intractable collatz

(background color: truly is the nightmare before xmas around here lately. seems as if santa died this year.) when in doubt, write some code. this idea occurred to me and is similar to hybrid3 experiment from 12/2018. the idea there is to just run the hybrid optimizer to find large ‘cm’ max glide index trajectories for fixed bit widths. here that is changed to ‘cg’ glide length. what will be the pattern of results? this generates 500 trajectories after 10K optimization iterations and then sorts results from longest to shortest by ‘cg’ and looks at the initial bit patterns. some of the hypothesis is confirmed. it turns out they are all utilizing a 1-lsb triangle to generate very long glides; this explains the lsbs. the msbs are harder to characterize. this grid plots the inverse of the bits and the msbs line up on the right in the same ‘cg’ order, longest to shortest bottom to top.

but generally this all fits into a brand new hypothesis. the idea is that generally the longest glides by bit width utilize the 1-lsb run structure. this would seem to explain a lot of other experiments that optimized but did not look further at the bit structure. the similar hybrid3 experiment did find/ notice the 1-runs but didnt notice at the time the probable lsb link. there was a series of experiments to find longest glides or trajectories limiting by 1-run lengths. my suspicion now is that for those longest trajectories found, the 1-runs tend to be in the lsb. it would have been simple to check at the time, but didnt. recall this experiment is showing that starting from random seeds, optimization of the longest glides all converge to this 1-lsb bit structure. therefore it seems likely (all) other optimization schemes (eg bitwise etc) would tend to follow the same pattern.

this is a very simple idea building on lots of prior analysis/ features but hasnt been surfaced from this angle until now. it also reminds me of the construct49 experiment from 9/2019. the basic idea there was that, iterating, the way 1-runs eventually intersect with the lsb seem to have the key effect on glide lengths.

some days it seems like a proof is still far off or unreachable, and other days it seems like just a slight twist or change of focus could bring out the path to solution. when simple unifying ideas are found that were previously missed, as here, its a cause for optimism.

💡 building on this, general proof idea: the longest glides have long lsb 1-runs, up to a point. all shorter glides have shorter or missing lsb 1-runs. this points to some kind of induction on the lsb 1-run length. this rough idea has come up previously. from prior observations there are instances of adjacent lsb 1-runs in trajectories, but maybe they are limited in size somehow? thinking it over, have not yet studied yet the scaled lsb 1-run… ❓




collatz unvanquished

(still hanging in there despite going thru/ in middle of tumultuous 1-time in lifetime personal hardship…) have been pursuing many miscellaneous ideas and firing off many different algorithms without a lot of particular direction and many mostly null results. then was looking closely at backtracking patterns of 1-run patterns (riffing some off the direction of last months backtrack21 code which found the stripe pattern in msbs) and then ran into this idea/ finding/ feature, maybe not obviously/ apparently/ closely aligned with overall proof ideas but definitely worth documenting. not sure how exploitable it is, but it was a little surprising. it uses a simple backtracking algorithm (with restarts) to create 20-length pre-trajectories on a 50-length 1-run in a 100 bit width seed with ½ density random background. then it was found that there is a signal that occurs about ½ the number of iterates of the initial 1-run, ie ½ * 50 = 25 in this case (graphed to 20).

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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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