I began by exploring if there was a bias to which "phase" numbers were in when they terminated, if 400 steps of 3n*1 or 2n are applied regardless of whether it took 3 steps or 303 steps to reach 1, the number at the end of the 400th cycle would be either 1,2 or 4. while I observed a visual pattern in this, there was no overall bias to the "phase".
Next I explored how consecutive numbers were in phase how many pairs of numbers both were in phase of the 4,2,1 cycle how many triplets, qudruplets etc. Up to strings of 32 consecutive numbers being in phase. Again I saw a general pattern but nothing mathematically defineable.
http://imgur.com/q3DVOWa You can clearly observe what i explored as consecutive integers being in phase, it is where the colours lined up
I then explored whether if at the 400th iteration would pairs more likely occur if they ended in 4,2,1. but again there is no bias, 1 appears to form longer pairs, but that may be due to sample size.
Next I explored if there was a "prime bias" as to if I started with a prime number, would it be 4,2,1 at the 400th iteration. But again there was no clear bias, the primes appear to terminate evenly across 4,2,1
On being surprised that what I expected to show bias didn't I began to investigate if the numbers had "forms" (this relates to previous work on Goldbach)
I realised I could ensure that every number was in "phase" if I started with N=1 --> infinity, if odd --> pause, if even divide repeatedly until odd was reached. There was then 100% certainty my current number was odd. Then I applied the algorithm.
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(here is my initial explanation which I wrote when I first went to reply to your thread, I have re-evaluated it to the 2nd version shown outside of these lines)
Final edit: added this: http://i.imgur.com/34kvzE2.png This is my actual first draft, I used 2X before I realised it was of the 12n+2 -->6n+1 odd
Your number can only be odd (N)
Multiply that number by 3 and add 1 3*(N)+1
Your number can only be even it has the form 6n+4
since 6n+4 is even, it can be halved to 3n+2 3n+2
3n+2 is either odd or even, it has 3 potential forms 3n+2 = odd -->3*(3n+2)+1 --> 9n+7 (has form 6A+4)-->3A+2
3n+2 = even (it is of the form 12X+8)-->6B+4 -->3B+2
3n+2 = even (it is of the form 12X+2)-->6C+1 -->18C+4 (which has form 6D+4) --> 3D+2
Here are some inequalities:
6n+4 = 3*(N)+1 > (N)
6n+4 > 3n+2 > (N)
6A+4 = 9n+7 > 6n+4 > 3A+2 = 4.5n+3.5 > 3n+2 > (N)
6n+4 > 3n+2=12X+8 > (N) > 6B+4 > 3B+2
6n+4 > 6D+4 > 3n+2=12X+2 > 3D+2 > (N) > 6C+1
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My current explanation: (about 8 hours later)
Since N was odd, and 3*N+1 was even, a smaller set of even numbers exist which would be made directly from this initial step from odd.
They are all of the "6n+4 form" which obviously is halved to "3n+2" and that is where the fun begins.
But the value at "3n+2" has three distinct forms
it is either odd of form "6W+5", of form "12X+8", or of form "12Y+2"
I.e. it is either odd (which has form 6W+5), so multiplied by 3 and 1 is added, transiently 18W+16 --> returning to the "6n+4" form. but with a different variable ("6A+4")
it is of the form 12Y+2 it can be halved only once to 6B+1 at which point it returns to the "6n+4 form" but is equal transiently to 18B+4 (also 6C+4)
it is of the form 12X+8, it can be halved to 6X+4, and halved again to 3X+2 where there are multiple paths it can take
I realised the route it took depended on the starting odd number, which can be split into 3 forms:
8n+1-->(12Y+2)
4n+3-->(6W+5)
8n+5 -->(12X+8)
if we use a small n, which can take values 0 to infinity and use it to construct the numbers:
(out of the first say 5000 numbers)
Total times
1250 8n+1-->24n+4 --> 12n+2--> 6n+1 A reduction of 2n (-2n)
2500 4n+3-->12n+10--> 6n+5 An increase by 2n+2 2*(2n+2)
1250 8n+5-->24n+16--> 12n+8--> 6n+4-->3n+2 A reduction of 5n+3 (-5n-3)
This leads to a net reduction. It is of "-3n+1"
The key principle is redefine each number at an appropriate step into ta different form. This prevents spiraling into decimal values of n + decimal values and huge values because the 6n+5 form increases, which I believe cannot be shown to decrease as it can always potentially create a larger odd number.
