Deep Manifold, Two Years Later

2024-2026

Deep Manifold began not with a grand design, but with an innocent and entirely unplanned phone call. What started as an isolated inquiry into the geometry of a single token gradually expanded into a vast landscape of rolling hills, steep peaks, and deep valleys. I was simply meandering through a forest of piecewise manifolds, captivated by their structural beauty and mathematical precision, only to find myself unexpectedly tracing the wave-like training dynamics and rippling contours of the neural fixed-point field.

It is only the beginning. If we think of the mathematics of AI as an oil painting, we have only laid down the preliminary sketch. If we think of it as a sculpture, we have only built the supporting framework.

• Deep Manifold Part 1: Anatomy of Neural Network Manifold, 2024

• Deep Manifold Part 2: Neural Network Mathematics, 2025

• Deep Manifold Part 3: Neural Fixed Point Field, TBD, 2026

Core

1. What is Deep Manifold ?

2. Neural Network Fixed-Point Field

3. Learning Is an Inverse Problem

4. Early Validations of Deep Manifold

5. Deep Manifold in the Real World

Architecture

1. Mathematical Considerations for Neural Network Architecture

2. Mathematical Considerations for Giant Monolithic Models

3. Mathematical Considerations for Omni-Model Architectures

4. Mathematical Considerations for Open-Ended Foundation Models

5. Mathematical Considerations for Self-Evolving Neural Networks

6. Mathematical Considerations for Manifold Federation

7. The Myth of Random Weight Initialization

Training

1. Mathematical Considerations for Neural Network Learning

2. Mathematical Considerations for Training Progression

3. Mathematical Considerations for Continual Learning

4. Mathematical Considerations for Learning Space

5. Mathematical Considerations for Training Dynamics

6. Mathematical Considerations for Contrastive Learning

7. Mathematical Considerations for Premature Pretraining

8. Mathematical Considerations for Learning Complexity

9. Mathematical Considerations for Transfer Learning through Category Theory

10. The Myth of Neural Network Optimization

Data

1. Dataualism

2. Single Token Geometry 03: Data Complexity

3. Mathematical Considerations for Expert Data Transfer

4. An Expert, They Say

5. Expert Curriculum

Complexity

1. Single Token Geometry 03: Data Complexity

2. Mathematical Considerations for Learning Complexity

Scientific and Engineering Intelligence

1. Mathematical Considerations for AI Scientific Discovery

2. Mathematical Considerations for Scientific and Engineering Intelligence

3. Mathematical Considerations for Neural Network Learning and Discovery

4. Mathematical Considerations for Complexity and Emergence

5. Mathematical Considerations for Learning Complexity

6. An Expert, They Say

7. Expert Curriculum

8. The Comedy of Neural Learning

9. Mathematical Considerations for Causal Inference

10. AI Through the Lens of Yu Deng’s 2026 Fields Medal–Winning Mathematics

Mechanistic Interpretability & Alignment

1. Mathematical Considerations for AI Alignment

2. Mathematical Considerations for Jagged and Ill-Posed Intelligence

3. Mathematical Considerations for Mechanistic Interpretability

4. The Myth of Hidden Space and Compression

5. The Myth of the Attractor

6. The Myth of Superposition, Circuit, Dictionary

7. Anthropic’s Global Workspace

8. Mathematical Considerations for Complexity and Emergence

9. AI Through the Lens of Yu Deng’s 2026 Fields Medal–Winning Mathematics

“Blue-Collar” Precision AI Self-Service Automation

1. An Expert, They Say,

2. Expert Curriculum

3. No Nobel Prize Winner Runs a Company

4. The Significance of MiroFish Through the Lens of Deep Manifold

History

1. Deep Manifold Was Not Planned

