The Torus Sieve: Dangerous Cells on (Z/3^kZ)²

Each pixel represents a cell (a, b) on the torus (Z/3^kZ)². The cell error a − log₂3·b determines whether the Collatz trajectory gains a large or small power of 2 at that step. Dangerous cells (red/yellow) have cell error near zero — the trajectory stays near the critical line with v₂ ≈ 1. Safe cells (blue/purple) push the trajectory away, giving v₂ ≥ 2. The green trajectory shows the Collatz walk for the chosen starting number. White dots mark continued-fraction convergents of log₂3, which are the hardest points for Baker-type bounds to repel.