  The Gell-Mann-Levy (GML), Schwinger and Standard Models were previously shown
to lack a Brout-Englert-Higgs (BEH) fine-tuning problem due to quadratic
divergences, with finite Euclidean cut-off \Lambda, because of the symmetries
obeyed by all O(\Lambda^2) contributions. We extend those results to finite
contributions from certain M_{Heavy}^2>> m_{BEH}^2 particles in SO(2) versions
of GML and Schwinger. We demonstrate explicit 1-loop physical naturalness for
two SO(2) singlet examples: a heavy real scalar S and a right-handed Type 1
see-saw Majorana neutrino. We prove that for low |q^2| the heavy degrees of
freedom contribute, at worst, marginal operators in spontaneously broken SO(2)
Schwinger.
  The key GML lesson from these examples is that the pseudo Nambu-Goldstone
boson (NGB) mass-squared must be properly renormalized. A true NGB value, m_3^2
= 0, is then protected by the Goldstone theorem. For the Schwinger model, two
crucial observations emerge: global Ward-Takahashi identities (WTI) force all
relevant operators into the pseudo-NGB mass-squared; and WTI enforce the
Goldstone theorem by forbidding all relevant operator contributions in the
spontaneously broken Goldstone mode, \pi_3 is a massless NGB there. Goldstone
mode, with weak-scale m_{BEH}^2 \& <H>^2, is not-fine-tuned even as a
low-energy effective theory with certain high-mass-scale extensions. Its
"Goldstone Exceptional Naturalness (GEN)," where all relevant operators vanish,
a powerful suppression of fine-tuning, is simply another (albeit un-familiar)
consequence of WTI, spontaneous symmetry breaking and the Goldstone theorem.
  If GEN can somehow be extended to the Standard Model (SM), there should be no
expectation that LHC will discover any Beyond the SM physics unrelated to
neutrino mixing, i.e. the only known experimentally necessary modification of
the Standard Model plus General Relativity paradigm.
