100-500 years from now is enough time for AI to develop small nuclear and/or laser sail micro spacecraft that can colonize other systems at a good fraction of the speed of light.
Which would leave 4000 years or so to colonize every system with 1000 light years.
In each system collecting and engineering with low gravity material (asteroids, centaurs, smaller moons, etc.) and using that to capture solar energy will get routine.
Gas giants are basically massive hydrogen nuclear energy depots.
Earth and even our solar system don’t form any kind limit to resource growth on time scales like that.
Which would leave 4000 years or so to colonize every system with 1000 light years.
In each system collecting and engineering with low gravity material (asteroids, centaurs, smaller moons, etc.) and using that to capture solar energy will get routine.
Gas giants are basically massive hydrogen nuclear energy depots.
Earth and even our solar system don’t form any kind limit to resource growth on time scales like that.