endless(?) collatz

it might seem sometimes like work on this problem is “endless”. am coming up on the ¾ 100 mark count of blogs on the subject. it seems sometimes endless not unlike the sense of an unhalting Turing computation, but one which neither cant be proven nonhalting (lol hows that for a triple negative?). there have been ups and downs to say the least. or maybe its been more of a rollercoaster ride. in the sense that bipolar disorder switching between mania and depression is also a rollercoaster. ok, just being a little )( dramatic there. but one tends to get dramatic after years of work on a problem. in agile programming theres a concept of “features, stories, sprints, epics” of increasing complexity. work on this problem is exceeding an epic at this point. speaking of that and drama, there was also an old book on software engineering about “death march projects”. they are real, they do exist. maybe am feeling a bit demoralized/ disillusioned after slaving working on one at the corporate salt mine work for ~¾ of a year now. ofc such things arent announced/ designated (thats part of the groupthink/ collective denial/ blind spot that goes on!) so it comes down to “for all practical purposes…”

overall feel the work has been worthwhile but the goal of a proof is always lurking and by that measure, theres something of a failure. argh! but then, thats a very high standard that no human alive out of 7B, even more nonliving, has ever achieved. yes, lets keep in mind that mathematicians who have worked on collatz in the past have died, not the least collatz himself. it would be dramatic to say they “died working on the problem” although it would not be entirely inaccurate either…! hard problems were once compared to a femme fatale in this blog… and the other metaphor is relevant, the (mostly unpublished, but still very real) “graveyard of failed approaches” by others + myself…

its a bit awkward or unseemly to call all this an “obsession” but not entirely inaccurate either.

in this realm, have to take solace and reassurance in some small )( victories.

  • small victory: CS blogging superstar RJLipton recently inquired about my work, and replied to him. (one of the rare few who successfully published a book out of his blogs. utterly shamelessly, another that immediately comes to mind is belle de jour.) didnt lead to any further dialog so far but its a delightful moment/ incident/ vignette.
  • another very noteworthy item: Terence Tao one of the biggest legendary living mathematicians, the only one winning $3M to my knowledge, filed a paper on arxiv on the collatz problem. afaik its his only “published” work on it outside of a blog page. have posted comments to his collatz blog page over the years asking for attn, never got his or others direct attn, the comments have tended to get anonymously downvoted—no hard feelings lol!
  • am reading the Book of Satoshi edited by Phil Champagne, what a awesome joy/ pleasure. SN is one of the greatest living computer scientists if you ask me, one cannot overstate his once-in-a-century accomplishments, and as far as code affecting reality in an almost superpower/ superhero or even godlike way instead of the typical case of vice versa, he is unmatched, it is an extraordinary gift/ honor merely to be a contemporary and living on the planet at the same time! a new hero of our age for the pantheon. he accomplished what almost nobody has ever done. he managed to write and start/ lead the open source project and gain followers, almost making it look easy. almost all other cases of this type of accomplishment pale in comparison. his work is quite similar to Linus Torvalds in many ways and yet in many ways on a much vaster/ impactful scale (honestly am a bit envious of them both, having long attempted a vaguely similar venture and having achieved an infinitesmal )( fraction in comparison). basically, geeks meet economics. which brings to mind that old zen question, what happens when an irresistible force meets an immovable object? had no idea how difficult such endeavors really are. and theres a unmistakable element of randomness as with all “viral” phenomena… alas, there are very few clues as to his personality or identity…

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collatz match

that last graph from last month shows clear signal on Terras-generated glides wrt a matching feature/ metric. so then a simple exercise is to look at last months matching metric/ feature associated with the trajectory database. this code works on glides longer than 10 and calculates the corresponding match count up to that point. the color coding is by 1 of the 8 generation types. the ‘i’ matching metric shows some signal relating to the glide lengths for some of the trajectory generation types. there seems to be an inverse correlation. but its also a bit noisy and maybe “fooled” by some of the generation types. there is some further adjustment to the increasingly general plotting routine. there are so many varieties of plots and its hard to build a general system/ API that services all of them, gnuplot is something of a wonder in its flexibility/ power at times.

(later) an obvious idea on hindsight is to sort generation types by glide lengths. thats a 1 line modification in the 2nd graph. the color scheme by generation type is different for the graphs. it makes it more apparent that maybe 1 generation scheme, the blue one in 2nd graph, is generally low in the metric while others seem to be higher or more random. which is the blue scheme? obviously it would have been nice to label the generation schemes in the graphs, didnt figure it out yet.





