Proof of the Triviality ofϕd4Field Theory and Some Mean-Field Features of Ising Models ford>4

It is rigorously proved that the continuum limits of Euclidean ${\varphi }_d^4$ lattice fields are free fields in $d>\mathrm{}4\mathrm{}$. An exact geometric characterization of criticality in Ising models is introduced, and used to prove other mean-field features for $d>\mathrm{}4\mathrm{}$ and hyperscaling in $d=2\mathrm{}$.