Ultimately when the 4,2,1 cycle is reached, it comes from 6*variable +4 --> 12*variable+2 --> 6*variable +1 where the variable is zero, at this point the variable cannot be converted into other forms, and this is why the 4,2,1 loop occurs, cannot be escaped, and why every integer will enter the entire Collatz cycle and eventually reach 1, before repeatedly cycling to 4,2,1. (this is my hypothesis anyway)
I believe this is where your tiers come in, you notice the + 32's, 64's etc powers of two. e factor, I look at it through the fundanental value of n which makes up the number. (I admittedly haven't spent enough time going over your work compared to my own, but this is how I perceive what you have presented. It describes the comb like graphs which can be generated at least)
Returning to my work:
8n+1 decreases by 2n (has n value of 4n and then n=n) (0,4,8,12...)-->(0,1,2,3....)
4n+3 increases by 2n+2 (has n value of 2n+1 and then only 3n+2) (1,3,5,7...)-->(2,5,8,11...) (1-->4-->2, 3-->10-->5,5-->16-->8,7-->22-->11)
8n+5 decreases by 5n+3 (has n values of 4n+2 --> n=n) (2,6,10,14...)-->(0,1,2,3,4...)
example from 8n+1
SEE IMAGE THE TABLE BROKE http://imgur.com/k7PQWTg
from 4n+3
SEE IMAGE THE TABLE BROKE http://imgur.com/k7PQWTg
(I believe it is only needed to go to the ODD, but I continued this for some reason) it demonstrates that the "n" values follow the collatz conjecture I think?
because N started off as odd, in converting a form, it causes the "n" values to be initiated into the Collatz cycle, just like we did by initially ensuring only odd values could begin the cycle. Because only n values here are odd, when manipulated, they enter the cycle. (This isn't coherent really, I'll return to it at some point)
from 8n+5
SEE IMAGE THE TABLE BROKE http://imgur.com/k7PQWTg
I think I've shown the Collatz is contained within the Collatz? (again I'll return to this, my work is as I say in it's infancy atm)
http://imgur.com/0EQymYv
The values in green are unique values, it shows that odd numbers enter via 8n+1, 4n+3, 8n+5 but can only exit via being even in the form 8n+5, they either at this point halve further to 1, or halve to an odd number which previously cycles to 5,7, etc. a number which can be written as 3n: 3,9,15,21 cannot re enter the cycle unless it is the result of a halved even number. But this has already occurred previously, because all halvable evens were halved to odd before the algorithm was applied. (also starting to lose clarity, I'll readdress this at some point)
http://imgur.com/3UmbjRz
This is the image I had obscured, I guess it didn't really detail to much, just demonstrated what I'd use as forms, of course with this information my work is repeatable but they've already been explained above.
http://i.imgur.com/ZZ6B1zl I generated this exploring average values (I won't detail the method, but this is the result of it across 400 passes as mentioned earlier)
http://i.imgur.com/08mZo5N has similar origin to the graph above, this is looking at the tail end of what occured in the first graph that seemed to reach almost zero.
I have an array of beautiful graphs, I have not seen replicated through an image search, but they don't really add too much of value, except it is evident that there is a cycle going on here and with each stage something is filtered out, leaving yet another unique graphical pattern (I like graphs, I guess that is also why I noticed a fair amount in a short time, as I just generate mass amounts of data and graph it, if there is a pattern I analyse it, if not I try something else)
This is quite a large part of my work, but not all of it, and I'm still looking to explore deeper. I would be interested your thoughts on how you perceive what I've presented relates to your "tiers", and perhaps if you have any questions it can guide my direction to further explore this. I suspect not much of this is novel to you, but I just wanted to demonstrate I really did formulate this independently of your work, although I've since seen the discussion here and on mymath. Which I guess means I can't claim anything I do from now i did wholely without influence. But I was just surprised that by Googling some of my findings I found this particular thread, and you had only just yourself made a video of your findings.
Good luck to you, and maybe we can collaborate on this? this is what I'm willing to release publicly at the moment,
Based on your statement that your only protection to your work is what you have time stamped through usinIg this, and my Math. I will do the same, which i why I used an initial reference number at the start of the message, It is so I can prove when I released my findings. Sorry if I came across negatively, but I'm sure you can understand my position.
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For my reference: 1928372