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This work is dedicated to the memory of R. Stora. The spontaneously broken(SSB) $U(1)_Y$ Abelian Higgs model (AHM) (the gauge theory of a scalar $\phi\propto (H+i\pi)= {\tilde H}e^{i{\tilde \pi}/<H>}$ and a transverse vector A)has a massless pseudo-scalar $\pi$ in Lorenz gauge. Physical states have aconserved global current and Goldstone theorem (GT). $\tilde \pi$ becomes aNambu-Goldstone boson (NGB). Slavnov-Taylor identities keep on-shell T-matrixelements of physical states independent of anomaly-free gauge, and global,transformations, yielding towers of $\phi$-sector Ward-Takahashi Identities(WTI), and constraining external $\phi$ dynamics. Ultraviolet quadraticdivergences (UVQD) contribute only to $m_\pi^2$, forced by the GT to 0, so allUVQD vanish. Weak-scale renormalized gauge-independent Higgs pole-mass and VEVare therefore not fine-tuned. The NGB is "eaten" and decouples, hiding the$U(1)_Y$ WTI from observable particle physics. Ourregularization-scheme-independent results are unchanged by the addition ofcertain heavy fields as the extended WTI and GT cause all relevant operators tovanish. We prove 5 SSB decoupling theorems, illustrating them with twoexamples: a heavy $>>m_{Weak}$ $Z_2$-symmetric singlet real scalar field with 0VEV; and a heavy singlet right-handed type 1 see-saw Majorana neutrino.Including all loops we prove that certain heavy degrees of freedom decouplefrom the low-energy effective Lagrangian, contributing only irrelevantoperators after renormalization. The $\nu_R^{M}$ cannot completely decouple,but becomes invisible in practice. The NGB decouples, but our hidden SSB$U(1)_Y$ WTI, and a ${\tilde \pi}$ shift symmetry, protect the low-energy SSBAHM physics from loop contributions of heavy particles! Gauge-independentobservable weak-scale $m_{H;Pole}$ and $<H>$ are Goldstone ExceptionallyNatural, not fine-tuned.

Hidden $U(1)_Y$ Ward-Takahashi identities in the spontaneously  brokenAbelian Higgs model and the decoupling of certain heavy particles in  its simple extensions