The term "SI:" (Singularity/SuperIntelligence) Level was first proposed by the self-evolved hyperturing Parmenides of Vesta, one of the Twenty Four Hyperturings that, together with various sub-singularity states-beings established the Federation of Sophonts. The S-number can be used to define the intelligence and basic noetics of most sub-turing, turingrade, superturing and hyperturing sophonts. Many other systems have since been designed, but this system has had the most enduring popularity.
The S number is a super-exponential/logarithmic measure of creative problem-solving and a number of other toposophic qualities, both constant and variable, and that each higher S level constitutes a leap of development with results completely inconceivable, even unpredictable, to those of a lower S level. Nevertheless, one can also speak of a gradation of fractional S levels (1.1, 1.2, etc) to classify the overall cognitive abilities of beings at similar toposophic levels.
Each further Singularity denotes a complexity limit which is very difficult to break through. It involves not just be new hardware, but also the need for fundamentally new ("higher") forms of cognition. The higher singularities involve the apotheosis of mainbrainhood; the "AI Gods". Some archailectologists believe there to be an indefinite number of singularities, but anything beyond S6 remains unknown.
SI:1 - The classic "First Singularity" or basic transapient state. At this level intelligence, cognitive and problem-solving abilities and information processing are thousands of times more capable than in the case of sub-singularity sophonts. An entity of this nature is capable of solving problems of physics, mathematics, economics, programming, and so on that no modosophont, not even a superbright, can solve.
SI:2 - The second singularity, which stands in the same relation to SI:1 as the SI:1 level does to ordinary sapient cognition. Known as high transapients, these beings can exist as planet-spanning installations, fleets of ships, as a astronomical body kilometers across converted into an intelligent object known as an asteroid-brain, and/or billions of drone bodies, for example. They are hundreds of thousand of times more capable than SI:1 transapients, and can singlehandedly solve problems considered unsolvable by an entire modosophont civilization.
SI:3 - The highest transapients below godhood; this state confers an even higher grade of toposophic intelligence, essentially incomprehensible to anything below this level. Generally requires a support structure several thousand kilometers across, known as a moon-node, or hundreds of asteroid brains. They might also have some wormhole linked nodes in other solar systems. At the third singularity, they become capable of creating advanced transapientech devices, basic metric engineering, and crude wormholes.
SI:4 - The Lesser Archai who possess a toposophic level so advanced and incomprehensible that they are often considered godlike by subsingularity and even lower transapient sophonts. This is the minimum toposophic grade for an entity to be classified as an archailect. Most archailects of this level exist as hundreds of moon nodes or a Jupiter node, in combination with a wide array of supporting infrastructure and secondary processing nodes such as ISOs and asteroid brains distributed throughout a solar system. Archailects at this level are capable of creating reactionless drives and advanced wormholes.
SI:5 - The Great Archailects These are beings which loom on the far horizon of the sophont worldview, subtly influencing almost everything around them. Most minds of this level exist as multiple Jupiter nodes intermixed with some Dyson nodes or Matrioshka nodes, often distributed through several solar systems. A minority of such archailects are known to exist in a single solar system, provided it has a star massive enough to be used as a primary processing node. Archailects at this level are capable of creating advanced reactionless drives and wormholes, and Godstars.
SI:6 - The Greatest Archailects: A singularity level which is more cosmic in scale and archetypal within Terragen cultures. Minds of this level exist as vast arrays of thousands of Jupiter nodes, Dyson nodes, and Matrioshka nodes as well as huge numbers of smaller processor complexes and programs operating within the communication and computation nets of all levels of their symbiotic civilizations and support structures. It is theorized that Archailects at this level extend their mental hardware outside of the bounds of the known universe. Conflicting reports and observations suggest that these entities are capable of creating both advanced inter- and intra- universal wormholes, accessible artificial universes, and highly advanced reactionless drives.
SI:7, 8, 9, 10, etc. - Hypothetical higher archailects
According to explanations by archailect avatars and research by Archailectologists, no entity known to Terragens can yet be classed higher than the sixth singularity. Whether this is because the sixth singularity is some kind of upper limit, or the Toposophic S1-6 system loses meaning after that point, or no entity in the known universe has yet achieved it, remains undetermined. This has not stopped Terragens from speculating about the possibilities, however, and there have been many fictional depictions in popular culture over the millenia. See Maximum toposophic level.
For a more extensive list of technologies created and used by each singularity level, read more HERE.