  Standard lore uses local anomalies to check the kinematic consistency of
gauge theories coupled to chiral fermions, e.g. Standard Models (SM). Based on
a systematic cobordism classification, we examine constraints from invertible
quantum anomalies (including all perturbative local and nonperturbative global
anomalies) for gauge theories. We also clarify the different uses of these
anomalies: including (1) anomaly cancellations of dynamical gauge fields, (2)
't Hooft anomaly matching conditions of background fields of global symmetries,
and others. We apply several 4d $\mathbb{Z}_{n}$ anomaly constraints of
$n=16,4,2$ classes, beyond the familiar Feynman-graph perturbative $\mathbb{Z}$
class local anomalies. As an application, for
(SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$ SM (with $q=1,2,3,6$) and SU(5)
Grand Unification with 15n chiral Weyl fermions and with a discrete baryon
minus lepton number $X=5({\bf B}- {\bf L})-4Y$ preserved, we discover a new
hidden gapped sector previously unknown to the SM and Georgi-Glashow model. The
gapped sector at low energy contains either (1) 4d non-invertible topological
quantum field theory (TQFT, above the energy gap with heavy fractionalized
anyon excitations from 1d particle worldline and 2d string worldsheet,
inaccessible directly from Dirac or Majorana mass gap of the 16th Weyl fermions
[i.e., right-handed neutrinos], but accessible via a topological quantum phase
transition), or (2) 5d invertible TQFT in extra dimensions. Above a higher
energy scale, the discrete $X$ becomes dynamically gauged, the entangled
Universe in 4d and 5d is mediated by Topological Force. Our model potentially
resolves puzzles, surmounting sterile neutrinos and dark matter, in fundamental
physics.
