  Two global symmetries are holo-equivalent if their algebras of local
symmetric operators are isomorphic. Holo-equivalent classes of global
symmetries are classified by gappable-boundary topological orders (TO) in one
higher dimension (called symmetry TO), which leads to a
symmetry/topological-order (Symm/TO) correspondence. We establish that: (1) For
systems with a symmetry described by symmetry TO $M$, their gapped and gapless
states are classified by condensable algebras $A$, formed by elementary
excitations in $M$ with trivial self/mutual statistics. Such classified states
(called $A$-states) can describe symmetry breaking orders, symmetry protected
topological orders, symmetry enriched topological orders, gapless critical
points, etc., in a unified way. (2) The local low-energy properties of an
$A$-state can be calculated from its reduced symmetry TO $M_{/A}$, using
holographic modular bootstrap (holoMB) which takes $M_{/A}$ as an input. Here
$M_{/A}$ is obtained from $M$ by condensing excitations in $A$. Notably, an
$A$-state must be gapless if $M_{/A}$ is nontrivial. This provides a unified
understanding of the emergence and symmetry protection of gaplessness that
applies to symmetries that are anomalous, higher-form, and/or non-invertible.
(3) The relations between condensable algebras constrain the structure of the
global phase diagram. (4) 1+1D bosonic systems with $S_3$ symmetry have four
gapped phases with unbroken symmetries $S_3$, $\mathbb{Z}_3$, $\mathbb{Z}_2$,
and $\mathbb{Z}_1$. We find a duality between two transitions $S_3
\leftrightarrow \mathbb{Z}_1$ and $\mathbb{Z}_3 \leftrightarrow \mathbb{Z}_2$:
they are either both first order or both (stably) continuous, and in the latter
case, they are described by the same conformal field theory (CFT).
