Convergence to cyclic fixed points implies attention patterns stabilize, which implies mixing-based stages of inference become constant across recurrences

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• 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 modelsclaim0.815
  Strong claim that inference stage structure is architectural rather than learned
• 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 spaceclaim0.795
  Theoretical framing that establishes cyclic fixed points as the meaningful limiting behavior
• 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 pointclaim0.779
  Formal proposition establishing that fixed-point convergence implies cyclic fixed points for all block permutations
• Fixed-point stability correlates with out-of-distribution performance when extrapolating to unseen numbers of test-time recurrencesclaim0.776
  Practical implication connecting mechanistic analysis to performance benchmarks
• Convergence on constraints across substrates happens because each develops cognitive glue satisfying the same coordination properties.claim0.772
• Cyclic Fixed Pointconcept0.769
  The key theoretical contribution: each layer in a cyclic recurrence converges to a distinct fixed point, tracing a consistent trajectory in latent space
• Huginn-0125 all-layer fixed point cosine similarities converge to 1 for all pairs, indicating convergence to the same fixed point rather than distinct cyclic fixed pointsfinding0.765
  Distinguishes Huginn's convergence behavior from the ideal cyclic fixed point behavior
• Randomly initialized (untrained) models exhibit the same cyclic fixed-point behavior as their trained counterpartsfinding0.763
  Suggests cyclic behavior is emergent from transformer architecture itself, not learned during training