We investigate the Hamiltonian formulation of 1+1-dimensional
staggered fermions and reconstruct the vector and axial charge operators,
originally identified by Arkya Chatterjee \textit{et al.},
within the Wilson fermion formalism.
These operators commute with the Hamiltonian and reduce,
in the continuum limit, to the generators of the vector and axial
$\mathrm{U}(1)$ symmetries.
A notable feature of the axial charge operator is that it acts locally on operators
and possesses quantized eigenvalues in momentum space.
Its eigenstates can therefore be interpreted as fermion states
with well-defined integer chirality, analogous to those in the continuum theory.
This structure enables the formulation of a gauge theory
in which the axial $\mathrm{U}(1)_A$ symmetry is promoted to a gauge symmetry.
We construct a Hamiltonian in terms of the eigenstates of the axial charge operator,
thereby preserving exact axial symmetry on the lattice
while recovering vector symmetry in the continuum limit.
As applications, we study the implementation of the Symmetric Mass Generation (SMG) mechanism
in the 3-4-5-0 models.
Our framework admits symmetry-preserving interaction terms
with quantized chiral charges,
although further numerical investigation is required to confirm
the realization of the SMG mechanism in interacting systems.