Fractal basins trap latent reasoning

Authors: The fetched body omits an author list; attribution is inferred from the acknowledgments ("W.G." — NSF DMS 2436233, NSF CMMI 2440490), the code repo (github.com/GilpinLab/loopscape), and the UT Austin computing facilities named (TACC, Biomedical Research Computing Facility). arXiv: 2609.04963v1, September 2026.

Reasoning models are widely observed to reason longer on harder tasks, but the general mechanism behind these slowdowns — "overthinking" — was unknown. This paper supplies one: reasoning models are dynamical systems whose basins of attraction become fractal as task difficulty rises, and whose slowdowns are the signature of transient chaos near saddle points.

The key move is a new probe. Treat the model and problem instance as an autonomous discrete-time dynamical system acting on the latent state, then vary the initial latent state along random 2-D slices and label each initialization by its convergence time. On hard tasks (Sudoku, mazes, Countdown, integer linear systems) these convergence-time maps are self-similar fractals, quantified by basin entropy; fractality tracks task difficulty across three distinct models (EqR, FPRM, Parcae) and architectures.

The mechanism: saddle points — weakly-unstable, nearly-correct attempted solutions (a Sudoku grid with repeated digits, a maze dead end) — scatter trajectories and delay convergence. Decoding slow trajectories shows they get trapped near almost-right answers before backtracking. The saddles are localized with the fast Lyapunov indicator, a metric borrowed from asteroid-orbit stability analysis.

The deepest result is a training experiment. A miniaturized looped transformer trained on integer linear systems starts with many stable incorrect fixed points; at a bifurcation (weights as control parameter) the incorrect solutions lose stability and become saddles, solvability and basin entropy jump together, and transient chaos appears — precisely when the model acquires the multi-step reasoning needed to escape incorrect solutions. Only the "core" variables (requiring Gaussian elimination, not direct substitution) produce positive Lyapunov exponents.

The conclusion: reasoning slowdowns are an inevitable consequence of problem hardness, not a training bug to be patched — and reasoning traces form a rich new class of dynamical system.