Limits of Transapient Power
The abilities of transapient beings seem miraculous to ordinary nearbaseline humans and other SI:<1 intelligences, but all of these miracles are thoroughly grounded in technology and science. It is apparent that even the Archailects must work within the rules of the universe they live in. Some actions might be impossible, because they contradict those rules. Others are impossible simply because they are logical contradictions. Many more are entirely possible in principle, but require physical or mental resources beyond those the transapient can reasonably muster, and are uneconomical. This category of "possible but impractical" problems is very large indeed. Most are related to "chaos" effects: extraordinarily fine measurements of a system's initial state and considerable computing power are required to make accurate long term predictions for many kinds of systems, even given the advanced pattern-finding abilities of the transapient mind.
While it is by definition impossible for SI:<1 beings to know all of the details of the limits faced by transapients, there are two sources of information. The first is observation: there are some things the trans-sapients have never been observed to do. The other is conversation: trans-sapients may simply say that they are incapable of some actions. Neither of these is entirely reliable, of course. As to the first, trans-sapients may take actions which they hide from baselines, or in some cases the actions themselves may be impossible to show or explain. As to the second, they may lie, or their explanations may be misinterpreted or simply incomprehensible.
Even given these limitations, lists of actions of which transapients are incapable are very popular with sapient level beings. These first appeared when sapient humanity realized that the first Singularity had been breached, and many have been devised since. The most complete and concise version that is comprehensible to sapient level intelligence is maintained at Ken Ferjik. It should be noted that every item on the list has been challenged many times, by sapient witnesses, researchers observing traces of transapient behaviour, or by some individual transapients themselves. The last case is difficult to evaluate; many individuals and even whole societies have been convinced by a transapient that one of the limits mentioned below has been surpassed, and some societies or religions are founded on precisely such a premise. However, such reports have not been reliably confirmed by external sources, even at the sapient level, and in many cases have been declared debunked by other beings of the same or higher toposophic level as the claimants.
Some of the things which transapients are not believed to be capable of are:
1) Detailed long-term memetic predictions. The smaller the system, the broader the nature of the prediction, and the higher the toposophic of the transapient making the attempt, the longer term and more accurate the predictions may be. However, not even an Archailect has the resources to comprehensively predict all the details of a polity's culture centuries hence. Most transapients simply do not bother to make the shorter term and smaller scale predictions they are capable of unless they have some good reason for doing so. Such prognostications require that extremely detailed data be collected, and are a tremendous sink for computronium and time. From the point of view of sapient level beings, of course, transapient prognostications can border on the miraculous. Also, if the goal is to interfere rather than merely predict they can set up extremely durable memetic structures.
2) Ecosystem resurrection without a backup. Though some ecosystems have been forensically reconstructed, and in some cases the differences are impossible for nearbaseline intelligences to detect without aid, higher level transapients assert that these are clever simulations, not equal to the original item.
3) Prediction of chaotic phenomena such as planetary weather, stock markets, and other nonlinear systems. Even Archailects, with all of their computational resources, cannot make long-term detailed predictions about chaotic systems. They can make general statements about the solution space of such systems, but details are not available beyond a certain point. For low-dimensional chaotic systems any available solutions tend toward abstract rather than concrete solutions. For chaotic systems that involve high-dimensional phase spaces even these kinds of answers may be out of reach unless the Archailect is willing to make extraordinary commitments of resources. So, not even an Archailect can know whether it will be raining on the north end of New London on the second day of the third week of the tenth month of the year a decade hence. Guiding or forcing a system toward the desired result through a series of shorter term predictions and interventions may be possible, however, assuming the necessary agents are in place. Chaos Wands are an example of this technique in action.
4) Predicting the transapient's own future behaviour, or that of other transapients of their own toposophic level. Accuracy and precision in these sorts of predictions is approximately what it would be for a human nearbaseline trying to understand eirself or other nearbaselines.
5) Violating known physical laws. This is a broad category. No transapient, not even the most powerful, is capable of exceeding light speed, violating conservation of mass and energy, travelling backwards in time, and so on. Note that it is possible for them to produce effects which appear to have violated such laws, but close investigation has always shown this not to be so. Note that in many cases they have found extensions of earlier understandings, just as Einstein extended classical physics without contradicting it, or have discovered entirely new realms of physical action, as with quantum mechanics in the late 1st century BT (20th century AD). They have also discovered new laws that are fundamental to this universe and which may not be violated by any being of any of the known toposophic levels.
6) Disproving mathematical proofs within the terms of their own definitions. This falls within the realm of self-contradiction. No transapient has disproved the Pythagorean Theorem for Euclidean spaces as defined by classical Greek mathematicians, for instance, or disproved Godel's Incompleteness Theorem on its own terms.
7) Finding fundamental observations to be false. Not even an Archailect can conclusively show that the value of pi is actually 3, or that evolution does not involve natural selection, or that the force of gravity does not move planets in their orbits.
8) Executing true "paranormal" phenomena. Though they can do miraculous things, transapient "telekinesis" always involves known forces, "telepathy" involves the careful use of nanites, powerful simulations, and reading of electromagnetic field (plus considerable computational power), and so on.
9) Resolution of the Fermi paradox. Even a single civilization comparable to the aggregate of present-day Terragens, if it had arisen at any time in the entire past of the Galaxy, ought to have swept the whole Galaxy and left nearly indelible traces in every star system, in much the same way that Terragens do now. Even if it subsequently became extinct, it would have left few star systems unaltered. Given their apparent rates of expansion the other High Energy Emission civilizations would have done the same. Apparently Someone or Something prevents this from happening. If any of the transapient entities has solved the riddle, they have not said so.
10) Conclusive proof or disproof of the existence and nature of God, or of the existence and nature of life after death. This is true of many other religious or philosophical assertions that might be true or false but cannot be shown from available evidence as understood by sapient minds. Transapients willing to discuss these matters have various answers. Some declare the questions themselves to be arrant nonsense. Others say that they are as baffled as any nearbaseline. Some give answers, comprehensible or otherwise, but are unable to explain their evidence or reasoning at the level of mere sapience. Yet others give definite and clear answers, persuasive to millions, but offer proofs which are not convincing to sapients who are well beyond their sway. Most are silent on these matters.
11) Resurrection without a backup. As with ecosystem reconstruction, such "resurrections" are in fact clever simulations. If the available information is sufficiently detailed, the individual and even close friends and associations are unable to tell the difference. However, transapient informants say that any being of the same toposophic level as the "resurrector" can see marks that the new being is not at all like the old one. "Resurrections" of historical figures, or of persons who lived at the fringes of Terragen civilization and were not well recorded, are of very uneven quality.
12) Some other memetic actions. This is not a well understood category of difficulties, but there appear to be a number of noetic limits which the transapients are unable to cross. One has to do with the limitations of any single paradigm or set of meta-paradigms. Apparently no transapient has been able to create a single overarching paradigm to describe the present state of knowledge without contradiction; instead even the Archailects use a series of mutually incompatible mental tools to understand the universe. This appears to be a more general extension of Godel's Incompleteness Theorem. In addition, there are a very large number of concepts that transapients have simply been unable to describe in ways simple enough for ordinary sapient beings (or even other transapients of a lower toposophic) to understand. Regardless of one's own toposophic level, some things are irreducibly complex, and appear nonsensical or contradictory if an attempt is made to make them understood to beings that comprehend the universe in a less complex way.