Diagonal Ramsey numbers R(k,k) are bounded above by 4^k classically, with recent work lowering the exponential base. The question is how far a fixed Ramsey upper bound can be improved by iterated descent through blue neighborhoods while keeping a second vertex set fixed.

A retained-set descent applies density bounds inside a vertex set X while keeping a second set Y fixed, yielding a weighted inequality in |X|^w|Y| that determines when descent stops. The infimum diagonal exponent over all finite derivations is characterized as a greatest fixed point of a monotone operator on a complete lattice. Concave polygon lower bounds and finite upper proofs are certified via exact interval arithmetic. A Python certificate checker and a Lean formalization accompany the rational inputs in the repository.

A finite derivation gives R(k,k) ≤ 3.69507^k for all sufficiently large k. The infimum diagonal exponent lies in [1.305, 1.307], with a concave polygon proving the lower bound at every finite depth.