Rado numbers of a(x-y)=bz, II: the golden-ratio classification, lower bounds for every number of colours, and a four-variable relative

Let 2 ≤ a < b be coprime and let R_k(a,b) be the least n such that every k-colouring of {1,...,n} contains a monochromatic solution of a(x−y)=bz. Up to renaming, these are all partition regular three-variable equations with non-trivial Rado numbers. This preprint continues "Rado numbers of a(x−y)=bz with a<b" (doi:10.5281/zenodo.23935587). (1) An exact classification: with s = √5(φa − b) one has D = a²+ab−b² = as − s²/5, and the classes of the recursive lower-bound colouring are exactly the windows 0<s<1, 1<s<2, 2<s<φ² and s>φ²; equivalently, the Galois conjugate of a+bφ is cut at τ, 2τ, 1−τ with τ = (5−√5)/10. (2) Explicit lower bounds for every coprime pair in the main region and every number of colours. (3) The one missing statement in the proof of the three-colour formula is certified with checked LRAT proofs for all 426 main-region pairs with b ≤ 50. (4) For the four-variable equation 2c(x−y)=b(z+w): R_2 = b(⌊b/c⌋+3)+1 (computer-assisted proof with checked DRAT proofs); for c = 1 and odd b, the same known bounds as x−y=bz for every number of colours, in particular R_3 = m³−m²−m−1 with m = b+2.