The exact attention lessons name multi-key softmax instead of decimalizing it. This bridge explains the cost of that honesty choice and points to Running a Real Model, where real floating-point softmax values are captured with byte provenance.
Why softmax stayed named
This book kept multi-key softmax as a named expression on purpose. The scores and mask are exact, but evaluating the exponential normalization produces a real number rather than an exact rational token. Showing that value as if it were exact would break the register discipline.
exact scores and mask→named softmax
Where the real number appears
In a real model, the softmax weight is a captured floating-point value strictly between zero and one. This lesson does not invent or pin such a value. Running a Real Model handles that register with raw bytes, trace hash, and approximation label.
real softmax value belongs to the captured register
Summary boundary
The boundary is the lesson: softmax is transcendental, so the exact wing names it instead of fabricating decimals. Book B shows the practical captured register. This bridge does not compute softmax, does not decimalize it, and does not claim learning or meaning.
named here; captured in the real-model bridge