On Rado Numbers for x + by = bz: The b^k Pattern and a Threshold Conjecture

For integers b >= 2 and k >= 1, let R_k(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 R_k(b) >= b^k via the b-adic valuation and computed R_k(b) = b^k for k <= 4 with various values of b. We contribute the following new results. First, we develop a Color Compression Lemma yielding a new self-contained analytic proof that R_2(b) = b^2 for all b >= 2, and a hybrid analytic-SAT proof that R_3(3) = 27. Second, we give a hybrid analytic-SAT proof that R_4(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 R_3(b) = b^3 for b in {4, ..., 10} and R_4(b) = b^4 for b in {3, 4, 5}. Third, we prove that the b^k pattern breaks: R_5(3) > 296 > 243 = 3^5, with an explicit public 5-coloring witness. Fourth, we identify a structural mechanism underlying the b^k pattern and propose a threshold conjecture: R_k(b) = b^k if and only if k <= 2(b - 1). Fifth, we prove a Lift Lemma and a Backward Tower Reduction that collapse the backward direction of the conjecture at any fixed b >= 2 to a single first-breakdown bound R_{2b-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 R_4(3) = 81 is the largest boundary instance verified here.