Category Archives: collatz

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.

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collatz killer

hi all here we go with the latest installment. trying to come up with new names/ themes. again theres a “pivot” going on at the moment but maybe there are now too many too count. time now for an intermediate retrospective/ pov. my recent physics blog talked about “killing the copenhagen interpretation” and thats my latest idea for this problem. the problem is definitely “killer” in many senses of the word. it kills all great ideas launched against it, its like an impenetrable fortress.

there was a tone of optimism in a lot of prior writing. now looking all that over, it was based on a longtime theme that was yielding fruit(s) of labor(s). the basic idea is that there are locally computable “features” that can, with enough ingenuity, predict longterm glide behavior with high accuracy, and also generally explain other basic trajectory dynamics properties. this clearly ties in with the machine learning approaches. this research theme has been pursued for several years now.

however, last month there was a massive setback on this particular theme/ direction. did you catch it? to summarize, the features being used, mostly based on (binary) density, were leading to a lot of insights and leverage on the problem. but there was a moment a few years ago when the research started to focus on generating density-based seed trajectories instead of more generally. that turned out to be a major detour bordering on a mistake (in 2020 hindsight). 😮 😳 😥 😡 👿 o_O

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collatz trap(s)

the last installment ended with some idea of possible “traps” in certain metrics. this idea occurred to me quite awhile ago and didnt work out under some examination previously but theres some new ways of looking at this. in the past, it is clear that eg there is no strict density trap in the sense of one iterates density bounding the next ones eg wrt to the density core. but the last experiments led to a different idea. what if a metric is bounded over some count of iterations, does that limit future glide potential? its a simple variation, seems to be quite related, but is maybe the key twist that looks more plausible as a measurable/ consistent property.

this new experiment simplifies the code a lot and bounds distance-from-core labelled ‘dc’ and a entropy metric. the entropy is counting total # of 0-to-1 and 1-to-0 transitions in the binary form, scaled by the bit width. the (scaled) inverse entropy formula (aka “order”) ends up as one minus sum of count of 0/ 1 runs/ groups divided by iterate bit width, labelled ‘e’. this upper bound on the inverse entropy is equivalent to a lower bound on the entropy (because as mentioned entropy increases as glides progresses, and the potential trap is at the end; also note a “low upper bound (on order)” is a “high lower bound (on entropy)”). it finds a sharp transition point ‘e’ ≈ 0.40. (as mentioned, suspect that both low and high entropy may be tending to bound glide length, therefore maybe glide bounding wrt density-distance-from-core is inversely related to entropy-distance-from-core?)

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collatz pivot/ new ideas

hi all, have been working on some other ideas re A(G)I, heavily promoting them all over cyberspace + analyzing/ collecting copious references, and havent been banging on collatz quite as much last few weeks. honestly its a bit of a (well deserved) break or respite. however, its always at the back of my mind. feel that am getting close to a solution but theres a lot of trickiness/ subtlety in the current stage.

here is a new analogy/ pov. the linear regression is finding a “global/ local gradient”. for the theoretical trajectory it is both, for the actual trajectory there are local perturbations/ disturbances/ noise fluctuations in the global trend. the picture is something like the wind blowing a leaf. the leaf has a very definite position but does a sort of multi-dimensional (3d) random walk in the wind. the wind is a general trend. now the basic idea/ question is whether the leaf will land at a given location/ circumscribed area given a predictable/ consistent wind dynamic.

further thought, another way of looking at it is that the leaf has a very dynamic/ even sharp response to the wind depending on what its current orientation is, and it also has an internal momentum. actually since the (real) wind is typically so dynamic whereas the linear regression is fixed (although arriving at the final regressions was dynamic, cf earlier saga of that), one might instead use the similar analogy of an irregularly shaped object in a (more consistent/ uniform) fluid flow, maybe even a field.

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collatz ad infinitum

threw this code together and it was unusually/ surprising timeconsuming to debug yet has rather little code/ logic. the difficulty is related to the early-discovered property of how skewed (nonlinear) a random sample of trajectory distributions is. nearly all are very short and it seems quite possible its in line with some power law (never tried to fit it, but do have the overwhelming urge to, and just go hunting for as many power-law properties in the data one can find, strongly suspect they might be quite widespread).

the idea is to try to bias (or unbias depending on pov) the sample of randomly chosen trajectories (based on initial density spread sample) so that they are nearly linearly distributed in length. was developing the test data to test the trajectory (meta function) calculation logic. to be more exact was partly working off of the matrix6 code and wanting to improve the aesthetic graph results with “more linearity” in the sample. did not end up with meta function calculation logic tied to it worth saving, but given how tricky it was, do want to save this sampling code for future reference! this generates 10 samples of 500 points over uniform density, and then finds the min and ¾ the max of trajectory lengths, and tries to sample linearly over that range (starting top down), and the result is quite smooth as in the graph (of trajectory length).

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