Refined Minor Arc Framework: Two-Parameter Vaughan Decomposition and Boundary Smoothing Techniques (Supplement to DOI 10.5281/zenodo.16106593

This supplement to "Toward an Unconditional Proof of the Binary Goldbach Conjecture" (DOI 10.5281/zenodo.16106593) provides two systematic refinements for minor arc analysis in the Hardy-Littlewood circle method: (1) A two-parameter Vaughan decomposition using mini-max balancing to redistribute logarithmic losses across Type I, II, and residual sums, and (2) A boundary smoothing technique employing linear tapering to improve Fourier decay from t⁻¹ to t⁻². These modules offer an analytic pathway to reduce the minor arc exponent from 12 to 11 while preserving the major arc infrastructure. The refinements require only reallocation of existing techniques and explicit recounting, without new deep analytic inputs or heavy computation.