A bifurcation in the miniaturized looped transformer occurs at ~150,000 training steps, when accurate integer-linear-system solving first emerges

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

• Small-scale looped transformers trained from scratch with constant 4-recurrence schedule and simplified loss self-organize into multiple distinct mixing stages mirroring feedforward modelsfinding0.789
  Evidence that stages of inference emerge without training biases from retrofitting, recurrence scheduling, or multi-recurrence losses
• Only the multistep 'core' variables (not directly-substitutable variables) produce positive FTLE in the trained looped transformerfinding0.769
  Localizes transient chaos to the sub-algorithm requiring multi-step Gaussian elimination.
• Learning to encode position for transformer with continuous dynamical model (Liu et al., 2020)concept0.762
  Prior work on learned dynamic position encodings; cited alongside Wang et al. as precedent.
• Looped transformers trained from scratch without feedforward-biasing training procedures still self-organize into multiple distinct mixing stages resembling feedforward stages of inferenceclaim0.757
  Establishes that stages of inference are beneficial even when repeatedly applied in recurrent depth
• Transformers almost surely maintain input-injectivity throughout training, not just at initialisationhypothesis0.754
  Conjecture supported by Nikolaou et al. 2025 for last-token hidden states
• 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 modelsclaim0.753
  Strong claim that inference stage structure is architectural rather than learned
• Transformers learn in-context by gradient descent, functioning as mesa-optimizers that learn internal models in real timefinding0.745
  Evidence that in-context learning is not mere pattern matching but genuine optimization, relevant to applying the thesis to inference
• Transformers use an anti-Markovian solution that recomputes relevant numeric information at each step in the Multi-Object taskclaim0.742
  Prior finding from Grant et al. 2025 used to interpret low MAS IIA for GRU-Transformer hidden state comparisons.