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

Abstract: 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) = ∑₏ ∈ ₏ zᵖ The Goldbach function G (2m) = (1P ★ 1P) (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 HG (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 (e^iθ) * on the unit circle to the eventual positivity of G (2m), via the Szegő condition: ∫₀^2π log |Π (e^iθ) |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