Can a shared-weight recurrent Transformer develop distinct internal roles without being partitioned into separate modules? We study this in Asymmetric Input Recurrence (AIR), a minimal two-state reasoning architecture in which the same Transformer model is reused for both updates (per literature, L and H) and the only built-in difference in the update rule is that the encoded input is injected during L-updates but not H-updates. Across Sudoku-Extreme and Maze, decoded rollouts reveal a consistent split: $\zH$ behaves like a fully committed proposal state, whereas $\zL$ retains local uncertainty and shifting intermediate structure. Freeze experiments show that this split is, in practice, related to the model's state dynamics: in Sudoku, freezing $\zH$ reduces $\zL$'s content changes whereas freezing $\zL$ increases $\zH$'s, while in Maze, freezing either state increases content changes in the other state. Ablations show that to induce specialization, the shared model needs to be able to tell the two update types apart, either from input injection asymmetry or from a separate level token. Mechanistically, attention analysis shows that L-updates are consistently more local than H-updates in both Sudoku and Maze. Together, these results show that, in a two-state recurrent setting, a clear state-identity signal can induce stable, related functional roles inside a shared-parameter recurrent Transformer. Code is available at \href{https://github.com/juchengshen/air}{\textcolor{blue}{https://github.com/juchengshen/air}}.