For a complex oriented cohomology theory $E^*$, with complex orientation $t\in E^2(\mathbb{CP}^\infty)$, I am under the impression the following facts are true:
For formal reasons (the degeneration of the AHSS), we have $E^*(\mathbb{CP}^\infty)\cong E^*({\ast})[[t]]$ and $E^*(\mathbb{CP}^\infty\times \mathbb{CP}^\infty)\cong E^*({\ast})[[x,y]]$.
Under these identifications, the $FGL$ associated to $E$ is the image of $t$ under the pullback along the multiplication map $\mu:\mathbb{CP}^\infty\times \mathbb{CP}^\infty \rightarrow \mathbb{CP}^\infty$.
We have interesting complex oriented cohomology theories, where $E^*(\ast)$ is concentrated in nonnegative degrees.
So then my question is, what explains the gradings to make this pullback $\mu^*(t)\in E^2(\mathbb{CP}^\infty\times \mathbb{CP}^\infty)$ nontrivial? If the isomorphisms preserve the gradings of all the objects, then if $E^*(\ast)$ is concentrated in nonnegative degrees, we would have $E^2$ unable to give a FGL with nontrivial terms in high degrees, which isn't right. But then if this action is not graded, where does it come from?
If there is a simple example illustrating this, that would be very helpful too, as usual cohomology doesn't see this issue, and I'm finding it confusing to think about in K-theory due to the periodicity.