I knew because I worked out every path in base 16
4#+1 goes up
4#+3 goes down
But if you got to 16# and look at the 4#+3 cases and work them all the way to end of information.
16#+3 only does down
16#+7 is conditional up conditional down
16#+11 is conditional up conditional down
16#+15 is unconditional up
Half the up cases turned down.
The simple to understand reason is as you look in more and more base 4 digits the up gets more numerous but the down get more numerous AND some of them get stronger.
This is because there is always a direct path that hits 2^# and falls all the way to 1.
So as you add digits the downward force become more and more powerful.
I now have the congruent number problem proof. But because I am older and Cornel participates in age discrimination I cannot get a ARXIV while they were giving them away to anyone with an email address they liked as recently as last year.
My congruent number proof is built on the 2 characteristic equations of the congruent number problem
n1 d1^2 +n2 d2^2 = n3 d3^2
and
n1 d1^2 +2*n2 d2^2 = n4 d4^2
t=n1 d1^2 / (n2 d2^2)
X=2+t
Y=2+2 / t
Z=X+Y-2
N=XY/2=n1n2n3n4#^2
This is really easy to test by picking n1,n2,and d1,d2 where n1 and d1 must be odd and all must be relatively prime. Then calculate n3,n4,d3,d4 and you can see it always works. I can prove that all triplets map to 2 characteristic congruent number problem equations like this but it is a whole different math set up and would take you a long time to follow. there are 2 paths to creating the 2 equations and one of them always allows n1,d1 to be odd.
This is a Legendre Diophantine equation pair and they can each be solved independently.
First guess n1,n2,n3,n4 which takes 4^(primes in N) guesses. So we treat that as known.
Then use the minimal solution that is bounded in pre known finite space for Legrande Diophantine equations.
The two solutions are not synchronized d1 is not the same as d1' but I have logic that gets a True/False test regardless of that.
You can find it here
https://www.linkedin.com/feed/update/urn:li:activity:7431582366856462336/
Thank you for your help in getting an endorsement.