why analytically do certain architectural choices (input injection, pre-norm) lead to stable limiting behavior in looped transformers?

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• is stable fixed-point limiting behavior desirable or restrictive for reasoning tasks in looped transformers?question0.815
  Open question about whether convergence to fixed points helps or hurts reasoning performance
• Input injection encourages fixed-point convergence in looped transformershypothesis0.815
  Hypothesis replicated from Bansal et al. and Anil et al. and further investigated with norm ablations
• Pre-norm without input injection reaches a degenerate fixed point where all layers converge to the same pointfinding0.793
  Pre-norm model reaches a fixed point without input injection but all layers converge to identical representations
• Input injection produces stable fixed point behavior for all norm types tested except Ouro norm on randomly initialized modelsfinding0.779
  Replicates and extends prior findings on input injection; tested on randomly initialized 12-layer models across three norm structures
• what is the full classification and explanation of orbit and slider non-fixed-point limiting behaviors in looped transformers?question0.762
  Left to future work after demonstrating these behaviors are rare but not explaining their mechanism
• Transformers almost surely maintain input-injectivity throughout training, not just at initialisationhypothesis0.757
  Conjecture supported by Nikolaou et al. 2025 for last-token hidden states
• Small-scale looped transformers trained from scratch with constant 4-recurrence schedule and simplified loss self-organize into multiple distinct mixing stages mirroring feedforward modelsfinding0.755
  Evidence that stages of inference emerge without training biases from retrofitting, recurrence scheduling, or multi-recurrence losses
• Different introspective tasks may preferentially use different path distributions in the transformer.claim0.744
  Interpretive claim connecting exponential path combinatorics to Lindsey's layer-dependent findings.