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.

bitwise39b.rb
bitwise39c.rb

bitwise39b

bitwise39cx

bitwise39cy

(8/10) coded up some multiple experiments in a dash and they all came up with null results. am thinking it is at least worthwhile to refactor +  consolidate them all because the analysis is nontrivial and the analysis subroutine grouping is coherent/ convenient so to speak (they “go together,” ie like tools in a toolbox). the idea is to look at ½-density iterate evolution for large 1000 bit width iterates. the idea with the large size is that breaking down into 0/1 runs gives more samples for histograms. there is a significant spread in the postdetermined iterates after 1000 iterations wrt eg bit size and other statistics such as glide peak index and glide length etc. but none of this seems to be quantifiable/ predictable in terms of the known binary features of the original seeds.

this is all again a very strong demonstration of the now-familiar curse of undifferentiability. many of the prior features/ analysis was applied and it all comes up empty. it is reminiscent of how in cryptography, arbitrary signals can be encoded in what appears, under statistical examination, to be uniform random noise. this seems somewhat in contrast to similar exercises for Terras-generated distributions studied at length on 2/2019. in particular the very notable construct9c experiment was able to find a very subtle but identifiable/ differentiable bias in the 0/1 runs histograms. it appears to me from latest analysis this doesnt occur in the random ½-density iterates. and it would be very worthwhile to try to pin this down a little further. it seems to show that even though Terras-generated iterates seem to run through the ½-density region theyre somehow different.

pondering these results am thinking more about the vary1 experiment from 12/2018. that showed some identifiable trend for ½-density seeds in density/ entropy distances. am thinking now that very few experiments have shown anything similar and want to re-survey prior analysis to find any, but think they are very scattered. this seems to be some key to understanding differentiability in ½-density seeds.

(8/12) ok here it is after substantial refactoring effort. this has some nice streamlining of a lot of scattered code. now have that feeling of something like cleaning up a room and being a little tired/ sweaty afterwards… some entropy of the universe has been converted to order thru human effort/ intervention. one nice feature is how it converts all the analysis code to output separate self-contained graphs on the filestream ie a more reusable pattern (and there is a lot of call for that given all the scattered/ specialized/ nonuniform related graph code). it also includes the "histcurve" algorithm associated with construct9c code to look at/ normalize histogram trends. theres also an idea about/ code for combining the 0/1 runs into pairs for similar analysis as was seen in an isolated older experiment. but, am omitting the graphs for now to save time because they seem to be mostly null results (but not to be taken as an indication that theyre not worth looking at, ie/ aka “exercise for the reader”).

some caveat applies to the known overplotting bias for some of the diagrams ie sometimes patterns seem to appear that dont exist. overall anyway these are all powerful statistical analysis routines that have all revealed significant structure in different contexts but, despite substantial variation in trajectory/ glide statistics incl initial postdetermined iterate bit width (as seen in the earlier experiment, 1st graph output), here they all come up empty. but from long experience, theres no shame in null results esp with this problem. done well, keeps the knife shiny/ sharpened.

construct31.rb

rethinking the vary1 code just mentioned. a basic property long noticed is a gradual uptrend in density and entropy variance at the end of trajectories. this can be seen in many experiments, eg review124 from more recently at 10/2018 sticks in my mind but was probably 1st noticed years ago almost immediately starting with density measurements. on closer examination it looks like vary1 code is just reflecting this in a slightly different way. apparently all trajectories tend to “end” in this similar way. it looks like nearly all the differentiation found in that experiment can be attributed to this “end differentiation.” but what is needed is some differentiable function for the beginning of the trajectory, not the end. there is some hint of this in the vary1 diagrams esp the end. it looks like its gonna take some more careful analysis to isolate something here. is any of this attributable to initial-trajectory instead of end-trajectory? ie exploitable…

(8/15) 💡 ❗ had some idea to look at this density/ entropy variance “wedge” in closer detail, dont recall doing so beforehand, although there were some similar experiments. this generates 100 ½ density seeds and sorts by trajectory lengths, color coding by that. then it overplots density and entropy trends in 1st 2 graphs, aligned by start of sequence. in 2nd two graphs the sequences are aligned by ends instead. the 3rd and 4th pair of graphs have a new idea. its interesting to look at density by seed width. and then one can also look at the 1st value by width or the last value by width using a hashmap for replacement. surprisingly in the 3rd pair for density, the 2 graphs are different! in the left leading top edge, from downsloping in 1st case to flatter in 2nd case. in the 4th pair by entropy, there is not much difference. this is suggesting there is some trend to density as it “enters and exits” by bit width of iterates! ie some new differentiable feature, in brief “entry has higher variance than exit”, and needs more investigation (it would definitely help to plot the diference in a single graph). this reminds me of stock market candlestick graphs and maybe will try to build something like that in this case. its also a surprise that the density/entropy-by-width graphs are generally differently looking/ shaped than the density/entropy-by-sequence. needs more investigation. ❓

construct39.rb

⭐ ⭐ ⭐

💡 😎 ❗ 😮 this is a very straightfwd idea carried out and seems to have deep implications. this generates the 100 ½ density distribution, sorts by trajectory length ‘c’ red, and then looks for trends in density effecting the overall outcome, graphed right side scale. the results were striking so quickly hooked up 100 sample averages with the nice/ very handy avgs routine to take out the noise. ‘da’ is average density in green and roughly tracks glide length, although glide length is a sigmoidal trend and ‘da’ is concave downward. an earlier experiment about ½ yr ago 2/2016 looked at density parity and this is now shown to be pivotal. density parity ‘dp’ blue closely tracks the total glide length ‘c’.

and heres a new extraordinary finding! the collatz operation was seen as a sampling operation from a random bit string, and this reveals it in a new way. a “power sequence” is constructed n·3x where ‘n’ is an initial iterate and ‘x’ is the sequence length (position). this sequence has a density (for each iterate) and a density parity (count or average) associated with it. the average density ‘d3a’ is flat, but the density parity ‘d3p’ closely tracks the collatz sequence density parity. remarkable! something very deep is revealed here. there must be big implications to all this, have to assimilate it and build on it. at least, it does seem to confirm that density spread at initial part of trajectory can discriminate trajectory lengths, for the drain “region” (or integer set). ❓

❗ 💡 immediate idea! these experiments seem to be showing that the drain set has a range of different nearly-linear slopes based on the initial entry iterate so to speak (probably around the 1st postdetermined iterate). this seems utterly basic, is hinted at in probably many prior diagrams, but has been missed until now. holy cow! 😮 ⭐

construct40.rb

construct40

 

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2 thoughts on “collatz microscope

  1. John Duffield

    HI vzn. I had to laugh at this from bolbteppa: “I agree, but the ‘naysayer’ arguments are veering close to an analogue of climate denial when they call things like susy/sugra/strings “failed science””. LOL. I agree with Sabine Hossenfelder about putting lipstick on a pig. Only I think it’s worse than even she thinks. See this: http://physicsdetective.com/supergravity/ : “Poor old Yuri Milner. I am reminded of Alfred Nobel, whose prizes have done more harm than his dynamite. Milner tried to do something good and noble to help physics along, but he didn’t keep control, and now his prize has been hijacked by the quacks and charlatans who caused all the problems in the first place.”

    Reply
  2. Alberto Ibañez

    hello vz.i am Looking a study or post about the properties of Collatz binary relation. If the conjeture is true there is just 1 class of congruence and it is equal to the quotient space.therefore is a disjoint union of subset,/subgraph. The function preserves the disjoint union. I think the binary region is the cornerstone of the Conjecture.thanks

    Reply

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