Spiral geometry, laminated arms and Kloosterman dispersion for Goldbach - An Archimedean–modular formulation with an eff ective threshold matched to the computational boundary

We present a conditional proof framework for the strong Goldbach conjecture based on a spiral-geometric reformulation of the additive prime problem. The method associates prime values with quadratic arms on an Archimedean spiral and converts the Goldbach correlation into a weighted intersection functional. The central analytic device is a laminated family of quadratic arms, designed to overcome the small-BBB bottleneck by introducing an additional long oscillatory parameter. After local projections, the remaining oscillatory contribution is reduced to a closed Kloosterman interface, supported by BridgeNorm stability, a rho-first normal form, and an explicit model certificate. The effective constants are tracked throughout the argument. In particular, the analytic threshold is shown to satisfy: N₀eff,new ≤ NG = 4 · 10¹⁸, while the finite range 4<N≤NG is covered by the known computational verification of Goldbach. Thus, conditional on the mathematical completeness of the technical appendices CKI/BridgeNorm/rho-first/ModelCert, the argument covers every even integer N>4.