It should be given when you cross each stage, but if you got to a stage while it was bugged (couple of patches ago) you may need to wait for it to appear naturally.

Or you can give yourself these techs:

tech_dyson_sphere_entry_1
tech_dyson_sphere_entry_2
tech_dyson_sphere_entry_3
tech_dyson_sphere_entry_4

Is there a way to trigger the event to proceed rather than just getting the tech? I got the first tech but the event ceased to progress after researching it, and I assume colonization of the sphere is the end of the event, considering the habitability rating, rather than 2 trivial techs with minor buffs.

If it's not proceeding and you have the right tech, then you probably don't have any ships inside?

The tooltip should explain when you hover over the progress rate box.

I have no progress box or ability to send ships inside. Does it proceed like a cluster exploration? I researched the first tech and absolutely nothing happened, no triggered message or additional abilities.

How can you have the situation and not have a progress box?

Or do you not have a situation for it? If so, run the edict that triggers it.

Ah I see now, didn't realize there was an edict attached to the tech, it doesn't have a description attached to show what it unlocks for me so I assumed it would trigger an event upon researching it. Appreciated!

It's a fair point. I'll review the flow when we add some more exploration events for it.