Resolution of the Double Sum Paradox: A Rigorous Framework for Nested Divergent Series — Introducing the Ω-Summation Operator

In this paper, I resolve the long-standing ambiguity in nested divergent series by introducing the ΠΩ-algebra, a rigorous algebraic structure that distinguishes between different types of infinity. I introduce the Ω-Summation Operator (also referred to as the Judsan Summation), which formally defines the summation of divergent sequences over a ring. Using this framework, I show that the expression∑ₙ₌₁^∞ ∑ₙ₌₁^∞ 1/n + 1/nis inherently ambiguous and that different interpretations yield distinct algebraic forms. Specifically, I prove that the nested interpretation equals ω(γ + 1) + ω^2 + γ, while the separate interpretation equals 2γ + 2ω, where ω is a divergence unit and γ is the Euler-Mascheroni constant. The difference is γ(ω - 1) + ω^2 - ω. I generalize this result to k-fold nested sums:S_k = γ + ω for k = 1, and S_k = γωᵏ⁻¹ + ω^k for k ≥ 2. I further expand the Ω-Summation to alternating, exponential, and factorial series, prove a sufficient convergence criterion, compute a concrete 2-loop Feynman diagram, and compare the framework to Connes-Kreimer Hopf algebras and Écalle's theory of resurgent functions. This framework provides a rigorous foundation for renormalization in quantum field theory and offers a new algebraic tool for handling divergent series.