The sigmoid link introduces the named-boundary register. Only the midpoint is exact; the other sigmoid entries stay symbolic.

highlighted = computed this step

The sigmoid link

The sigmoid maps a score into the open unit interval. Its formula contains an exponential, so the curve is a named boundary guide.

σ(z)=11+ez\sigma(z)=\frac{1}{1+e^{-z}}
Named sigmoid guideThe table and guide pin only the exact midpoint; other sigmoid samples stay named.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zerosigmoid named-boundary guide1/2GUIDEzσ(z)-2σ(-2)-1σ(-1)01/21σ(1)guide curve named; only z=0 pins 1/2

The one exact midpoint

When z=0, sigmoid equals 1/2 exactly. The other sampled points are named symbols, not decimal values or exact claims.

σ(0)=1/2\sigma(0)=1/2
Named sigmoid guideThe table and guide pin only the exact midpoint; other sigmoid samples stay named.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zerosigmoid named-boundary guide1/2GUIDEzσ(z)-2σ(-2)-1σ(-1)01/21σ(1)guide curve named; only z=0 pins 1/2

Summary

Sigmoid and logarithm are named boundary operations here; the rendered structure is exact, but no decimal value is pinned for them. The threshold decision from the previous lesson did not need sigmoid.

named boundary: σ\text{named boundary: }\sigma
Named sigmoid guideThe table and guide pin only the exact midpoint; other sigmoid samples stay named.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zerosigmoid named-boundary guide1/2GUIDEzσ(z)-2σ(-2)-1σ(-1)01/21σ(1)guide curve named; only z=0 pins 1/2