That's okay, thank you for everything you do 🤗 

 It's alright mate, thanks 

  https://image.nostr.build/46dd1d4a60df819a12222751342e4ddc2026b32b5147404b4a1a46516b7f7da4.jpg  

 there is zero guarantees about where your solution will fall, it just is probably going to fall around the middle of the bell curve for your target partial hash collision

my research and experiments with this kind of process show there is ways to make the curve taller and the edges lower (decreasing variance) but i don't think this can apply very easily to an irregular timing system like you have with nostr note ID hashes, the methods i discovered had to do with target variation over time, one that worked was a doubling series with averaging across multiple targets and another was the use of integrals, and i have read more about PID controllers and possibly derivatives can improve the timing of distribution if the integral has a noise filter added to it

prior to my foray into this area i knew about the principle of "mass action" in chemistry and it is also a similar thing that is the basis of statistical probability and the gaussian distribution (bell curve) but the only thing that improves timing other than varying your target in response to historical timing is increasing the number of attempts at finding a solution

so, i'm just telling you this because reality is not going to be what you expect, the numbers are too small to get such smooth predictable results 

 This seems completely wrong to me. How could it be anything other than Geometric?  I don't see how an experiment would even be necessary, it is self-evident. 

If hash functions are good than doing one hash will not help with the next, which will mean each is an independent trial from an identically disteibuted  bernouli distribution with p = 1/2^target_difficulty

where am I wrong?