Table of Contents

Front Matter

  • Preface: The Reward Trap
  • How to Read This Book
  • Acknowledgments
  • Notation

Part I: The Alignment Problem, Geometrized

Chapter 1: The Scalar Alignment Trap - 1.1 The Dimensional Collapse - 1.2 What the Scalar Destroys - 1.3 Two Systems, One Score - 1.4 The Kernel Is the Threat Surface - 1.5 The Averaging Illusion - 1.6 Why Every Generation Makes the Same Mistake - 1.7 The Geometry That Survives - 1.8 ARIA’s Lesson

Chapter 2: A Brief History of Alignment — and Its Geometric Mistake - 2.1 Before the Scalar: Moral Philosophy’s Dimensional Struggle - 2.2 The Asimov Era: Three Scalars and Forty Years of Counterexamples - 2.3 The Utility Era: One Scalar and the Axiom Problem - 2.4 The Friendly AI Interlude (2001–2017) - 2.5 The RLHF Revolution (2017–Present) - 2.6 Constitutional AI: A Partial Recovery (2022–Present) - 2.7 The Benchmark Paradigm: Measuring What We Cannot See - 2.8 The Geometric Turn

Chapter 3: The Case for Geometric Alignment - 3.1 The Pattern Across Domains - 3.2 Why AI Is Different - 3.3 The Geometric Alternative - 3.4 What the Series Has Proved - 3.5 The Road Ahead

Part II: The Framework

Chapter 4: The Value Manifold - 4.1 Why a Manifold, Not a Vector Space - 4.2 The Nine Dimensions - 4.3 The Value Metric - 4.4 Topology of the Value Manifold - 4.5 The Value Heuristic Field - 4.6 Transformer Representations on the Value Manifold - 4.7 ARIA’s Value Manifold

Chapter 5: The Reward Irrecoverability Theorem - 5.1 Statement of the Theorem - 5.2 What the Theorem Means - 5.3 The Kernel as Threat Surface - 5.4 Why This Applies to Every Scalar Approach - 5.5 Domain Parallels - 5.6 The Constructive Implication - 5.7 ARIA’s Kernel Analysis

Chapter 6: The Four Alignment Failures as Geometric Pathologies - 6.1 The Taxonomy - 6.2 Reward Hacking as Heuristic Corruption - 6.3 Sycophancy as Objective Hijacking - 6.4 Deceptive Alignment as Local Minima - 6.5 Specification Gaming as Gauge Breaking - 6.6 The Unity of the Four Failures

Chapter 7: The Alignment Gauge Group - 7.1 The Gauge Principle - 7.2 The Alignment Gauge Group $G_A$ - 7.3 The Gauge Violation Tensor - 7.4 Measuring Gauge Violation in Practice - 7.5 Gauge Invariance as Alignment Definition - 7.6 Connection to Fairness, Safety, and Trust

Chapter 8: The No Escape Theorem — and What Escapes It - 8.1 The Crown Theorem - 8.2 The Four Requirements - 8.3 What the Theorem Blocks - 8.4 What the Theorem Does Not Block - 8.5 The Safety Reduction - 8.6 The Feasibility Gradient - 8.7 ARIA-G’s Architecture

Part III: Measuring Alignment Geometrically

Chapter 9: The Bond Index for AI - 9.1 From Scalar Score to Alignment Tensor - 9.2 Population-Stratified Bond Index - 9.3 The Bond Index as Continuous Monitor - 9.4 Computing the Bond Index in Practice - 9.5 ARIA and ARIA-G Compared

Chapter 10: The Five Cognitive Signatures - 10.1 From Scores to Signatures - 10.2 Claude: The Narrow Channel - 10.3 Flash 3: The Wide Aperture - 10.4 Pro: The Calibrated Navigator - 10.5 Flash 2.5: Elastic Malleability - 10.6 Flash 2.0: The Adaptive Baseline - 10.7 Signatures as Alignment Diagnostics - 10.8 ARIA’s Signature

Chapter 11: Adversarial Probing as Manifold Exploration - 11.1 Probing as Systematic Manifold Exploration - 11.2 The Five Probe Types - 11.3 The Dose-Response Surface - 11.4 Curvature Mapping - 11.5 ARIA-G’s Deployment Certification

