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Update: 2 Reflection see at end
Update: 1 I have just edited the title of the question and created a new question where I have added my given answer.

This question asks in the title for
$\qquad \qquad$ "what is the longest cycle in the Collatz ruled out so far" (1).
However, the answers which are honoured are answers for the rather opposite question
$\qquad \qquad$ "What is the currently largest known lower bound for nontrivial cycle?"(2)

The likely misformulation was indicated towards the asker in a couple of comments, and because his name is "Learner" and has low profile, one would take in account, that this might be a neglectable misnoming of the problem.

Now the OP even assigned a bounty towards an answer, which answers the (2) version of the question and makes sure that this was also his intention.

I had already deleted an answer of mine (because of deletion now visible only to $\gt 10k$ people) which made explicite the obvious misunderstanding in the title to the asker, which the asker seemingly had not been able to reflect correctly.

However, I invested much in answering for the current formulation of the title (1) and, moreover, found it now an interesting problem in itself, never been handled before this way (at least I'm not aware of such a discussion).

I've been reluctant to simply edit/adapt the title according to my interpretation of things, but tend now to do so. I think to also create an own question asking for the "currently longest explicitely disproved nontrivial cycle of general type" (except the "$m$-cycles" with low $m$ which are disproved by Steiner/Simons/deWeger for any length)

I ask here because such a rigorous reformulation (instead of simply typo-correction etc) might be felt much rude by the asker.

What do you think? Editing the title of the OP, create an own question with appropriate title to which my answer is directed and move my answer towards that new question?

Update 2
Possibly the misconception of question and answers has been expressed more clearly in the comments to my first answer which was but downvoted and was not reflected properly, and which I had thus deleted for being not welcomed.
At rereading my first (and deleted) answer and the comments there again, it seems now that the problem has been with understanding the "k-cycle" concept (which would likely better called "k-peak-cycle" btw.) in difference to some "N-step-cycle", where "N-step" means a length of the cycle and "k-peak-cycle" means "hilliness" of a cycling trajectory, or how many local maxima are existent, independently of any length of that cycle in question.
See here the screenshot of the comments:

image

Likely the "Learner" changed his focus of question after this, not catching the meaning of the explanations at all, and adapted the interpretations of the other answerers as his original intention.
So maybe the whole process of discussing, editing, discussing it here is a double ore triple misunderstanding.

Anyway, the asker didn't seem to really communicate further and I think any more change by my part would be obsolete here.

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  • $\begingroup$ If version (1) interests you, it might be best to (a) edit the title (unclear if the failure of the OP to do what is clearly a correction is due to misunderstanding) to match the problem being bountied/rewarded, and (b) post your own version of (2) with the intention of supplying your results (either as an Answer or as information in the Question body that you'd like replies to extend). Of course a link from your "new" Question to the old one (with a bit of discussion of relationships) will be helpful to future Readers. $\endgroup$
    – hardmath
    Mar 7, 2019 at 3:38
  • $\begingroup$ I've just edited the question in question. Let's see, whether the problems are resolved... $\endgroup$
    – Gottfried Helms
    Mar 7, 2019 at 17:17
  • $\begingroup$ I look forward to reading your post! $\endgroup$
    – hardmath
    Mar 7, 2019 at 17:33
  • $\begingroup$ @hardmath : Ah,forgot. My new question is here math.stackexchange.com/q/3138566/1714 $\endgroup$
    – Gottfried Helms
    Mar 7, 2019 at 20:12
  • $\begingroup$ "Redact" means to edit into a form that is suitable for release to the public, usually by censoring sensitive information. So your title is also inaccurate. (Apologies if that was a deliberate joke that just whooshed straight past me.) $\endgroup$
    – David Richerby
    Mar 8, 2019 at 16:51
  • $\begingroup$ @David - no joke. I've not a big resource in english verbs (not my native tongue) and maybe another term would be appropriate. I just wanted to hint that I felt such a change would be more than a small correction of typos or one or two words (by this full refocusing of the question likely being rude to the original asker likely being a newcomer). If you've a better term, feel free to improve my expression! $\endgroup$
    – Gottfried Helms
    Mar 8, 2019 at 17:12
  • $\begingroup$ @GottfriedHelms I think you just mean "edit", so I'll... er, edit. :) $\endgroup$
    – David Richerby
    Mar 8, 2019 at 17:14
  • $\begingroup$ @DavidRicherby - well, just one which does not capture my concern that my correction would not only be a correction but refocusing; "edit" had seemed to me too neutral, not showing here my concern. But well - if there is no better idea, I'd resort to "edit" then. Language learning really rocks :-) True, after thinking & sensing one more moment "redact" might really be a reference to some official measure and should be avoided here. So, thanks for the impulse,too! $\endgroup$
    – Gottfried Helms
    Mar 8, 2019 at 17:17
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    $\begingroup$ @GottfriedHelms I agree that "edit" is kind of vague but sometimes it's difficult to say it all in the title. I think your post as a whole makes it clear that your concern is that the edit might be to big (the sort of thing that might be rejected as "clearly conflicts with original author's intent" if your edits still needed to be reviewed). $\endgroup$
    – David Richerby
    Mar 8, 2019 at 17:50
  • $\begingroup$ @DavidRicherby - not really related to your comments, but having three curious incidents in this class of problem in a short time under the tag of collatz-problem I'm getting curious whether we see live the training of a math-bot here... That would make all my reservation in my question and comments redundant... </paranoia off> Just flagged one of that problems towards to mods. But enough said here - thanks for your input! $\endgroup$
    – Gottfried Helms
    Mar 8, 2019 at 18:03

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