Triangle Restoration and Packing–Covering Stability in Tuza's Conjecture

For a finite simple graph G, let ν(G) be the maximum number of edge-disjoint triangles and τ (G) the minimum size of a triangle edge cover. We integrate triangle-restoration criteria and packing–covering stability results. Restoring one triangle to a remainder with tripartite full triangle hypergraph preserves Tuza’s inequality; consequently the inequality holds for odd-cycle transversal at most four and, by an adaptation of Liu–Zeng, for sufficiently large graphs with e(G) ≥ n2/4− cn for every fixed c < 5/2. For two restored triangles we give complete proofs for exact-equality and essential-edge tripartite remainders, flat remainders, three specified nonflat star cores, and exact two-apex bowties with shared apices or mixed triangles. We prove closed- boundary deletion and stability criteria and a compression preserving four exact optimization values. Selected deductions are conditionally checked in Lean with their external mathematical inputs explicit; the remaining results have audited ordinary proofs. Unrestricted two-triangle restoration, OCT-five, and the fixed-c < 3 density extension remain unproved.