  There are two physically different interpretations of ``triviality'' in
$(\lambda\Phi^4)_4$ theories. The conventional description predicts a
second-order phase transition and that the Higgs mass $m_h$ must vanish in the
continuum limit if $v$, the physical v.e.v, is held fixed. An alternative
interpretation, based on the effective potential obtained in
``triviality-compatible'' approximations (in which the shifted `Higgs' field
$h(x)\equiv \Phi(x)-<\Phi>$ is governed by an effective quadratic Hamiltonian)
predicts a phase transition that is very weakly first-order and that $m_h$ and
$v$ are both finite, cutoff-independent quantities. To test these two
alternatives, we have numerically computed the effective potential on the
lattice. Three different methods were used to determine the critical bare mass
for the chosen bare coupling value. All give excellent agreement with the
literature value. Two different methods for obtaining the effective potential
were used, as a control on the results. Our lattice data are fitted very well
by the predictions of the unconventional picture, but poorly by the
conventional picture.
