The 4-color Rado number of x+y+c=z: general case

The claim is $R(c) = 40c+41$ for every $c \ge 2$, reduced to three finite facts: the base value $R(2) = 121$, and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every $D \ge 1$; the two unsatisfiability results carry DRAT proofs. This completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a scaling lemma; that lemma is now one of three legs, covering the branch where $d$ is divisible by 3. The supporting results are worth more than the headline for anyone deciding whether to believe it: the paper also shows every band relaxation is satisfiable, which is why previous attempts stalled, and that the affine method alone is exactly sharp and can never finish.

The claim is $R(c) = 40c+41$ for every $c \ge 2$, reduced to three finite facts: the base value $R(2) = 121$, and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every $D \ge 1$; the two unsatisfiability results carry DRAT proofs. This completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a scaling lemma; that lemma is now one of three legs, covering the branch where $d$ is divisible by 3. The supporting results are worth more than the headline for anyone deciding whether to believe it: the paper also shows every band relaxation is satisfiable, which is why previous attempts stalled, and that the affine method alone is exactly sharp and can never finish.