For integers b >= 2 and k >= 1, let Rₖ (b) denote the k-color Rado number for the equation x + by = bz, defined as the least positive integer n such that every k-coloring of 1, 2,. . . , n contains a monochromatic solution. Under the substitution d = z - y, this equation is equivalent to x = bd, placing it in the family studied by Chang, De Loera, and Wesley, who established Rₖ (b) >= bᵏ via the b-adic valuation and computed Rₖ (b) = bᵏ for k = 2, and a hybrid analytic-SAT proof that R₃ (3) = 27. Second, we give a hybrid analytic-SAT proof that R₄ (3) = 81, in which the structural reduction is analytic and a single finite key lemma is verified by SAT; the same Distance Pair Lemma also independently verifies R₃ (b) = b³ for b in 4,. . . , 10 and R₄ (b) = b⁴ for b in 3, 4, 5. Third, we prove that the bᵏ pattern breaks: R₅ (3) > 296 > 243 = 3⁵, with an explicit public 5-coloring witness. Fourth, we identify a structural mechanism underlying the bᵏ pattern and propose a threshold conjecture: Rₖ (b) = bᵏ if and only if k = 2 to a single first-breakdown bound R₂₁-₁ (b) > b^2b-1; the backward direction at b in 2, 3 is thereby established in the threshold regime, and b >= 4 reduces to one finite witness per b. The case b = 3, k = 4 is the boundary case k = 2 (b - 1) at b = 3, so R₄ (3) = 81 is the largest boundary instance verified here.
Li Alex (Fri,) studied this question.