A continuous thought machine does not search per image: it rides one shared fixed-point curve, and its attention schedule is nearly generic

When the Continuous Thought Machine (CTM), a recurrent network that predicts from the synchronization between its neurons, classifies ImageNet images, its internal state does not take a separate path per image; it follows an attracting fixed-point curve that moves with each tick, and most of that curve's motion is shared across images. The CTM was known to show unexplained emergent behavior across several tasks, including low-frequency traveling waves across its neurons, but what kind of object its internal trajectory is had not been identified. With the official ImageNet checkpoint and no further training, freezing the memory turns one tick into an ordinary map; iterating it converges to a fixed point at most image-tick pairs, and the Jacobian there has a median spectral radius well below one, with a maximum of about 0.9, so each tick has a locally unique attracting fixed point.

The curve moves because the memory updates at every tick, so the target keeps changing and the state trails it by one update rather than sitting on it. With the memory held fixed, a tick is an ordinary map whose state converges to a unique attracting fixed point; in actual inference the memory changes each tick, so the fixed point is redefined continuously. The mean distance from the state to the previous tick's fixed point is about seven times smaller than to the current one (about 0.05 versus 0.35 over several images), and the offset points along the fixed point's own motion with a length of about one full step.

Where the model looks is essentially a schedule shared across images, and the image itself only decides which features that schedule lands on. The attention trajectory was previously described as emerging through training with visually diverse, multiscale dynamics, without separating how much of it is an image-specific search. Handing each image another image's query at the matching tick costs only about 7% of accuracy; substituting the population-average query still recovers the image's own attention map at cosine about 0.99, and the current query plus the shared average increment forecasts the next attention map at cosine about 0.9 on held-out images.

Class information rides the curve from the first tick, and classes do not occupy different regions of the curve but sit displaced sideways along the same one. Because every trajectory starts from the same learned initial state, class identity must enter the curve along the way, and where and how it enters had not been located. On a sample of validation images per class, a linear probe decodes class from the fixed point at the first tick above the state and the readout memory, and above the mean of the image's feature bank; the shared curve carries about 97% of the variance, class offsets about 2%, and almost all of the offsets' variance lies outside the curve's leading two-dimensional plane.

The results come from the official ImageNet checkpoint without further training, evaluated on a subsample of the validation set; apart from the class-identity analysis, which uses its own sample of images per class, all ImageNet results are on that subsample or subsets of it. The subject is image classification, and the method locates and linearizes fixed points by treating one tick as an ordinary map with the memory frozen. A reader interested in other tasks or checkpoints would need separate validation; the source proposes no next experiment.

Fixed-point iteration failed to settle within the given number of steps at about 10% of image-tick pairs, which have no fixed point to analyze and are excluded from the spectral statistics. Non-normality is reported as the median of the per-fixed-point ratio, which is right-skewed across ticks and therefore exceeds the ratio of the marginal medians. The functional role of balanced amplification is left open, and no coupling above a permutation null was found between the traveling waves and the attention schedule. The results are limited to one checkpoint, a subsample of validation images, and image classification.