The Perfect-Square Law for Distinct-Variable 2-Color Rado Numbers of x + by = bz

We prove that R (x + by = bz, 2; distinct) = b² for all integers b >= 4, where R is the 2-color Rado number with distinct-variable convention. The lower bound follows from the canonical 'multiples of b red, non-multiples blue' coloring. The upper bound combines a Structure Lemma (in any 2-coloring of 1,. . . , b²-1, all multiples of b in this range share a single color) with a Two-Triple Blocking argument. The Structure Lemma is proved by an Alternating Cascade for generic (b, k) pairs together with explicit case analyses for the three exceptional pairs (b, k) in (4, 1), (4, 2), (5, 1). The proof is fully analytical with no SAT-only steps; SAT verification for b = 4. . 50 is provided as a cross-check. This resolves Conjecture 2 from the companion Rado-families paper (Towell, 2026, Zenodo DOI 10. 5281/zenodo. 19212447). v2 (April 27, 2026): Revised version following internal review.