The Conversion Vertex: Lagrangian Structure, S-Matrix Derivation, and Element Production Thresholds from Γ

We derive the formal Lagrangian and S-matrix element for gravitational phase conversion (hν → mc²) within the Unified Light Framework. The conversion vertex is identified with the centrifugal zero—the specific radius in Kerr geometry where the photon’s angular momentum is fully absorbed by the gravitational field, tangential velocity vanishes, and the photon undergoes instantaneous radial conversion to matter. This condition uniquely determines the vertex structure, yielding a single-degree-of-freedom Lagrangian at the conversion point. From the vertex amplitudes, we formally derive the baryon branching ratio f_b = √(α_em/α_pc) = 0.162 (observed: 0.156, 3.5% agreement), previously motivated by analogy but here obtained from the S-matrix element directly. We then apply the vertex physics to stellar-scale conversion, deriving element-specific production thresholds from Γ = 0.53. The gravitational vertex enhancement factor (3+Γ²)/[2(1+Γ²)] = 1.281 reduces the required photon frequency for each element by 22% relative to free-space estimates, yielding minimum progenitor masses for element production that match supernova remnant observations: titanium requires ≥15 M☉ (Cassiopeia A progenitor: 15–25 M☉, titanium detected), while the Crab Nebula progenitor at 8–10 M☉ falls below the titanium threshold (titanium absent). The mass-to-light inversion process is derived as the time-reverse of the vertex, occurring when angular momentum is returned to matter by colliding frame-dragging fields during black hole mergers.