  We analyse the stability lower bounds on the Standard Model Higgs mass by
carefully controlling the scale independence of the effective potential. We
include resummed leading and next-to-leading-log corrections, and physical pole
masses for the Higgs boson, M_H, and the top-quark, M_t. Particular attention
is devoted to the cases where the scale of new physics \Lambda is within LHC
reach, i.e. \Lambda\leq 10 TeV, which have been the object of recent
controversial results. We clarify the origin of discrepancies and confirm our
earlier results within the error of our previous estimate. In particular for
\Lambda=1 TeV we find that
  M_H[GeV]>52+0.64(M_t[GeV]-175)-0.50\frac{\alpha_s(M_Z)-0.118}{0.006}.
  For fixed values of M_t and \alpha_s(M_Z), the error from higher effects, as
the lack of exact scale invariance of the effective potential and higher-order
radiative corrections, is conservatively estimated to be \simlt 5 GeV.
