The Bell Angle Transform: A Normalized Framework for Bell-Like Functions and Tail Behavior

This preprint introduces the Bell Angle Transform and the Bell Angle which are a normalized scalar shape statistic for symmetric functions. that are bell-like and shaped The method uses peak height, half-width at half-maximum, and alternating periodization to compare normalized bell shapes on a common scale. The paper derives a selected interval formula for the Bell Angle, proves that its value lies strictly between 0 and 1, and studies several reference families, including heavy tailed cases. It also shows that the Bell Angle behaves affinely for matched width mixtures, with an application to a peak normalized pseudo-Voigt family. A second construction, called the Bell Vector, gives a complete representation of the normalized bell using a compactified Faber-Schauder expansion. The Bell Angle is intended as a comparison and screening statistic rather than a unique identifier of a distribution.