Figure coordinates · A Small Switch in a Vast Machine

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The same probability state follows a straight path in the three-class simplex and a curved path in log-odds coordinates. Equal-time markers and probability bars stay synchronized. — Dual coordinates · Exact mathematical model
A straight path in probabilities generally curves in log-odds. Both panels describe the same changing distribution. This softmax coordinate duality is distinct from the relationship between weights and activations.
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Exact three-class softmax toy, not a Qwen landscape. Natural coordinates are the log-odds $h_i=\log(p_i/p_3)$ ; expectation coordinates are probabilities on the simplex. The curved path in natural coordinates and the straight path on the simplex are synchronized with the same probability bars.
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