The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7

Twenty-eight individual exact values, each proved by SAT certificate (coloring at n-1, UNSAT at n). The synthesis theorem covers every c >= 2 whose c+1 is divisible by 3, 4, 5, or 7 (~66% of integers). The prime-reduction corollary shows the full conjecture (R(c)=40c+41 for all c >= 2) is equivalent to checking primes p >= 89; all primes through 83 are settled. What stays open: the conjecture at c=88 (p=89) and every larger c whose c+1 has all prime factors >= 89. The scaling lemma's attribution is hedged relative to Malo 2000 (full text not accessed). No Lean formalization; the SAT certificates and dual-encoder architecture are the verification tier.

A ten-year-old conjecture with real standing - posed independently in a Math. Comp. paper and a Rutgers thesis, in the Schur/Rado tradition - advanced to two thirds of all cases with a clean reduction of the remainder to primes. Specialist territory, and the conjecture itself stays open, which caps it: level with the a >= 4 dihedral theorem (8), above the finite-cell bundles (5), below the Erdos entries at 10.