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 space

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• 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 spacequote0.833
  Caption of Figure 1 summarizing the central empirical finding of the paper
• 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 pointclaim0.829
  Formal proposition establishing that fixed-point convergence implies cyclic fixed points for all block permutations
• Convergence to cyclic fixed points implies attention patterns stabilize, which implies mixing-based stages of inference become constant across recurrencesclaim0.795
  Core mechanistic claim linking fixed point theory to observable inference stage behavior
• 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 modelsclaim0.789
  Strong claim that inference stage structure is architectural rather than learned
• Predictable stages of inference in looped models offer actionable pathways for efficient architectural design including stage-dependent attention sparsification and leaner middle-stage MLP parameterizationclaim0.775
  Practical design implication of the paper's mechanistic findings
• Randomly initialized (untrained) models exhibit the same cyclic fixed-point behavior as their trained counterpartsfinding0.766
  Suggests cyclic behavior is emergent from transformer architecture itself, not learned during training
• Cyclic Fixed Pointconcept0.755
  The key theoretical contribution: each layer in a cyclic recurrence converges to a distinct fixed point, tracing a consistent trajectory in latent space
• Looped models learn stages of inference that closely mirror those of feedforward models, repeating these stages in depth with each iterationclaim0.755
  Central empirical claim of the paper supported by ColSum concentration analysis across multiple architectures