Attained signed duality for fractional triangle packings

Every finite simple graph $G$, including an edgeless graph, admits a feasible fractional triangle packing $w$ and a feasible signed triangle dual $y$ such that

Here $m(G)$ is the number of unordered edges, the packing is nonnegative with total load at most one through each edge, and $D_G(y)$ is the total signed edge weight. Dual feasibility requires an upper bound of one on each edge weight and on each triangle's total edge weight; negative edge weights are allowed.

The equality is the attained primal–dual certificate for the exact fractional partition problem with edge and triangle pieces. It provides the graph-specific interface between finite LP duality and chordal signed-weight estimates. No chordality or minimum-order hypothesis is required.

Formalization Note The platform already contains the proved general theorem SmaleNinth.lp_strong_duality. This target concerns its finite graph-incidence specialization, including the conversion to the existing packing representation.