Hello. Long-ish time wiki reader here.

[Drashner1]

Okay then. My main questions would be:
What amount of resources (EG computronium, support) would a S4 intelligence need, and what would that translate to in terms of the amount of modosphonts that could be supported by the resources used separately?

Assuming the S4 is 'self-ascended' rather than being the creation of another being of an even higher S-level, the mass required to support an S4 mind translates into about 300x the mass of the Earth. This can take various forms:

a) A Jupiter Brain - a concentrated intelligence, a sphere of computronium, about the size of Jupiter. About 3 Earth masses of that is a 'CPU' made up of plasma processors - processors that use a mix of magmatter and high temperature plasma to perform computations at such high speed and temperatures that it's rather like a stable nuclear explosion or the inside of a small star.

b) A Dyson Brain - the mass of a Jupiter Brain, but distributed in the form of a swarm of networked processors surrounding a star, usually at about same distance that Earth orbits the Sun.

c) A Matrioshka Brain - Much the same as a Jupiter Brain, but the processors are individually smaller and occupy a much larger volume extending beyond something the size of our solar system.

A single kg of plasma processor can basically run an entire S3 - and and S3 can be discussed in terms of having the processing power of 2.97e20 modosophont minds. Since the CPU of an S4 is some 3 Earth masses, that works out to:

2.97e20 modos/kg of S4 processor x 1.7916e25kg = approx 5.32e45 modos equivalent. By way of comparison, the entire non-transapient population of the setting maxes out at roughly 3e21 modos.

What sort of truly new mental capabilities (beyond being able to design and use new technologies) would emerge at the S4 level?
What about S3 or S5 intelligences?

We don't honestly know if what you mean by 'truly new' is some form of mental capability that has no equivalent in human minds. Rather by definition, it is literally beyond our ability as humans to conceive of what transapient minds can do. It's sort of like asking a creature that lacks self-awareness to conceive of and describe it. Or creatures that are blind to imagine and describe sight.

That said, it is something of an ongoing occasional hobby among the membership to try to imagine truly qualitatively different mental abilities for transapients (as opposed to quantitative differences, which are much easier to imagine). So far we haven't has a lot of luck in this area, but it's fun to keep trying. Most of what we've come up with is down at the S1 and S2 level however.

In terms of mental capabilities for S3 and above beings - whether they are truly new or qualitative or quantitative, some ideas that we've come up with include:

1) Creating new AIs/new minds - An S3 can do this as easily as a human would speak a single sentence. An S4 can imagine a new human level mind into existence as easily as a human would speak a one syllable word. At the S5 level it is not uncommon for individual thoughts to achieve self-awareness/sophonce and form complex societies before being reabsorbed or fading out of existence as the S5 thinks of other things. At the S6 level, it has been suggested that all of Terragen civilization is just a figment of an S6's imagination.

In a related area, it has been suggested that if an S5 or higher pays sufficient attention to a modo it may - without really trying and as a side effect of creating a mental model of the modo in order to predict its behavior - create anything up to trillions of copies of the modo, each copy being fully self-aware and sophont, and subject to a huge number of simulated life scenarios - some of them painful or deadly.

2) Rewriting - It is possible for S3 and above to rewrite the minds of modosophonts entirely and en masse. Changing their memories, personality, most deeply held beliefs, etc. and replacing them with a perfectly synthesized set of artificial memories, personality, most deeply held beliefs etc that are more to the transapient's liking. Modos and lower transapients can do this to individuals and groups - but S3 and above can do this to entire civilizations if they are so inclined.

3) An S5 or higher thinking about a concept even slightly - for example, if it hears the word 'dinosaur' - may imagine entire planetary biosphere's into existence, simulated down to the level of individual cells and micro-organisms and up to the level of planetary weather and climate - and encompassing the entire evolutionary history of dinosaurs - and then either unmake them again as it thinks about something else - or cause the figment of its imagination to be constructed via nanotech in the real world.

There's probably others, but those are what come to mind most immediately.

Hope this helps,

Todd