At the equator the Robertson bound for S_x, S_y is zero — but the actual uncertainty product is ħ²/4. Zero bound ≠ zero uncertainty.
Bloch Sphere Heat Map (click to inspect)
Click any point on the sphere to see values.
Bound vs Actual as Function of θ
For spin-½: actual σ_{Sx}σ_{Sy} = ħ²/4 everywhere
Robertson bound = (ħ/2)|⟨S_z⟩| = (ħ²/4)|cosθ| — varies
Equality (saturation) only at poles (θ=0, π)