Proposition 4.1: If a (l,k)-Recurrent block reaches a fixed point S_k(X')=X', then any cyclic permutation of blocks 1,...,k will also have reached a (different) fixed point

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• If a looped model with cyclic recurrence reaches a fixed point, either each block's contribution vanishes asymptotically or the sequential application traces a constant cyclic trajectory in latent spaceclaim0.829
  Theoretical framing that establishes cyclic fixed points as the meaningful limiting behavior
• latent states after each block in a recurrent model frequently tend towards separate fixed points, meaning that the application of a recurrent block tends towards a consistent trajectory in latent spacequote0.810
  Caption of Figure 1 summarizing the central empirical finding of the paper
• Convergence to cyclic fixed points implies attention patterns stabilize, which implies mixing-based stages of inference become constant across recurrencesclaim0.779
  Core mechanistic claim linking fixed point theory to observable inference stage behavior
• Cyclic Fixed Pointconcept0.764
  The key theoretical contribution: each layer in a cyclic recurrence converges to a distinct fixed point, tracing a consistent trajectory in latent space
• Cyclic fixed point behavior and corresponding stages of inference appear to be emergent from the Transformer architecture itself, arising in both trained and randomly initialized modelsclaim0.756
  Strong claim that inference stage structure is architectural rather than learned
• Fixed-point stability correlates with out-of-distribution performance when extrapolating to unseen numbers of test-time recurrencesclaim0.734
  Practical implication connecting mechanistic analysis to performance benchmarks
• Ouro 1.4B does not converge to a fixed point even after 128 recurrences, despite showing small successive differencesfinding0.732
  Key negative result showing that not all looped models reach a true fixed point, contrasting with retrofitted models
• Retrofitted Llama and OLMo recurrent blocks closely follow their respective base model stages of inference while repeating middle stages in the recurrent blockfinding0.731
  Shows that retrofitting preserves base model inference stage structure in the cyclic blocks