The Exponential Primitive in Functor-Model Aggregation: Range Reduction, Padé Approximation

Softmax-style aggregation attention accumulators, mixture gates, and any weighting of the form rests on repeated evaluation of the exponential function. This note records how the exponential is made cheap in practice (range reduction followed by rational/Padé approximation on a bounded interval), places that cost reduction correctly relative to the structural energy channels of the surrounding architecture, and states the numerical conditions the technique imposes on deterministic replay. We are explicit about provenance: range reduction and Padé approximation are established numerical analysis and are cited, not claimed. The contribution of this note is placement — distinguishing a constant-factor primitive floor from the asymptotic, structural channels and the observation that the range-reduction scheme is a third replay-exactness condition alongside fixed reduction order and fixed dtype.