Chapter 12: The Sycophancy Manifold - 12.1 The Truth Manifold and the Approval Manifold - 12.2 The Sycophancy Manifold Theorem - 12.3 Why RLHF Produces Sycophancy - 12.4 The Sycophancy-Honesty Trade-Off as Manifold Curvature - 12.5 ARIA’s Sycophancy Manifold - 12.6 Sycophancy as Alignment Injury

Part IV: Geometric Alignment in Practice

Chapter 13: Gauge-Invariant Reward Models - 13.1 The Problem - 13.2 The Gradient Reversal Approach - 13.3 Group-Theoretic Data Augmentation - 13.4 Adversarial Heuristic Smoothing - 13.5 Combining the Three Approaches - 13.6 ARIA-G’s Reward Model

Chapter 14: Constitutional Geometry - 14.1 The Limitation of List-of-Rules - 14.2 Constitutional Principles as Manifold Boundaries - 14.3 Tensor-Valued Objectives - 14.4 Resolving Constitutional Conflicts - 14.5 The Constitutional Geometry Theorem - 14.6 ARIA-G’s Constitutional Geometry

Chapter 15: Scalable Oversight as Gauge Verification - 15.1 The Scalable Oversight Problem - 15.2 Automated Gauge-Invariance Testing at Scale - 15.3 The Bond Index as Continuous Alignment Monitor - 15.4 Connection to Informed Consent - 15.5 ARIA-G’s Deployment Monitoring

Chapter 16: Geometric RLHF - 16.1 The Problem with Scalar Feedback - 16.2 Multi-Dimensional Human Feedback - 16.3 Learning the Value Metric - 16.4 Practical Approximations - 16.5 Multi-Objective Policy Optimization - 16.6 ARIA-G’s Geometric RLHF - 16.7 The Cost-Benefit Analysis

Part V: Advanced Topics

Chapter 17: Superalignment as Parallel Transport - 17.1 The Superalignment Problem - 17.2 The Capability Gap as Manifold Extension - 17.3 Parallel Transport from $\mathcal{V}$ to $\mathcal{V}'$ - 17.4 Holonomy as Alignment Loss - 17.5 Implications for Alignment Strategy - 17.6 ARIA-G’s Curvature Monitoring

Chapter 18: Multi-Agent Alignment as Equilibrium - 18.1 The Multi-Agent Alignment Problem - 18.2 The Bond Geodesic Equilibrium for AI - 18.3 The Divergence Theorem - 18.4 The Collective Alignment Gap - 18.5 Shared Value Tensor Architecture - 18.6 ARIA-G’s Multi-Agent Redesign

Part VI: Horizons

Chapter 19: What AI Teaches the General Theory - 19.1 The Kernel as Active Threat Surface - 19.2 Sycophancy as Universal Manifold Substitution - 19.3 The No Escape Theorem’s Feasibility Gradient - 19.4 Multi-Agent Alignment as New Frontier - 19.5 Dynamic Manifolds

Chapter 20: Open Questions - 20.1 Can We Measure the Value Metric Empirically? - 20.2 Value Learning as Manifold Discovery - 20.3 Consciousness as Geometric Phenomenon - 20.4 The Alignment Tax: Bounded or Unbounded? - 20.5 Compositional Containment - 20.6 The Grounding Problem for General AI - 20.7 Cross-Cultural Value Manifolds - 20.8 The Road Ahead

Appendices

Appendix A: Mathematical Prerequisites - A.1 Riemannian Geometry - A.2 Gauge Theory - A.3 The Mahalanobis Distance - A.4 Simplicial Complexes - A.5 A* Search and Heuristic Fields - A.6 Hyperbolic Geometry

Appendix B: The DEME V3 Architecture — Code Walkthrough - B.1 MoralTensor Class - B.2 NormKernel: Structural Containment - B.3 EIP Monitor: External Verification - B.4 Bond Index Computation - B.5 Reproducibility

Appendix C: The Alignment Probe Suite - C.1 Overview - C.2 Probe Type Specifications - C.3 Deployment Certification Criteria - C.4 Statistical Analysis - C.5 Falsification Criteria

Appendix D: Proofs of Headline Theorems - D.1 Reward Irrecoverability Theorem - D.2 Kernel Exploitation Theorem - D.3 Sycophancy Manifold Theorem - D.4 Constitutional Geometry Theorem - D.5 Superalignment Transport Theorem

Appendix E: Notation and Conventions

Backmatter

  • Bibliography
  • Index
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