collatz standing on shoulders of giants (not!)

RJL has a nice new blog that cites collatz/ Taos recent work on it. wrote a comment on it and it led to a negligible # of hits. lol!

(uh oh!) wordpress has a new editor. always run into funky quirks in the past, some of them nearly showstoppers eg incorrectly handling escaping of code blocks thereby screwing them up/ aka corrupting them every time saved. but maybe this one will be slightly better. you cannot imagine how many times have edited posts, saved them, then had to scroll back to the point where was previously editing. omg, who the @#%& asked for that? what an unbelievably tedious “feature” that shouldnt have gotten past UAT even one nanosecond and yet was default functionality for years, lol! (argh already found a code-set shortcut doesnt work, sigh) oh and it only took me about 30m to figure out how to insert the excerpt indicator… thanks guys! and it took yet more “fiddling” to find the rearranged controls for image embedding. welcome to 2020 the age of ubiquitous smartphones and endless fiddling, lol! Nero would be happy! also the idea of going back to old blogs to edit them, esp long ones, and trusting wordpress to “do the right thing” is a bit terrifying right now. welcome to modern software which is that weird mix of both extremely powerful and extremely fragile at the same time… o_O

this is some recent hybrid code that has some minor improvements and didnt want to lose the changes. the basic idea was to start with limited 1-runs and see how much the trajectory size could be optimized. it has to throw away candidates that exceed the 1-run limit. the limit is determined by average statistics with ½ density iterates. the graph code is a little improved by numbering multiple graphs etc. it was found the prior “init” routines were not working as expected and/ or had low probability of creating sub-threshhold runs so they were adjusted.


(9/12) this is another idea that just occurred to me. it seems not to be ruled out by existing experiments.

consider all glides longer than some short threshhold. they all seem to have “semiorderly non ufo drains”. in other words ~½ density parity sequences without outlier long runs.

some of the idea here is that the outlier ufos were found in drains but never found in any drains for non-short glides. there is a long earlier conjecture that the postdetermined glide drain is semiorderly, this extends it to the full drain. also all this time “postdetermined glide” has been used but it needs a little more distinction. once again evoking the slipperiness of words and this problem. it could mean either

  1. the postdetermined section inside the glide. this is the typical usage. may be empty if the glide terminates in the predetermined range. doesnt include much of the drain, ie specifically excludes postglide drain.
  2. the entire postdetermined trajectory after the glide which includes the drain.

(9/16) this blog is copious with ideas now and my brain also, and its like RAM deciding on what to try/ pursue next. lately the image from construct101b and construct101c from 1/2020 has been on my mind some. it was alternatively called a blob, cloud, or an arch. the other idea that occurred to me at the time but did not mention it was a brainlike structure where the stem is the drain region. just had an idea to look at the postdetermined glide only. the prior code aligned the trajectories based on endpoint of trajectory. but what about aligning on endpoint of glide? that change along with omitting the predetermined range led to this striking, extraordinary result. ofc its both unexpected and unsurprising at the same time. there has been some usage of the “alignment technique” ie hortizontal or vertical alignment of trajectories based on various criteria and here it turns out to be utterly pivotal.

the 1st graph looks at the parity sequence and generates a random walk with increments -1, 1 for the 0, 1 parities. it has a lot of signal, but on wondering about it, then came up with the idea of assigning increments 1, 3 instead. these may seem like magic numbers but they make a lot of sense. in a log2 graph, as long mentioned/ utilized (this shows up also decades ago in the Terras ideas), the semicompressed mapping leads to increments of roughly -1/2 and +3/2. simply multiplying by 2 leads to -1, 3 increments and the 2nd graph which is tighter.

this is a different way of looking at a phenomenon that has long been noticed/ examined: in short, (postdetermined intraglide) drains are nearly linear random walks. one of the earliest revelations of it was the “rosetta diagram”. even earlier were a series of experiments that looked at “slope correlation” or the linear correlation in the drain and found it to be very high. but these results are reaffirming and point to some new ideas/ angles about the “core inductive structure” of the problem. in a sense this is a self-similar/ scale invariant structure for the “remainder” of trajectory glide verification calculation.


