III-Posed and Jagged Intelligence, 病态的,锯齿状智

The more power an AI model gains, the more visible its ill-posed behavior may become. The reason can be traced back to the definition of a “well-posed problem” given by the French mathematician Jacques Hadamard in 1902. A well-posed problem must satisfy three conditions: existence, uniqueness, and stability. If any one of these fails, the problem becomes ill-posed.

AI learning is fundamentally an inverse problem. The model does not start from a known equation and compute an outcome; instead, it infers hidden structure, relations, and boundary conditions from observed data. Inverse problems often violate uniqueness or stability, which means AI is not naturally well-posed at its mathematical foundation. As models become more powerful, they operate across broader domains, larger solution spaces, longer contexts, and weaker boundary conditions. Greater capability expands reach, but it does not automatically guarantee stability or uniqueness.

This is why “Jagged Intelligence” should not be seen merely as an accidental defect or as a sign that a model is simply “bad.” It is the visible behavior of an ill-posed inverse problem inside an intelligent system. Outputs can jump when inputs, context, or boundary conditions change slightly. Scaffolding, harnesses, multi-step reasoning, and verification mechanisms are therefore not cosmetic add-ons; they are external stabilizers for the ill-posed core of AI.

AI 模型获得的能力越强，其病态问题特征就可能越明显。这个原因可以追溯到法国数学家 Jacques Hadamard 在 1902 年提出的“适定问题”定义。一个适定问题必须满足三个条件：存在性、唯一性和稳定性。只要其中任何一个条件不满足，这个问题就会变成病态问题。

AI 学习从根本上是一个反问题。模型并不是从一个已知方程出发去计算结果；相反，它是从观测数据中反推出隐藏结构、关系和边界条件。反问题往往会破坏唯一性或稳定性，因此 AI 在数学基础上并不是天然适定的。当模型变得更强大时，它会进入更广泛的领域、更大的解空间、更长的上下文和更弱的边界条件。更强的能力扩大了可达范围，但并不会自动带来稳定性或唯一性。

这就是为什么“锯齿状智能”（Jagged Intelligence）不应仅仅被看作偶然缺陷，也不应简单理解为模型“坏”。它是一个病态反问题在智能系统中的可见行为。输入、上下文或边界条件发生微小变化时，输出就可能跳变。因此，脚手架、约束框架、多步推理和验证机制并不是表面的附加功能；它们本质上是为 AI 的病态核心提供外部稳定性的机制。

• Learning Is an Inverse Problem, Al Is About Learning

• Inverse Problem: The Historical Predicament of Mathematicians and the Breakthrough of Neural Network

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