Riding the Fixed-Point Curve: Image Classification in the Continuous Thought Machine

The Continuous Thought Machine (CTM) is a recurrent network that attends to inputs and makes predictions based on the synchronization between its neurons. On several tasks and domains, it displays emergent thought processes that have not been explained. We give an account of how the CTM classifies images, focusing on ImageNet. Each thinking step is a spectrally contractive but strongly non-normal map exhibiting balanced amplification (Murphy & Miller, 2009). By Banach’s theorem the neural state has a single attracting fixed point, which moves at every step, and the thought trajectory is the curve these fixed points trace, followed one tick behind. Notably, what drives that curve is a preset, largely image-independent schedule of where to look — regular enough that the next glance can be predicted before the model takes it, and visible as a previously unexplained traveling wave of activity across neurons. The prediction sharpens along the way, because the syn- chronization memory accumulates and lowers the entropy landscape itself, while the state’s motion stays orthogonal to the uncertainty gradient.