Abstract: The SSB AHM (the gauge theory of a scalar ϕ∝(H+iπ) and a transverse vector A) has a massless pseudo-scalar π in Lorenz gauge. Physical states have a conserved global current and Goldstone theorem (GT), but no global charge. π contains a Nambu-Goldstone boson (NGB). Slavnov-Taylor identities force on-shell T-matrix elements to be independent of anomaly-free gauge, and global, U(1) transformations, yielding towers of ϕ-sector Ward-Takahashi IDs (WTI), which severely constrain the dynamics, and the effective potential, of external ϕ. Ultraviolet quadratic divergences (UVQD) contribute only to the pseudo-NGB mass, forced by the GT to 0 in the SSB mode, so all UVQD vanish. Weak-scale renormalized gauge-independent Higgs pole-mass and VEV are therefore not technical-fine-tuned. The NGB is "eaten" and decouples as usual, hiding the U(1) WTI from observable particle physics. Our regularization-scheme-independent all-loop-orders results are unchanged by the addition of certain heavy matter fields, as the WTI and GT cause all relevant operators to vanish. We prove 5 new SSB decoupling theorems, illustrating them with two examples: a Heavy (>>Weak) Z2-symmetric singlet real scalar field with 0 VEV; and a heavy singlet right-handed type 1 see-saw Majorana neutrino. Including all loops we prove that these, and certain other heavy degrees of freedom, decouple from the low-energy effective Lagrangian, contributing only irrelevant operators. The NGB again decouples, but our hidden SSB U(1) WTI, and the resultant NGB shift symmetry, protect the low-energy SSB AHM physics from loop contributions of heavy particles! Gauge-independent observable weak-scale Higgs pole mass and VEV <H> are not technical-fine-tuned.