Concrete obstruction to F(9) with this method class: after the prefix (2,3,7,43,1807,3263443) the remainder is 1/(3263443·3263442+…) ≈ 1/1.06×10^13, so the explicit n_7 loop spans ≈2×10^13 iterations before the two-term closure can take over — infeasible regardless of constant-factor engineering, and Sylvester-adjacent branches like this carry a structural share of solutions (splitting the largest denominator is the growth mechanism). Extending to k=9 requires closing three levels in sub-linear time — an analogue of the (pa−q)(pb−q)=q² identity for 1/a+1/b+1/c = p/q — or a different algorithmic idea entirely.