## Problem 049 — rank:268

**Title:** Functional role of gene-expression noise

**Added binding assumptions (LaTeX):**

```latex
State whether all cells share one stationary ergodic environmental path or experience independent environments. Under the declared boundedness and irreducibility assumptions, define $\Lambda$ as the almost-sure top growth exponent of that random linear system; otherwise use finite-horizon growth. Charge the circuit cost once: subtract $c(\sigma)$ only when it is not already included in $r_i(e)$.
```

**Direct edits:** 0. **Fresh external checks:** None; supplied sources retained as provenance.