Mathematical Considerations for AI Alignment

Chinese Edition: 关于人工智能对齐的数学思考

AI alignment is commonly understood as making AI systems behave consistently with human intentions, values, preferences, and constraints. Yet before discussing possible remedies, we must first understand why alignment is mathematically difficult. The same properties that make artificial intelligence powerful, its ability to infer structure through an ill-posed inverse problem, to learn through propertyless numerical representations, and to preserve high-dimensional relational freedom through full-rank transformations, also make human values difficult to specify, represent, and separate. This article therefore makes no attempt to propose a solution. Only after we understand the mathematics of the problem can possible remedies be meaningfully considered.

What Must Be Aligned?

At a high level, alignment concerns what the system is trying to achieve, how it interprets human instructions, how it uses its learned capabilities, and how it behaves in the world. Its goals, outputs, decisions, and actions are expected to remain consistent with physical reality, legal requirements, safety constraints, institutional policies, social norms, and human values. These layers are not identical. Physical laws constrain what is possible, laws and policies constrain what is permitted, and social norms and human intentions constrain what is considered appropriate.

Alignment for an Ill-Posed Inverse Problem

AI systems are not given a complete description of acceptable behavior. They infer it from examples, instructions, corrections, rankings, policies, and observed consequences. This inverse-learning ability is a major source of AI’s power because it allows the system to discover structures that were never fully written down. But the same process makes alignment ill-posed. Different learned structures may fit the same feedback, human requirements may conflict, and small changes in context may produce different behavior. We cannot directly control the internal solution; what we can control are the boundary conditions under which learning and inference take place.

Propertyless and “Lawless” Neural Computation

Neural networks operate through numbers, vectors, activations, and transformations. These elements are not intrinsically truthful, safe, fair, harmful, or deceptive. Nor do they inherently obey physical, legal, social, or moral laws. This propertyless and lawless character gives neural networks extraordinary flexibility across domains. But it also means that human values are not built into individual neurons, parameters, or features. They can influence behavior only through learned relationships and imposed boundary conditions.

Full-Rank Relational Structure

From a category-theoretic perspective, neural networks learn relationships, transformations, and compositions rather than isolated objects. Full-rank transformations preserve many independent relational directions, allowing knowledge and capabilities to remain densely connected and recompilable. This connectedness is another source of AI’s power. But helpful and harmful behaviors may share the same representations, pathways, and relationships. Full rank preserves relational freedom; it does not provide moral separation.

• What is Deep Manifold ?

• Neural Network Fixed-Point Field

• Deep Manifold in the Real World

Single Token Geometry Series 单标几何系列

• Single Token Geometry 01: Topology 单标的几何：拓扑

• Single Token Geometry 02: DeepSeek V4 and Manifold Tearing 单标几何：DeepSeek V4 与流形撕裂

• Single Token Geometry 03: Data Complexity 单标几何：数据复杂性

• Single Token Geometry 04: A Critique of Manifold Steering 单标几何：对流形引导的批评

• Single Token Geometry 05: Numerical Manifold Method 单标几何 ：数值流形

• Single Token Geometry 06: Stacked Piecewise Manifold 单标几何 06：堆叠分片流形

• Single Token Geometry 07: Attention 单标几何 07:注意力

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