We hypothesize a new and more complete set of anomalies of certain quantum
field theories (QFTs) and then give an eclectic proof. First, we propose a set
of 't Hooft anomalies of 2d $\mathbb{CP}^{\mathrm{N}-1}$-sigma models at
$\theta=\pi$, with N $=2,3,4$ and others, by enlisting all possible 3d
cobordism invariants and selecting the matched terms. Second, we propose a set
of 't Hooft higher anomalies of 4d time-reversal symmetric SU(N)-Yang-Mills
(YM) gauge theory at $\theta=\pi$, via 5d cobordism invariants (higher
symmetry-protected topological states) such that compactifying YM theory on a
2-torus matches the constrained 3d cobordism invariants from sigma models.
Based on algebraic/geometric topology, QFT analysis, manifold generator
dimensional reduction, condensed matter inputs and additional physics criteria,
we derive a correspondence between 5d and 3d new invariants, thus broadly prove
a more complete anomaly-matching between 4d YM and 2d
$\mathbb{CP}^{\mathrm{N}-1}$ models via a twisted 2-torus reduction, done by
taking the Poincar\'e dual of specific cohomology class with $\mathbb{Z}_2$
coefficients. We formulate a higher-symmetry analog of "Lieb-Schultz-Mattis
theorem" to constrain the low-energy dynamics.