The Burr-Erdős-Graham-Sós conjecture for the seven-cycle

For a graph H, let f(n,e,H) be the least number of colors in an edge-coloring of some n-vertex graph with at least e edges in which every copy of H is rainbow. Burr, Erdős, Graham, and Sós conjectured that f(n,lfloor n^2/4rfloor+1,C_{2k+1})=(1/8+o(1))n^2 for every fixed kge3, and Bucić, Chen, and Ma recently proved this for all kge4. We prove the remaining case k=3: \[ f\left(n,\left\lfloor n^2/4\right\rfloor+1,C_7\right) =\left(\frac18+o(1)\right)n^2. \] The lower bound rests on a weighted palette inequality, which we prove with an exact rational certificate on five sampled vertices. Its main ingredients are a fractional matching of compatible triangular edges and private resources attached to nontriangular edges. A stable form of the inequality, combined with regularity, triangle removal, and a direct argument for graphs close to bipartite, transfers the bound to arbitrary edge-colorings. We also describe a Lean 4 formalization of the conjecture for every fixed kge3, which combines the new seven-cycle proof with a formalization of the Bucić-Chen-Ma argument for kge4.