2. Mathematicians Are Not Trained This Way

3. Mathematical Considerations for “Scaling”

4. Inverse Problems

5. Zienkiewicz Moment, History Repeating Itself ?

Open Source Model Series

1. Mathematical Considerations for DeepSeek V4

2. Mathematical Considerations for Inkling

3. Mathematical Considerations for Kimi K3

Deep Manifold Reading Series

1. Deep Manifold Interpretation 2026 Q2 Collection

2. Deep Manifold Interpretation 2026 Q3 Collection

3. Anthropic’s Global Workspace

4. Single Token Geometry 04: A Critique of Manifold Steering

5. Single Token Geometry 02: Manifold Tearing

Myth Series

1. The Myth of Hidden Space and Compression

2. The Myth of the Attractor

3. The Myth of Neural Network Optimization

4. The Myth of Random Weight Initialization

5. The Myth of Superposition, Circuit, Dictionary

6. The Myth of Where Dimensionality Resides

7. The Myth of Reasoning

Mathematical Considerations Series

1. Mathematical Considerations for Jagged and Ill-Posed Intelligence

2. Mathematical Considerations for AI Alignment

3. Mathematical Considerations for Mechanistic Interpretability

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4. Mathematical Considerations for Neural Network Architecture

5. Mathematical Considerations for Omni-Model Architectures

6. Mathematical Considerations for Giant Monolithic Models

7. Mathematical Considerations for Manifold Federation

8. Mathematical Considerations for Open-Ended Foundation Models

9. Mathematical Considerations for Self-Evolving Neural Networks

10. Mathematical Considerations for “Scaling”

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11. Mathematical Considerations for Training Progression

12. Mathematical Considerations for Learning Space

13. Mathematical Considerations for Contrastive Learning

14. Mathematical Considerations for Training Dynamics

15. Mathematical Considerations for Continual Learning

16. Mathematical Considerations for Premature Pretraining

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17. Mathematical Considerations for Neural Network Learning

18. Mathematical Considerations for Learning Complexity

19. Mathematical Considerations for Transfer Learning through Category Theory

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20. Mathematical Considerations for AI Scientific Discovery

21. Mathematical Considerations for Scientific and Engineering Intelligence

22. Mathematical Considerations for Neural Network Learning and Discovery

23. Mathematical Considerations for Complexity and Emergence

24. Mathematical Considerations for Causal Inference

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25. Mathematical Considerations for Expert Data Transfer

26. Mathematical Considerations for DeepSeek V4

27. Mathematical Considerations for Inkling

28. Mathematical Considerations for Kimi K3

Single Token Geometry Series

1. Single Token Geometry 01: Topology

2. Single Token Geometry 02: Manifold Tearing

3. Single Token Geometry 03: Data Complexity

4. Single Token Geometry 04: A Critique of Manifold Steering

5. Single Token Geometry 05: Numerical Manifold Method

6. Single Token Geometry 06: Stacked Piecewise Manifold

7. Single Token Geometry 07: Attention

*** Deep Manifold, Two Years Later on X ***

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深度流形的起点并非某种宏大的设计，而是一通单纯且完全出乎意料的电话。最初对单标几何结构的一次孤立探究，逐渐展开为一片由起伏丘陵、陡峭峰峦和幽深谷地构成的广阔景观。我只是漫步于分片流形的森林之中，沉浸于它们的结构之美与数学精确性，却意外地发现，自己正在追踪神经不动点场中波浪般起伏的训练动态与层层荡漾的等高轮廓。

这仅仅是一个开始。如果把AI的数学看成一幅油画, 我们只是打了一个油画的底稿, 如果看成是雕塑, 我们只是做了一个雕塑的框架.

二年的努力, 和石根华 老师一起找到神经网络后面的数学. 这些数学是由中国数学先辈, 江泽涵和陈省身开创性的工作：不动定群, 分片光滑.

深度流形*漫谈 视频

神经网络无意中推进了数学: 和石根华老师漫谈《AI和数学》*** 中国数学前辈们对此的贡献 **** 神经网络本身无意推动了数学

• 国内:https://www.bilibili.com/video/BV1U2j863EsL

• 海外:

• 陈省身的一个疑问，被两个世界同时回答