Uniform cluster (φ~U[0°,90°])
Random cluster φ = 45°
O → centroid
0.707
= std_radius
centroid = midpoint
Sphere radius
0.707
EXACT — all binary pts
on sphere from midpoint
Intra-cluster d
1.000
= standard distance
(reference)
d_intra / sphere r
√2
right-angle theorem
holds exactly
Uniform cluster φ ~ U[0°,90°]
O → centroid
0.866
= √3/2
further than R centroid
Sphere radius
0.500
statistical mean
smaller than R sphere
Intra-cluster d
0.707
= 1/√2 sub-random
uniform-phase signature
cR ↔ cU distance
0.500
centroid separation
= U sphere radius
Key asymmetry:
The random cluster centroid is the midpoint — its sphere radius equals its arm length (both 0.707, exact).
The uniform cluster centroid sits further from the origin (0.866) but has a smaller sphere (0.500, statistical).
Cross-cluster point-to-point distance = 1.000 (standard distance) — any answer vector from a random token
is always a standard distance away from any answer vector from a uniform token, because E[(v_R−v_U)²] = 0.5 per bit
regardless of the uniform phases.
· Arm angle = arccos(√(2/3)) = 35.26°
(vs 48.19° between two uniform tokens).
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