom A3 | Cyclic Ordering | Modeling Axiom | Axioms A1–A2 | §12.3 |
| Axiom A4 | One Continuous Conserved Quantity | Modeling Axiom | None | §12.3 |
| Axiom A5 | Commutativity of Discrete and Continuous | Modeling Axiom | Axioms A1–A4 | §12.3 |
| Def 12.1 | Moral Action (Lagrangian) | Modeling Axiom | Defs 4.4, 5.1 | §12.4 |
| Def 12.2 | Harm (Noether Charge) | Modeling Axiom | Thm 12.1, Prop 12.1 | §12.6 |
| Def 12.3 | Harm Ledger | Modeling Axiom | Def 12.2 | §12.6 |
| Thm 12.1 | Moral Noether’s Theorem | Conditional Theorem | BIP + C² Lagrangian | §12.5 |
| Thm 12.2 | Discrete Symmetry = D₄ | Conditional Theorem | Axioms A1–A3 | §12.3 |
| Thm 12.3 | Gauge Group Uniqueness: G = D₄ × U(1)ₕ | Conditional Theorem | Axioms A1–A5 | §12.3 |
| Prop 12.1 | Continuous Component is U(1) | Conditional Theorem | Axiom A4 | §12.3 |
| Cor A.1 | Number of Conservation Laws | Conditional Theorem | Thm A.1 | §11 TA |
| Prop A.2 | Discrete Conservation from D₄ | Conditional Theorem | Thm 12.2 | §11 TA |
| Thm A.1 | Moral Noether (Formal Statement) | Conditional Theorem | BIP + C² Lagrangian | §11 TA |
| Lemma A.2 | Aut(C₄) ≅ D₄ | Proved | None | §11 TA |
| Lemma A.3 | Compact Connected 1-D Lie Group = U(1) | Proved | None | §11 TA |
| Def 13.1 | Moral Hilbert Space | Conjecture/Proposal | Def 5.1 | §13.2 |
| Def 13.2 | Moral Observable | Conjecture/Proposal | Def 13.1 | §13.2 |
| Def 13.3 | Moral Density Matrix | Conjecture/Proposal | Def 13.1 | §13.3 |
| Def 13.4 | Moral Decoherence | Conjecture/Proposal | Def 13.3 | §13.4 |
| Def 13.5 | Stratified Lagrangian | Modeling Axiom | Def 12.1, Def 8.1 | §13.5 |
| Def 13.6 | Junction Conditions | Formal Definition | Defs 12.5, 8.11 | §13.5 |
| Def 13.7 | Moral Hamiltonian | Conjecture/Proposal | Def 13.5 | §13.6 |
| Def 13.8 | Moral Tunneling | Conjecture/Proposal | Def 13.7, Def 8.7 | §13.7 |
| Def 13.9 | Moral Entanglement | Conjecture/Proposal | Def 13.3 | §13.8 |
| Thm 13.1 | Decomposition of H by Strata | Conditional Theorem | Def 13.1, Def 8.1 | §12 TA |
| Prop 13.1 | Moral Uncertainty Principle | Conditional Theorem | Defs 12.1–12.2 | §13.3 |
| Prop 13.2 | Conservation of Probability | Conditional Theorem | Def 13.7 | §12 TA |
| Prop 13.3 | Tunneling Rate Through Moral Barrier | Conditional Theorem | Def 13.8 | §12 TA |
| Prop 13.4 | Entanglement Entropy | Conditional Theorem | Def 13.9 | §12 TA |
| Def 14.1 | Collective Agency Tensor | Modeling Axiom | Defs 6.1–6.4 | §14.2 |
| Def 14.2 | Collective Decomposition | Formal Definition | Def 14.1 | §14.2 |
| Def 14.3 | Structure Tensor | Modeling Axiom | Def 14.1 | §14.4 |
| Def 14.4 | Responsibility Tensor | Modeling Axiom | Def 14.1 | §14.5 |
| Prop 14.1 | Emergent Obligation Criterion | Conditional Theorem | Def 14.1 | §14.3 |
| Prop 14.2 | Emergent Moral Property | Conditional Theorem | Defs 13.1–13.2 | §14.3 |
| Prop 14.3 | Decomposition Theorem | Conditional Theorem | Def 14.2 | §13 TA |
| Prop 14.4 | Responsibility Inequality | Conditional Theorem | Def 14.4 | §13 TA |
| Prop 14.5 | Structural Coupling | Conditional Theorem | Def 14.3 | §13 TA |
| Def 15.1 | Contraction Loss | Formal Definition | Defs 6.1–6.4 | §15.2 |
