Symbolic Resolution of the Odd Perfect Number Problem and the Collatz Conjecture via the Abdeslam Irreducible Order of Naturals (AION)

We apply the Abdeslam Irreducible Order of Naturals (AION) framework to resolve two of the longest-standing problems in mathematics: the existence of odd perfect numbers and the Collatz conjecture. AION defines natural numbers symbolically from unity using prioritized multiplication and fallback addition. Under this structure, we prove that any odd number with divisor symmetry must be symbolically reducible, contradicting the irreducibility required for perfect balance. For the Collatz iteration, we show that every path undergoes symbolic compression and ultimately folds to 1. These results demonstrate that symbolic ancestry and irreducibility are sufficient to resolve both problems deterministically.