On Odd Perfect Numbers - The Dyadic Fabric: A Deterministic Refinement and Elimination Framework

This manuscript studies the odd perfect number problem using congruences modulo powers of 2 and an inverse-limit formulation of coupled residue data for integers and their divisor sums. It develops a refinement framework that tracks how the condition σ(A)=2A behaves under increasing dyadic resolution, isolates the Euler-form regime at the first nontrivial level, and proves an exact halving law for admissible classes under subsequent refinement. The paper then interprets these dynamics in an inverse-limit setting to describe the structural persistence required for any odd perfect number to survive all dyadic levels.