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Reasons from convexity class, KKT/complementarity, and LP/MIP relaxation gaps through interior-point and branch-and-cut (Gurobi, CPLEX, MOSEK, Ipopt) while treating loose big-M, IntegralityTol cheaters, IIS-hidden infeasibility, nonconvex KKT-as-global, and MIPGap-at-TimeLimit-as-optimal as first-class failure modes.