Marginal-Cap Lifting and Vanishing Connected Matchings in Graphs of Independence Number Two

This preprint studies connected matchings and Hadwiger numbers in finite graphs with independence number two.Building on OpenAI's September 2026 construction and accompanying Lean formalisation, we establish that for every integer q ≥ 1, every sufficiently large integer m is the order of a graph G with independence number α(G) = 2 and connected-matching number cm(G) < m/q. Consequently, the minimum connected-matching density tends to zero as m tends to infinity.The results imply counterexamples to the linear connected-matching conjectures of Füredi, Gyárfás and Simonyi and to the corresponding question attributed to Seymour. They also establish the asymptotic extremal Hadwiger numbers H₂(n) = n/3 + o(n), H₃(n) = n/4 + o(n), and H₄(n) = n/6 + o(n).The argument extends the marginal-cap quantification of the upstream supersaturation theorem, adapts its sampling parameters, and transfers the construction to all sufficiently large orders. The work relies on three imported upstream analytical theorems; it does not independently reprove them.The accompanying supplementary archive contains Lean 4 formalisation materials, verification records, and reproducibility evidence. The formal verification was performed in an author-run environment and has not been independently replayed.This is a research preprint and has not undergone journal peer review.