An Exact Weak-Saturation Family from a Two-Fan Construction

For every integer s >= 3, we determine wsat(2s+3,Ks,s+2) = 2s^2+s+1. The upper bound comes from an explicit complement with 4s+2 edges: two fans and four connecting edges admit a uniform chronological erasure certificate. The matching lower bound is Theorem 1.2 of Akhmejanova, Vorobyev and Zhukovskii. Relative to the cases stated in that preprint, the new numerical range is even s >= 10; the construction also recovers the known even cases s=4,6,8 and gives a uniform realization of the source's s=4 example. Every connector a_s b_r with 1 <= r < s works, while r=s fails initial freeness. Two broader block constructions yield isomorphism-disjoint classes of extremal complements, including an infinite four-fan family. An iterable conditional lift from (s,t) to (s,t+s) is proved, together with obstructions and a complete finite rejection of its rooted-prefix hypothesis inside the cut-vertex amalgamation class at (4,6). The lift yields no additional noncoprime exact values here. This is an unrefereed, AI-assisted research paper; priority and arbitrary noncoprime pairs are not claimed.