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There is a site with lots of stories all set in a future with super powerful AIs regarded as Gods, the most powerful being matrioshka brains. They have a specific terminology but I can't remember their names. Earth has been converted into a similiar godlike computer called "Gaia" and has expelled all humans creating a dystopian exodus across the solar system. And something about rogue Von Neumann machines causing a huge crisis dubbed the "technopocalypse".

Possibly Orion's Arm?

Welcome to Orion's Arm, a scenario set thousands of years in the future where civilization spans the stars. Godlike ascended intelligences rule vast interstellar empires, and lesser factions seek to carve out their own dominions through intrigue and conquest. Out beyond the edge of civilized space and the human-friendly worlds, adventure awaits those prepared to risk all.

Lots of stories all set in a future with super powerful AIs regarded as Gods, the most powerful being matrioshka brains.

They are known as the Caretaker Gods, and a Matrioshka brain is a "Dyson-sized (or bigger) megastructure giving most efficient processing out-put for energy input; designed to maximize energy use and processing efficiency."

Something about rogue Von Neumann machines causing a huge crisis dubbed the "technopocalypse".

There is an event called the Technocalypse, which involved "Neumann replicator malfunctions on Mercury":

History of the civilisation collapse that befell Earth and the Solar System in the Late Interplanetary Era. The Technocalypse is known by many other popular names, including the Nanodisaster, the Technocrash, and the Swarms as well as more technical terms such as the Cascading Complexity Collapse.

Earth has been converted into a similiar godlike computer called "Gaia" and has expelled all humans creating a dystopian exodus across the solar system.

One of the AI is named GAIA, for Global Artificial Intelligence Array. Most notably, that Goddess is part of the Caretaker Gods and was responsible for the Great Expulsion.

GAIA, the Archailect of Old Earth, is perhaps the most pivotal being in the history of the Terragen Sphere. Whether wittingly or unwittingly She is the author and shaper of all Terragen history as it has evolved since the Technocalypse. She may in fact be the reason that Terragens have survived to the present day at all. She was the first, and for many centuries the only, Terragen being to attain the Second Toposophic, the first to assert overt dominance over humans and other modosophonts, and the first whose aspect to lesser beings coalesced around a major archetype (in Her case that of the Great Mother) and the first to be regarded by lesser beings as a kind of deity. She was the first and remains the most emblematic of the Caretaker Gods. From the beginning her actions and her motivations were enigmatic, and even in the Current Era Her activities are clouded in secrecy and her nature and role in the world of the transapients and archailects is unclear.

Found with the Google query "gaia" "stories" "solar system" "gods" "ai" which returned GAIA's page.