(9/17) 😳 oops, math/ numeric typos are so easy to make, the actual increments should be log(1/2) and log(3/2), or working backwards (as is case here), log(2) and log (2/3), and both of the prior sets do not match that, and what the plots show is that if one of the increments dominates the other, the end result will be more nearly linear. using the correct increments displays a wide spread in the subtrajectories random walks that closely matches the log-scaled iterate plot. another glitch is at line 46, carried over from the prior code and not spotted; the line is redundant because all trajectories are glides and the last glide index is always found.

some other experiments reveal the following. the just cited construct101c led me to some question, it wasnt examined too closely at the time. one can see some “sideways” glides generated at random from ½ density iterates. but how does this mesh with the long noticed/ known finding that “most ½ density iterates chosen at random are drains”? it turns out, via simple code/ analysis, the larger the iterates, the less likely glides are chosen at random, ie the more likely the iterate is a drain. however, from Terras ideas its clear there are many glides for higher iterates, and its also known how to generate many with close to ½ density starting seeds. eg the recent 0.64 glide parity density construction does this.

so a theme long examined is, what is it about ½ density seeds with glides that makes them different from nongliding, aka draining ½ density seeds? after many months of analysis, maybe even years, it seems am coming up mostly emptyhanded on this basic question. it seems there is something very deep yet to find here, but it has been extraordinarily elusive so far. despite having quite an array of powerful tools, have not been able to turn up anything definitive so far. yet again more thinking outside the box is apparently required, even despite copious amounts/ pivots/ efforts/ time/ energy now expended so far.

(9/21) the 3-power sequence is starting to figure again in my thinking. there has been a relative lull in it over last ~½ year of blogs but its reemerging in my attn at the moment. last month had the crucial “twist/ turnaround” in that it reevaluated results that earlier seemed to come out as null but the null finding was in retrospect due to a miscalculation/ defect/ bug. had the urge to do an experiment but then it reminded me of prior ones, ie algorithmic/ logic/ code deja vu, and so then not remembering exactly, had urge to do a quick survey. its not easy to survey old blogs anymore, but heres a quick list. the sizeable list reflects an increase in apparent significance/ interest. many of the experiments are over the related match ratio. in short a lot of signal has been found in these different angles/ povs but some “mixed signal” in the form of limitations were found also.

8/2019 construct140, bitwise41, bitwise42b, construct41

9/2019 review134, review134b, construct42, construct43, construct45

12/2019 review140

1/2020 construct107, review152, review156, review159

8/2020 review134c

this is a sort of warmup exercise very similar to construct42 graph #3. am interested in the 3-power sequence associated with >0.64 Terras density iterates. a basic factoid seems to have been overlooked/ forgotten in all the other directions now “re-dawning” on me: the 3-power sequence density seems to be a basic “explanation” for a big “near-plaguing” question over years, ie the difference in ½ density iterates either gliding or draining.

(as mentioned/ mused recently eg 6/2018 blog title, these days the phrase “plaguing question” takes on distinctly new meaning…) this code does an exponential moving average instead of a running average as in construct42 and gets a significantly different result in the early trendlines, starting out in an increasing curve instead of a lower range moving into local maxima. so the density variations seem to have some deeper structure and need more attention to analyze/ isolate esp in the initial range. there are also some clear wavelike-trends for the higher densities in this graph somewhat similarly to the earlier construct42 graphs. something is going on here…

( 😡 re the new editor now just discovered that it locks/ loses scrolling on the ipad somehow due to a bug. @#%&, geez, argh! also, Evil Big Corp has announced they will no longer allow web mail access, which screws up my method of copying and pasting code, ie via saving drafts; they have years-long rejected pasting it into pastebin. they continue to tighten the screws, its like the scene with luke skywalker et al in the garbage compactor… pasting into a wordpress draft still works, lol sssh dont tell anyone!)