| Def 15.2 | Moral Residue | Formal Definition | Def 15.1 | §15.3 |
| Prop 15.1 | Inevitability of Residue | Proved | Defs 14.1–14.2 | §15.3 |
| Prop 15.2 | Harm Survives Contraction | Conditional Theorem | Defs 14.1, 11.2 | §15.4 |
| Prop 15.3 | Non-Commutativity of Contraction | Proved | Def 15.1 | §14 TA |
| Prop 15.4 | Residue of Maximin Contraction | Conditional Theorem | Defs 14.1–14.2 | §14 TA |
| Prop 15.5 | Transparent Contraction and BIP | Conditional Theorem | Def 15.1, BIP | §14 TA |
| Rmk A.1 | Residue as Moore–Penrose Pseudoinverse | Formal Definition | Def 15.2, metric gµᵥ | §14 TA |
| Def 16.1 | Theory Covariance Tensor | Modeling Axiom | Def 6.4 | §16.3 |
| Def 16.2 | Robust Obligation | Formal Definition | Def 16.1 | §16.4 |
| Prop 16.1 | Stability of Robust Obligations | Conditional Theorem | Def 16.2 | §16.4 |
| Prop 16.2 | Structure of the Robust Core | Conditional Theorem | Def 16.2 | §15 TA |
| Prop 16.3 | Irreducibility of Theory Uncertainty | Conditional Theorem | Def 16.1 | §15 TA |
| Prop 16.4 | Hedging as Minimax Regret | Conditional Theorem | Def 16.2 | §15 TA |
| Def 18.1 | Grounding Adequacy Predicate | Modeling Axiom | Def 5.1 | §18.7 |
| Thm 18.1 | No Escape Theorem | Conditional Theorem | Reqs 1–4, BIP | §18.6 |
| Thm 18.2 | No Escape with Formal Adequacy | Conditional Theorem | Thm 18.1, Def 18.1 | §18.7 |
| Lemma 18.1 | Coverage Completeness | Conditional Theorem | Def 18.1 | §18.8 |
| Lemma 18.2 | Robustness Implies Lipschitz Stability | Conditional Theorem | Def 18.1 | §18.8 |
| Prop 18.1 | Invariance Violation Decomposition | Conditional Theorem | Thm 18.1, BIP | §17 TA |
| Prop 18.2 | Contraction Mismatch Bound | Conditional Theorem | Thm 18.1 | §17 TA |
| Prop 18.3 | No Escape — Conditional Guarantee | Conditional Theorem | Reqs 1–4 | §17 TA |
| Prop 18.4 | Alignment Gap Convergence | Conditional Theorem | Thm 18.1, BIP | §17 TA |
| Prop 18.5 | Testability of Adequacy Clauses | Conditional Theorem | Def 18.1 | §17 TA |
| Def 19.1 | Norm-Constrained Scoring Game (NCSG) | Modeling Axiom | Def 5.1 | §19.5 |
| Def 19.5 | Rank-2 Canonicalization Tensor | Conjecture/Proposal | Def 4.6, Thm 18.1 | §19.13 |
| Def 19.6 | Rank-3 Uncertainty-Aware Canon. Tensor | Conjecture/Proposal | Def 19.5, Def 6.4 | §19.13 |
| Def 19.7 | Rank-5 Coalition-Parsing Canon. Tensor | Conjecture/Proposal | Def 19.5, Def 14.3 | §19.13 |
| Def 19.8 | Rank-4 Temporal Canon. Tensor | Conjecture/Proposal | Def 19.5, Def 10.3 | §19.13 |
| Prop 19.1 | NCSG Well-Posedness | Conditional Theorem | Def 19.1 | §18 TA |
| Prop 19.2 | Compilation Residue | Conditional Theorem | Defs 18.1, 14.1 | §18 TA |
| Prop 19.3 | Democratic Aggregation | Conditional Theorem | Def 19.1 | §18 TA |
| Prop 19.4 | Bond Index Convergence | Conditional Theorem | Def 19.1, BIP | §18 TA |
| Prop 19.5 | Gauge-Fiber Severance | Conditional Theorem | Def 19.5 | §19.13 |
| Prop 20.1 | Curvature Detection Criterion | Conjecture/Proposal | Def 10.3 | §19 TA |
| Prop 20.2 | Torsion Detection Criterion | Conjecture/Proposal | Def 4.4 | §19 TA |
| Prop 20.3 | Meta-Metric Existence | Conjecture/Proposal | Def 9.1 | §19 TA |