The O(N)-symmetric generalization of λ φ 4 theory is studied in the Gaussian-effective-potential (GEP) approach. It is shown that the GEP encompasses and transcends the leading-order 1/N expansion results. We find two distinct, nontrivial versions of the (3+1)-dimensional theory: (i) ‘‘precarious φ 4 theory,’’ with negative, infinitesimal λ B , which coincides with the 1/N result; and (ii) ‘‘autonomous φ 4 theory,’’ with positive, infinitesimal λ B and an infinite wave-function renormalization, which is inaccessible to perturbative or 1/N methods, and could well have eluded lattice-based approaches. ‘‘Autonomous φ 4 theory’’ can exhibit spontaneous symmetry breaking, and we speculate on its relevance for the Higgs mechanism.