Hello. The question of collective intelligence is an interesting one, though I'm not sure it has much to do with the official topic of this contest (it would have better fit with last year's contest).
However I am skeptical about the possibility to modelize it and establish any explicit formula about it. All what I consider to be clear is that increasing the population also increases the rate of technological progress but slower, and at a rate that is relatively lower as the population is big (i.e the advantage of a more numerous population vanishes when the population is already big enough so that being bigger no more changes much).
About your formula, I disagree with your way to derive it. First if we believe your assumptions, that any individual has a rate of solving problems that is independent of the rest of the population, that all problems solved by individuals are actually different problems (not the repetition of each other), and that the solutions are going to spread in the way you describe, then what we get is that the number of solved problems is proportional to n (population) that are produced at a time and that take the time log(n) to spread. Then we would seem to get a flow of problems that would be not log(n) as you wrote but rather n/log(n) (dividing the number n of problems solved in parallel by the time log(n) it takes to spread) but that is not even true because log(n) is not a factor of slowing down of the flow, but of delay after which the solutions would reach the whole population. During this delay, the flow of solutions goes on. In a large time interval (much larger than spreading time), if solutions add up and do not repeat each other and are all going to be spread then the slowing factor is irrelevant, so that the factor is n instead of n/log(n).
But I would also question your assumptions. It is common in science to have co-discoverers of important results, so that adding more researchers in parallel does not help. I discussed some disadvantages of overpopulation, including for speed of progress, in this text.
Also, I consider that your model of spreading is not realistic. Seriously, do you know any example of solutions to problems that are spread in this way ? Human attention cannot be extended endlessly, to simultaneously verify many solutions found by others. The only process I see roughly behaving in the way you describe, is the process of Darwinian evolution, with the results of beneficial genetic mutations spreading across a population.
Instead, if things happen ideally in an ideal world, any solution once found and verified by a few people comes to be published and thus instantaneously spread across the world, bypassing peer-to-peer communication.
But we are not in an ideal world, so that in many cases, solutions once found are not spread at all and remain unknown. Especially because intelligence is not something common, and intelligent people are not always well-connected to each other, so that even after someone finds a solution, and even once it is explained to a few people, the information might not go further, as the mediatic space is occupied instead by a lot of rubbish which looks much more interesting in the eyes of the less intelligent people. I reported here my experience about this.
Also, there are cases where a solution becomes known by the relevant community, but we have a persistence of a large separate community that remains ignorant about the solution. For example, the problem how to understand fundamental physics for so many purposes, has been solved by the community of physicists in the first half of 20th century, with general relativity and quantum physics, but still we have a persistence of a large community of cranks, including the majority of authors of essays in this very contest, who failed to understand these solutions because they are not good at maths, but to not feel ashamed of their failure to understand maths they need to mistake things as if the known mathematical solutions were wrong for no real reason but the fact these solution are mathematical so that it makes them obscure in their eyes.
Similarly, in the 19th century and until the 1930's there were big monetary instabilities. Economists found ways to stabilize the money by central banks. However the Bitcoin community still never heard about that problem and solution, and keeps blindly believing that the instability of the value of the bitcoin is just due to the fact it is not popular enough and it will naturally become stable by the magic of being popular, just because, as usual currencies are now relatively stable since a few decades, the very existence of the problem remains completely ignored by people who specialize in cryptography and have no clue about finance.
It is also possible that a larger population is a direct obstacle to innovation, as the larger number of people obliges a standardization of work where any deviation from the standard is becoming impossible. This particularly happens with the teaching system.
On the other hand there are cases where a large population induces a collective form of problem solving where solutions work without being understood by individual members (or at least not by any majority of members). This especially happens with the "invisible hand" of free market that provides a sort of global optimization that does not need to be understood by any individual in order to work.
