I think at least where I personally realized where a dead end was, about when everyone stopped is from how a better bound on the irrationality measure of log base 2 of 3 would help to or be able to disprove loops if it turned up to have a measure of pretty much 2, but I’m not sure if it would drop to 2, even if we could brute force a near infinite amount of accuracy/precision or find the exact measure, let alone better bounds.
Following this thread, I went looking for a case where the two-member difference method fails, but combining four cycle members succeeds.
Background: The subtraction method finds two cycle members whose difference can’t be an integer. In the extreme case of the high cycle, the difference is \cfrac{2^{l-2}}{2^l - 3^s}. Since the numerator is of the form 2^i 3^j, we can consider this an elementary divisibility-type result, not dependent on the magnitude of 2^l -3^s or the density of odds in the cycle.
Consider the family of parity vectors 01 (10)^b 001 (10)^b.
For example, with b=3, we have v = 01.101010.001.101010.
No two rotations of v have a \beta-difference of the form 2^i 3^j.
But
(\beta_2 - \beta_4) + (\beta_{11} - \beta_{13})
= 2^4(2^9-3^4) - 2^4(2^8-3^4)
= 2^4(256)
= 2^{12}.
So the combination of these four cycle members is
\cfrac{(\beta_2 - \beta_4) + (\beta_{11} - \beta_{13})}{2^{17}-3^8}
= \cfrac{2^{12}}{2^{17}-3^8}
which can’t be an integer.
Although, with two members, you can still get there, if you are trickier:
545 \cdot \beta_{11} - 529 \cdot \beta_{13} = 2^{12} 3^4
which again, 2^l - 3^s can’t divide.