A Spectral Framework for the Goldbach Conjecture: Natural Boundary Barriers and Winding Invariants

A Spectral Framework for the Goldbach Conjecture We develop a frequency-domain framework for the strong Goldbach conjecture based on the additive convolution structure of the prime indicator series: Π(z) = ∑p ∈ P z^p The Goldbach function G(2m) = (1_P ★ 1_P)(2m) is expressed exactly as the inverse discrete Fourier transform of the power spectral density |Π̂(k)|², yielding an exact DFT convolution identity. Key Contributions Topological Invariant (Winding Structure): We introduce an analytic signal z(m) = G(2m) + i H{G}(2m), together with an associated winding invariant. This provides a topological characterization of the non-vanishing of G(2m) and isolates a precise obstruction to converting this structure into an unconditional proof. Abel-Regularity Framework: Under an explicit Abel-regularity hypothesis (H) on the boundary behavior of Π, we establish a conditional chain linking the non-vanishing of Abel boundary values Π(eiθ)* on the unit circle to the eventual positivity of G(2m), via the Szegő condition: ∫₀2π log |Π(eiθ)|² dθ > −∞* Analytic Barriers We analyze the zero-density support of primes and show that the unit circle acts as a natural boundary for Π, forming a fundamental analytic obstruction. Three primary barriers to an unconditional proof are identified: Failure of Π to lie in the Hardy space H²(D) The lacunary natural boundary induced by zero-density prime support Connections to Siegel zeros at rational frequencies Unconditional Contributions This work establishes: The exact DFT convolution identity for Goldbach representations The formulation and rigorous analysis of the winding invariant A precise decomposition of analytic and arithmetic obstructions within a unified spectral framework