(9/22) forgot that density is quite noisy in those sequences and it explains some of the trends, (rerunning shows) they are somewhat random and related to initial noise in the sequences. old experiments involved eg high smoothing and one sometimes tends to misleadingly/ near-mistakenly picture trends in the smoothed form, and smoothing can disguise different types of noise.

this is some further analysis of the 3-power sequence starting from Terras density iterates, surprisingly a fairly basic idea that hasnt been looked at in prior results, although there are related povs; it could be related to/ nearly guessed from construct107 and the idea that longer glide Terras seeds have higher density, seen in the epic/ pivotal 2/2019 findings eg construct5, and the density/ entropy trajectory converge trends were seen in construct5d. 100 width iterates, 100 length sequences (predetermined range), 400 sample average (unusually large to overcome the noise), 4 different starting Terras densities 0.65, 0.75, 0.85, 0.95, and the density, entropy, and same for the 3-power sequence. the 3-power sequence is not “parity aligned” to the collatz sequence where parity alignment typically tends to give better fits/ correlations.

the 3-power sequence converges to center density and entropy faster, and the concavity is reversed. but theres clearly a sense in which the 3-power sequence is approximating the collatz sequence. this analysis for the collatz sequence is quite similar to construct15 on 2/2019. not remarked on at the time, it is somewhat striking how the density and entropy converge to the center at exactly the 1st postdetermined point.

💡 an immediate idea comes to mind. what if one tried to construct an approximation of the collatz sequence from the 3-power sequence using current iterates? it seems in principle it could/ would/ should be possible to devise something, albeit subject to a lot of noise constraints.


(later) there is a remarkable finding in construct15 that is overlooked and now just stumbled across it again. it is long known the low vs high bits sometimes have different characteristics, seen early on eg in construct5. did not notice this much earlier but just now found over trajectories for Terras density iterates, it is the case that the low density is sometimes stable at center while the high density varies!

💡 this led me to some remarkable idea. how could this be measured as a continuum instead of 2 separate values, one varying and one unvarying? it appears they are pointing to the idea of a center of weight. consider the 1 bits as having a weight (as the density concept already considers with hamming “weight”), then for a ½ density iterate, there will be approximately ½ of the width comprised of 1 bits. so where are they distributed? an even/ “uniform” distribution would put ¼ of them (of all bits that is, ie ½ of ½ all bits) low (half, lsb) and ¼ of them high (half, msb). but the density low and high differences show that this is not the expected distribution, ie the center of weight is not at the center of the width.

so then came up with this simple metric, it measures center of weight as a fraction of the width, either the “left or right” (msb vs lsb) by counting ¼ of the 1 bits counting from left or right. then applied it to the existing code, and got high signal. there is also a lot of correlation going on. this needs a lot more analysis and investigation, but am just posting it in relatively raw form for now. this also uses parity alignment on the 3-power sequence. as in the last series, the noise tends to decrease for the higher Terras density iterates. another observation is that ‘mr’ roughly tracks entropy ‘e’ and ‘m3r’ roughly tracks 3-power entropy ‘e3’ as in prior graphs, but this center of weight metric would still tend to be near ½ if entropy and density was evenly distributed in the high/ msb vs low/ lsb ends of the iterate. another strange twist/ finding is that ‘m’, ‘m3’ center of weight is ½-random measured from the left, and contains signal measured from the right. this seems to show the msb bits are roughly evenly distributed and the lsb ones are not, ie tend to “bunch” to one side, even if both sides have ½ density/ entropy. ❓

so anyway yeah its raw, but nevertheless…

😎 ⭐ birth of a new feature!


(9/23) this is an analysis with large 800 width trajectories to allow a long 500 width running average on individual trajectories, 40 trajectories above ½ Terras density. the density and entropy are extremely noisy hence requiring the large average span to smooth out, and the averages come out very close to the center also as seen in construct15 but they nevertheless have strong signal. the density and entropy parity on the other hand have very strong signal away from the center ½. the postdetermined return-to-center tendency seen in recent series of graphs and construct15 is not much observed/ evident at all in averaging in this dimension/ direction. so have to think more about why thats happening and come up with some kind of picture/ explanation for that, its not a trivial thing to understand.

❗ bottom line though, given “sufficient samples,” these 4 metrics seem to be sufficient to approximately determine the “relative location” of the iterate in Terras density glide “spectrum.” sounds relatively innocuous right? but that could be a very big deal because it possibly essentially leads to a general formula for “current glide location”…


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