Morphological Completeness of Positional Bases: Classification, Minimum Threshold, and the Structural Obstruction of Historical Number System

Every positional expansion of a unit fraction 1/n in base b belongs to exactly one of six morphological forms, determined by three independent binary parameters with two structural collapses. We define a base b to be morphologically complete if all six forms are realized for some n ≤ b, and prove that the minimum complete base is b = 15. We establish a necessary and sufficient arithmetic criterion for completeness depending on the joint factorization of b and b − 1. No composite base with b − 1 prime can be morphologically complete — a condition that excludes every historical number system. Part of the sub-limit dynamics research series.