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 (with the compression step covering the case b ≥ 3 analytically and b = 2 by direct case analysis), 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 (a self-loop step combined with a tree-shaped obstruction graph G*) is analytic and a single finite key lemma is verified by SAT with a DRAT-certifiable proof. Third, we prove that the b^k pattern breaks: R_5(3) > 243 = 3^5, with an explicit (SAT-discovered) 5-coloring witness, and incremental SAT solving confirms R_5(3) > 296. Fourth, we identify a structural mechanism (the Distance Pair Lemma) underlying the b^k pattern and propose a precise 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 cases b ∈ {2, 3} are thereby complete 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 verified boundary instance of this conjecture.