I have some potential answers for those of you wondering about the fate and nature of the people of the Dyson Sphere.
WARNING: Spoilers for a Star Trek Adventures episode follow. (Though I don't think it's an official Modiphius episode.)
It's going to be kind of long because, if you're interested, I want you to be able to really get a feel for it.
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In the STA episode my gaming group played tonight, we encountered a glowing haze with distributed mass of ½ solar mass over a diameter of ¼ AU, composed of trillions of 30cm cubes. A tiny M class star (a tenth the size of Sol) is at the center of this haze, all of which is surrounded by a subspace field. Inside the field, there are time dilation effects, which increase as you approach the center, where it is running approximately 4 million times faster than it is outside the field. The cubes, composed of an unknown material, absorb 95% of the solar radiation and emit it as heat. They're communicating with each other on a range of frequencies. The communications are both patterned and complex enough to indicate that vast amounts of information are being transmitted, faster than the computer can process. The signals are base 8 math, isolinear not binary, and the code is so complex it can’t be interpreted. This is a distributed computational network, like a giant brain.
We isolate the frequency one cube is using and hail on that frequency. Immediately, a number of the cubes begin to move and eventually take the shape of our ship, including navigational lights. The lights flash the first five numbers of the Fibonacci sequence. We respond with the next five. The simulacrum ship's lights flash with the first five prime numbers and we respond with the next five. The lights shut off and the area correlating to where engineering would be starts to take on the environmental conditions required to sustain humanoid life, just with very low gravity (15% of normal).
The away team beams over into a space that roughly corresponds to Engineering, though without any detail like LCARS displays. As soon as we've had a chance to look around, alarms start blaring and Lt. Cmdr. Gar, our Chief Engineer, is trapped in an impenetrable bubble that, we discover, is a manipulated subspace field. She only has ten minutes of air. A console near Lt. Zepht, our Chief Science Officer, has started displaying what appears to be an alien language. After about 8 minutes, Lt. Zepht decodes the language and discovers they are simple logic puzzles. After solving four of these puzzles, Gar is released. Suddenly, a light is emitted from the region of [what would normally be] the warp core and a smell the team associates with sickbay permeates the room. A being is constructed before us, roughly humanoid in appearance. The being says, “I have prepared for this moment. It is good that my preparations were not in vain. You are explorers, yes? We are also explorers. You may refer to me as Mercury. You are now welcome here for a time but that time is short.” Mercury explains that trapping Gar was a test of intelligence and compassion; the alarm was meant to cause stress but they would not have actually allowed harm to come to her. After some very cordial negotiations about correcting their course to avoid destroying a Federation colony and some confusion about the time difference and our level of technology, we get around to a cultural exchange.
Mercury's people once existed outside the time dilation as a biological species from a star system outside the galactic ecliptic plane. They achieved space flight 22 million years ago (our time). There were only four other nearby systems, all resource poor, so they gave up on space travel. However, as technology advanced, they discovered replicators that resulted in cultural revolution. They began dissolving asteroids and moons for minerals and manufacturing orbital habitats. After many centuries, they deconstructed their homeworld, saving as much wildlife as possible, and constructed a Dyson Sphere. It took tens of thousands of years to build. 100 trillion beings lived comfortably there; civilizations rose and fell over millions of years, dominant sentient species (with the same opossum-like common ancestor) rose and fell, swaths of sphere-dwellers even fell into barbarism at times. Eventually, The Enlightened took over. Mercury's people are The Children of The Enlightened. They wanted to explore, so they downloaded their minds into computer systems to extend life indefinitely, took the stellar corpse of a star, and used subspace field manipulation to set off into the galaxy. They constructed four separate “brains” around stars and began their mass exodus into simulated reality. Some members wished to remain behind on the Dyson Sphere and it is feared they have perished. The four "brains" chose different paths; one chose to explore this galaxy, one may be headed to the Andromeda Galaxy, the other two have been gone for millions of years.
Each cube is thousands of individuals. Though they do experience emotions, they don't remember what it was like to be biological. They sometimes take "the best parts" of themself and "the best parts" of another to create a new individual to interact with but this reproduction is for interest rather than from necessity. We share our respective cultural databases and they become enamored with the idea of eating for pleasure and cooking as art, something their species never did.
Eventually, he tells us that his body was created without necessary systems like immune function and says he must depart. The body is deconstructed in front of us and we beam back to our ship with petabytes of star charts and cultural data streaming into our computer.
This story isn't canon but it's a fun way to imagine the builders of the Dyson Sphere and what might have happened to them.