The One-Hot Target Selects One Term - Cross-Entropy, Exactly

A one-hot target collapses the sum before any logarithm is evaluated. The zero target entries drop structurally, leaving only the correct class probability.

highlighted = computed this step

The zero target terms vanish

The target is one-hot, so two entries are exactly 0. Those entries multiply their log terms by zero and drop structurally. The correct class has target entry 1, so only its probability remains in the loss.

y_i=0\Rightarrow y_i\log(p_i)\text{ drops}

One term is selected

The selected probability is 1/2. Therefore the full sum collapses exactly to a single named term: minus log of 1/2.

H=-\log(1/2)

Summary

The collapse is exact and does not require evaluating a logarithm. One-hot selection is arithmetic structure; the log value remains outside the exact register.

\text{one-hot selection is exact}