A Formal Resolution of the Beal Conjecture via Symbolic Ancestry and Arithmetic Embedding

We present a formal resolution of the Beal Conjecture by embedding symbolic ancestry from the AION framework into classical number theory. This framework constructs all natural numbers through symbolic closure under multiplication and addition. We demonstrate that any exponential identity involving three numbers raised to powers greater than two must involve a shared symbolic ancestor. This shared ancestry guarantees a common divisor in standard arithmetic, thereby confirming the Beal Conjecture structurally. No counterexample can exist under this construction.