The context can change each time. To implement a UTM only requires 2 states and 3 symbols or the reverse.
So if you use the context not as a tape in itself, but as an io channel to read/write an infinite tape, the context for any given step can remain fixed in size, and so the step itself can be O(1).
So if you use the context not as a tape in itself, but as an io channel to read/write an infinite tape, the context for any given step can remain fixed in size, and so the step itself can be O(1).