u/Particular-Cut-5982

My investigation has focused on strings of E and O operations (i.e., E, O, EO, EEO, EEOEO, or OEOEOEEOEOEOEEEE, etc.) and their potential to be Loops (i.e., {0}, {-1/2}, {-1,-2}, {4,2,1}, {-20,-10,-5,-14,-7}, or the 18-cycle Loop, respectively).

For any given string of E and O operations... such as EEOEOEOEEEOEOEEO...
...where E=10 and O=6, and L=E+O=16
...and (2^E)-(3^O) = (2^10)-(3^6) = (1,024)-(729) = 295 > 0
...there exists one and only one rational number that can be both an input to the string, and an output.
...and since (2^E)-(3^O)>0, that rational number must be positive.

In the case of EEOEOEOEEEOEOEEO, that number is...
N = Σ / (2^E - 3^O)
N = ((2^2)*(3^5) + (2^3)*(3^4) + (2^4)*(3^3) + (2^7)*(3^2) + (2^8)*(3^1) + (2^10)*(3^0)) / ((2^10) - (3^6))
N = ((4*243)+(8*81)+(16*27)+(128*9)+(256*3)+(1,024*1)) / ((1,024) - (729))
N = (972+648+432+1,152+768+1,024) / (295)
N = 4,996/295, or approximately 16.9356...

I also understand that the first positive integer that can successfully transit that string, and beget another integer, is 732 which begets 526 through EEOEOEOEEEOEOEEO.
732->366->183->550->275->826->413->1240->620->310->155->466->233->700->350->175->526

I also understand that the next higher input is 2^E greater, while the next higher output is 3^O greater.
732->526
732+2^10=1,756
526+3^6=1,255
...and sure enough, 1,756->EEOEOEOEEEOEOEEO->1,255.
(Note: I understand that 1,255 is a Collatz-INappropriate output, but that's irrelevant to this small point I'm getting at. But if it bothers you, you can increase the increment to 2*(2^10) to get 2,780 as your next input, and 1,984 as your next output.)

Meanwhile, the next lower input is 2^E smaller, while the next lower ouput is 3^O smaller.
732->526
732-2^10= -292
526-3^6= -203
...and sure enough -292 ->EEOEOEOEEEOEOEEO-> -203
(Same note as before.)

Having fun, so far? Are all your calculations aligning with mine? Great! Here's the next part...

Let's call the linear difference between that input and output "the gap".
-3,364 -> -2,390 ___gap=974
-2,340 -> -1,661 ___gap=679
-1,316 -> -932 _____gap=384
-292 -> -203 _______gap=89
xoxoxoxoxoxoxoxoxoxoxoxox
732 -> 526 _________gap=206
1,756 -> 1,225 _____gap=501
2,780 -> 1,984 _____gap=796
3,804 -> 2,713 _____gap=1,091

Here's my question: Are there any strings where the smallest gap is more than 2^E away from the zero line?

Or, rephrased: Are there any strings where the first positive integer input to yield a positive integer ouput, has a gap that is greater than the gap of the second positive integer input to yield a positive integer output?

My intuition is telling me that there aren't... but I'm not sure why.

Does anyone know if a study has been conducted that asks participants to reveal their memories of *multiple* ME's from a long list? The idea would be to see if people who have the Berenstein memory, also have the C-3PO silver leg memory, but not (say) the FotL Cornucopia memory.

If this survey happened, would we expect groupings of MEs together? Or would it be evenly distributed amongst all the MEs? Perhaps certain regions or age groups would be more likely to have some ME memories than others?

If anyone has a link, I'd appreciate it. Otherwise, maybe I need to get to work on it, myself.... for fun-I MEAN DATA POINTS!! FOR DATA POINTS!!