Sometimes I read this thread. That's what I see here for the first time - numerological fortune telling. Well, this can have absolutely nothing to do with the puzzle.
Everything that is known about the puzzle and the ways to solve it can be reduced to such conclusions:
1. The puzzle keys in the range from 20 to 2160 are not randomly distributed, because they obey the rule - every next key after x1 in the range 2n, the value of x2 is located in the next range 2(n+1).
2. The keys are located randomly inside the ranges. That is why two adjacent keys can be in such extreme positions as xn-1 and xn+1, as well as in positions xn+1 and x(n+2)-1.
That is, the law of normal distribution does not generally apply to all ranges, but conditionally applies if we consider ranges as separate elements. Indeed, if we take base 2 logarithms from known keys and consider their decimal part (you can multiply by 100% for clarity), then we can see that the location of the keys inside the ranges is closer to the middle. However, the ranges themselves are not equivalent.
This is all that can be accurately noted about the puzzle keys.
What are the possible solutions. For now, let's consider only the case of public keys, since going through addresses is uninteresting and useless, so there are two options:
A. Fast discrete logarithm algorithm. Whether it exists is unknown, just as it is unknown whether the expression P=NP is true. But there are two algorithms that, in a sense, can be considered as such: kangaroo and BSGS, since they reduce the complexity to O(n1/2).
B. Probabilistic approximation, that is, in other words, a reduction in the search space, where the above algorithms can already be effective, working on reasonable resources, and not on all video cards in the world.
There are two ways to make a probabilistic approximation: searching for a function that reflects a pattern among the distribution of private keys and searching for an approximate discrete logarithm function. The first seems possible, because the keys as a whole are not randomly distributed, but are subordinated to the patterns from paragraph 1. In fact, searching for patterns among already known keys is obviously a dead end option.
The reason for this is a very small sample. If there were 16000 or at least 1600 ranges instead of 160, then this might make some sense. You can set a function that connects certain reference numbers-arguments in ranges (for example, their midpoints) with known keys and approximate the function, but this is a false dependence.
As a result, the task of simplifying the search is to find an approximate function that establishes the relationship between the private key and the public and the further application of the already known algorithms "meeting in the middle" or "birthday attack" (BSGS or kangaroo). There are no other options yet. And these are definitely not magic circles or "solstice of prime numbers" tables.