  The weak-scale $U(1)_{Y}$ Abelian Higgs Model (AHM) is the
spontaneous-symmetry-breaking gauge theory of a complex scalar $\phi =
\frac{1}{\sqrt{2}}(H + i \pi)$ and a vector $A_{\mu}$. Global $U(1)_{Y}\otimes
BRST$ symmetry emerges: when it is realized that on-shell T-matrix elements
enjoy an extra $U(1)_{Y}$ global symmetry beyond the Lagragian's BRST symmetry.
The symmetries co-exist: $U(1)_{Y}$ generators $\delta_{U(1)_Y}$ commute with
BRST generators s and $[\delta_{U(1)_Y},s]{\cal L} = 0$. Two towers of Ward
Takahashi identities (WTI), which include all-loop-orders quantum corrections,
emerge: a tower of relations among off-shell 1-$\phi$-I (but
1-$A_{\mu}$-Reducible) Green's functions; another tower of Adler-zero WTI for
on-shell T-matrix elements. The T-matrix's LSS theorem forces tadpoles to
automatically vanish (equivalently $m^{2}_{\pi} = 0$) by symmetry alone. We
show that, when the full symmetries of Lorenz gauge AHM are enforced on the
scalar-sector effective potential, the vacuum state of the theory is
specified/decided by symmetry alone. We use recursive WTI relations among
Green's functions to include opeators of dimension $\geq 1$. We express the
fully renormalized scalar-sector effective potential in a form which shows
explicitly that, for small scalar field values, the gauge-independent vacuum
state of the theory $\langle H\rangle_{renormalized} = Z^{1/2}_{\phi}\langle
H\rangle_{bare}$ is determined by $U(1)_{Y}\otimes BRST$ symmetry alone,
without minimizing the effective potential.
