the word “search” is something to ponder wrt this effort in math “re-*search*“. there are different optimization algorithms as a theme but in general theyre different types of searches for trajectories. in a way its all about searching for some kind of invariant property of trajectories, both with the computer/ algorithm and more generally in the research (“program”). have come up with many ingenious search methods and that effort continues.

# Category Archives: collatz

# collatz marathon

yep, its definitely a marathon at this point, maybe now mostly comprised of month-long sprints (applying “agile” philosophy to mathematics/ research in the weird hybrid style typical of this idiosyncratic approach… *thats not a bug its a feature™*… or maybe in some sense that the historic Thai cave rescue was also *marathon*…). heres another new idea building on prior ones. earlier analysis looked at the “sibling trajectory” and found a strong correspondence with the trajectory. a fairly basic idea is to compare how much of the sibling trajectory matches wrt the climb length of the glide. in a sense the sibling trajectory “covers” some percent of the climb length.

using the latest generated glides (`mix26`

scheme) this finds that in ~40% of the cases the sibling trajectory covers the full climb. in other words in 40% of the cases the computation to find the highest point of the glide is exactly repeated in a “lower” glide/ trajectory. in the remaining “uncovered” cases theres still 1.0-0.3 = ~70% coverage at worst and more typically/ on avg in the ~95% range. the graph is the ‘icm’ variable in impulse format which is matching length of trajectory and sibling divided by ‘cm’ glide climb minus 1, here 0 means matching the exact length. as seen in the graph the non-fully matching siblings still cover most of the climbs.

# collatz new strategy

hi all. on vac this week & doing some new stuff (happy BTD US). there is a semifamous thousands-year old quote by sun-tzu maybe not yet contained in this blog (its been going thru my mind for quite awhile now, but wasnt able to find it in the blog via google). it is a quite favorite quote of business consultants which might tell you something about modern *“leave no prisoners”* business attitudes/ culture in our at-times militaristic/ hypercapitalistic modern age. (dramatic alpha-male stuff, but to put it more bluntly, one with a conscience/ empathy/ independent mind might wonder about the *“fine print,”* ie how many *“enemy…”* ** men **did sun-tzu kill personally or oversee killing as a general? …or even

**which includes**

*humans***? oh but ofc its utterly**

*women/ children**metaphorical*right?) 😮 😳

Strategy without tactics is the slowest route to victory. Tactics without strategy is the noise before defeat. –Sun Tzu.

collatz has been described as the very impenetrable/ unconquerable adversary. strategy/ tactics both play a key role and have commented at length on both. they are like a yin-yang combination. victory will likely not come without some kind of balance between the two.

have been more/ very tactical for quite awhile but have been musing on some overarching strategy/ perspective/ pov lately, thinking it all over at current point. this involves more abstraction.

couldnt find this basic idea pointed out in old blogs. was it? the key question is to prove *f(x) < g(x)* for all *x*. here *f(x)* is collatz stopping distance or some similar metric and *g(x)* is “any recursive function” (either time/ space bounded). now apparently *f(x)* in many related forms has extreme entropy, the “needle in haystack” property, and also “fat/ long tails” distribution, and fractal. earlier blogs have outlined the idea that it appears that victory seems to lie on the path of decreasing or minimizing entropy somehow. *g(x)* can be regarded as an orderly function from analytic mathematics and *f(x)* is “far from it” in the sense of being extremely disorderly.

# collatz shift

this months title is hopeful and in line/ theme with the last entry title. the disordered climbs were found to be a major wrench in the works of many months of analysis. however there are now some signs that even though seemingly without signal, maybe there are some angles to leverage on the disordered glides. am starting to get some general ideas. have developed/ honed some even stronger tools/ techniques. will they be enough to crack the problem?

this code is a modification of `review51`

to extract the recent/ latest generated glides from `mix25g`

. it turned out extracting usable climbs from the table/ database was really nontrivial. 1st this code looks for the most common bit width encountered in the glides (climbs). this ends up to be exactly 200 the same as the bin count. this is interesting and maybe worth exploring further: all the lower glides are tending to cross (repeatedly) through that section. there are ~10K total iterations. then the climbs containing at least 1 iterate with that width are selected.

# collatz new highs

am still clawing/ scrounging for any “big picture” leverage, resuscitating old leads. after some extended musing of new findings, came up with this latest idea. there are lsb triangles even in the disordered climbs for the uncompressed mapping. do these mean anything? seemed to see some cases/ trend where the triangle sizes successively decrease in the climb, ie dont successively increase. can this be quantified? this is somewhat similar to the old workhorse “nonmonotone run length”. instead, its something like “count of new highs” (in bit run lengths either 0/1). that statistic is (maybe not uncoincidentally) used in stock market analysis, which has to deal with some of the most wild/ intractable fractals of all.

it was not hard to wire up the last “dual/ cross-optimizer” (ie both within fixed bit sizes and also increasing bit sizes) to calculate this metric, named here ‘mc’, serving its intended purpose of trying out new ideas quickly. ran it for 100 bit width seeds and then it did indeed seem to flatline somewhat. upped the seed size to 200 bits and then more of a (very gradual) trend is apparent. it looks like a logarithmic increase (‘mc’ red right side scale, other metrics left side scale). ‘mw’ is the # of iterations since last max/ peak run length. the optimizer ran for a long ~650K iterations.