Four researchers have published a mathematical theory that formally derives "slow thinking" processes in large language models, offering a first-principles approach to understanding and designing AI systems that deliberate before responding.

The paper, published in the Journal of Machine Learning, introduces "active lifting" — a framework based on sampling latent sequences with an intrinsic drive to reduce uncertainty at maximum rate. The authors include Hongkang Yang, Zhi-Qin John Xu, Feiyu Xiong, and Weinan E.

The theory positions slow thinking models within what the researchers call a "static theory" that creates representation and sampler hierarchies. Models can be upgraded by climbing these hierarchies, potentially improving their reasoning capabilities.

Technical Framework and Applications

Active lifting derives an inference process with an internal time axis and a training objective that resembles minimum-length coding. The researchers describe this as similar to "the invention of languages," characterizing how perception develops agency.

The framework produces several technical applications:

  • A three-stage pathway for improving slow thinking models
  • Unified construction methods for encoders and generative models across data modalities
  • Formation of human-like visual representations
  • A potential solution to policy collapse in AI training

The theory encompasses design, training, and inference of slow thinking large language models. It starts with lifting and projecting probability distributions between observable and latent spaces, aiming to represent complex data distributions using simple function families like neural networks.

The research builds on growing interest in AI systems that engage in deliberative reasoning rather than immediate response generation. Companies like OpenAI and Anthropic have explored similar concepts in their recent model releases.

The paper was submitted to arXiv in July 2026 and represents part of a broader series on first-principles modeling of cognitive functions. The work provides mathematical foundations for understanding how AI systems can develop more sophisticated reasoning processes through structured uncertainty reduction.