Distinct-Variable Rado Numbers for x+y=kz: Computation, Closed-Form Formula, and Complete Proof

We determine the 2-color Rado numbers for the equation x+y=kz with distinct variables. Using SAT solvers, we compute R(x+y=kz, 2; distinct) for k=1,...,500. The values are sporadic for k<=7, but for k>=8 they follow a simple parity-dependent formula: R = k(k+3)/2 for odd k, and R = (k^2+2k+2)/2 for even k. We prove this formula for all k>=8. For odd k, the lower bound comes from an explicit residue-class coloring; the upper bound uses a Double Blocking Lemma that combines pair-sum constraints with a Cauchy-Davenport pigeonhole argument. For even k, the lower bound comes from a threshold-based staircase coloring; the upper bound uses a Two-Triple Blocking Lemma in which two specific triples trap the final element. v8: corrected partition regularity claim, fixed unified formula, harmonized 500/200 values, added 6 citations.