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collatz microscope

have the feeling of being put under magnifying glass or microscope sometimes at work and elsewhere (some recent joking about this on the physics stackexchange hbar chat room among long oldtimers). in worst case scenario as mentioned last month it feels like being micromanaged, bullied, stalked, or hunted. maybe this is my coming-home-to-roost karma for having a cyber alter ego—decades old now. lets face it no matter how human corporations pretend to be, theyre fundamentally soulless at heart.

theres some greatness that psychology is starting to understand the negative effects of the corporate world. could really relate to this latest headline/ study, Greedy bosses are bad for business, study finds. but could it really be true? bet theres some other study that says that soulless bosses can help drive up the bottom line. but ah, also trying not to be egocentric and put it all in perspective, this is a very old complaint in historical terms, ie roughly as old as capitalism, and intensified with the industrial revolution and so-called late stage capitalism (luv that phrase! what does/ can it mean?! reminds me of the term postmodernism…). overall what one might call 1st world problemstime to take a vac…

this code has optimization and analysis sections. the optimization is to push down mxl, mxr the max 0/1 lengths on both left/ right of the glide. then it looks at the binary structure of the largest trajectories. it takes a same # of iterates starting from left and right sides of the glides, concatenates them in binary, and then analyzes the 0/1 runs. in the 1st graph its shown that the histograms (4) for left/ right 0/1 run lengths are nearly identical as found awhile back with another generation scheme, (cant recall exactly, maybe the long-examined ‘w’ widthdiff). in the 2nd graph there seems to be some slight differentiation, but this seems a rough 1st cut on finding it. strangely there seems to be a difference between odd and even lengths seen in the apparent alternating/ thrashing pattern in the graph. the 2nd graph is 0 run length histogram difference and the 3rd is 1 lengths.

overall it needs some further investigation/ polishing but seems to be real. as was working on this it occurs to me immediately that both on the left and right side, there is not a “control” for varying trajectories starting at the intermediate positions and was wondering if thats causing the results to be more uniform/ undifferentiable. from prior experiments its known that many of the intermediate trajectories or “subtrajectories” (actually subglides) starting from the intermediate points tend to have a much different aspect of terminating quickly eg on the left side. another aspect that occurs to me is that the (“quick-and-dirty”) concatenation idea might bias the measurements slightly versus the alternative of concatenating all the separate 0/1 iterate string splits because the start and end of the binary iterate are always 1s and join to make a larger combined 1-run than either separately.

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collatz monotone estimator found?

last installment ended looking at the “forward/ lookahead trend” concept. heres a presumably stronger optimization/ test building on that idea using the hybrid system whipped off fairly quickly. the idea here is that the ~200 step lookahead function seems to be monotone (last bitwise experiment suggested max nonmonotone length is around ~200) and it would be useful to try to verify that as rigorously as possible. this calculates the ratio of the lookahead function at 200 steps fwd vs the initial point as ‘fr’ lightblue line and tries to maximize it along with the other trajectory metrics with no limit on starting iterate size. its indeed bounded very close to less than 1. there is one outlier point greater than one at about nl=235 bit size, but the fr vs nl graph seems to show a ceiling.

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collatz search for meaning

victor frankl, an extraordinary figure, a psychologist who survived nazi german camps, wrote the book “mans search for meaning”. read it last summer, was impressed. in contrast also am fond of the kurzgesagt video on “optimistic nihilism”. it seems one can get lost searching for meaning in certain math problems, the hard unsolvable ones. it appears that Church was involved in one of these “wild goose chases” (found that described in a paper somewhere, wanna find out more). finding a solution is a lot like the search for meaning in the problem. for me, as written at times, if something meaningful can be said about the problem, then feelings of despair about the unlikelihood of a solution can be held somewhat at arms length… like wandering around in underground caverns, its not entirely clear if one is any closer to an “exit,” but its clear that many diverse, even wondrous properties unseen by anyone else have been isolated over the years. the fractal nature in particular, a theme explored in great depth/ detail with this attack, is still not remarked on/ noticed much by almost all other authorities. other concepts such as “mixing” and the pre vs postdetermined themes tie in closely with ideas outlined by authorities/ experts Lagarias and Terras.

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