Mathematical Considerations for AI Alignment

English Edition: Mathematical Considerations for AI Alignment

AI alignment is commonly understood as making the behavior of an AI system consistent with human intentions, values, preferences, and constraints. However, before discussing possible remedies, we must first understand why alignment is mathematically so difficult. The very properties that make AI powerful — the ability to infer structure through ill-posed inverse problems, to learn from attribute-free numerical representations, and to preserve high-dimensional relational freedom through full-rank transformations — also make human values difficult to specify precisely, represent accurately, and separate cleanly. Therefore, this article does not attempt to propose a solution for AI alignment. Only after understanding the mathematical nature of this problem can we meaningfully consider possible remedies.

What Must Be Aligned?

At a high level, alignment involves what the system aims to achieve, how it understands human instructions, how it uses the capabilities it has already learned, and how it behaves and performs in the real world. The system’s goals, outputs, decisions, and actions should be consistent with physical reality, legal requirements, safety constraints, institutional policies, social norms, and human values.

These levels are not the same. Physical laws constrain what is possible, laws and policies constrain what is allowed, while social norms and human intentions constrain what is considered appropriate.

Alignment as an Ill-Posed Inverse Problem

AI systems do not receive a complete description of acceptable behavior. They can only infer from examples, instructions, corrections, rankings, policies, and observable consequences. This ability to learn in reverse is an important source of AI’s power, enabling systems to discover structures that have never been fully written down. But this same process also makes alignment an ill-posed problem. Different learned structures may be equally consistent with the feedback received, human-provided requirements may conflict with each other, and small changes in context can lead to different behaviors. We cannot directly control the solution formed inside the system; what we can control are the boundary conditions under which learning and reasoning occur.

Attribute-Free and “Lawless” Neural Computation

Neural networks operate through numbers, vectors, activations, and transformations. These numerical elements themselves have no inherent truth, safety, fairness, harmfulness, or deceitfulness, nor do they naturally obey physical, legal, social, or moral laws. It is precisely this attribute-free and “lawless” nature that gives neural networks extraordinary flexibility across domains. But it also means that human values are not built into any neuron, parameter, or feature. They can influence system behavior only through learned relationships and externally imposed boundary conditions.

Full-Rank Relational Structures

From the perspective of category theory, neural networks do not learn isolated objects but rather relations, transformations, and compositions between objects. Full-rank transformations preserve multiple independent relational directions, allowing knowledge and capabilities to remain highly connected and recombined across different contexts. This connectivity is another source of AI’s power.

But beneficial behavior and harmful behavior may share the same representations, pathways, and relationships. Full-rank preservation preserves relational degrees of freedom, not moral separation.

• What is Deep Manifold ?

• Neural Network Fixed-Point Field

• Deep Manifold in the Real World

Single Token Geometry Series

• Single Token Geometry 01: Topology
• Single Token Geometry 02: DeepSeek V4 and Manifold Tearing
• Single Token Geometry 03: Data Complexity
• Single Token Geometry 04: A Critique of Manifold Steering
• Single Token Geometry 05: Numerical Manifold Method
• Single Token Geometry 06: Stacked Piecewise Manifold
• Single Token Geometry 07: Attention

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