This work presents the 0:8 Dyadic Symmetry Kernel, a minimal, unique involutive mapping defined on the 9-element ordinal set D={0,…,8}. By imposing symmetry about the central index 4 and requiring the mapping to be closed, bijective, and involutive, we show that the reflection map f(n)=8−n is the only function satisfying all constraints. The construction produces four dual boundary pairs—(0:8),(1:7),(2:6),(3:5) —and one self-dual midpoint (4:4), yielding a complete and irreducible symmetry cycle. We further prove that the system is minimal (no smaller cardinality admits an equivalent structure) and unique (no alternative mapping satisfies the same axioms). The resulting dyadic kernel provides a foundational template for recursive symmetry analysis, non-associative algebraic structures, dimensional reflection systems, and invariant information mappings. This document formalizes the construction and presents a complete mathematical proof of involution, closure, uniqueness, and minimality.