Looped transformers reuse a single block across iterations, raising a natural question: how far can a block trained at one depth be safely extrapolated to greater depths?
We study this question for all-pairs graph connectivity, where harder graphs require longer-range information propagation.
We show that under exchangeability and symmetric positive initialization, population training at depth K converges to an update rule governed by a single effective parameter that balances graph propagation against fitting structure unresolved at training depth.
This trade-off creates a finite extrapolation window: too few test-time loops provide insufficient propagation, while too many eventually create false connections.
We derive explicit endpoints for this window and bound the graph hardness solvable beyond training as a function of training depth and hardness.
In particular, a test depth chosen solely from the training distribution can guarantee recovery on substantially harder graphs, with a certified depth gain that grows exponentially as training depth outpaces training hardness.
Experiments validate the predicted windows and show how compute budget and initialization affect the optimal training depth.