Nielsen Ninomiya Fermion-doubling problem has known to be a challenge to construct a chiral fermion or chiral gauge theory on the lattice.
There is a proposed resolution to use so-called two mirrored worlds, say the original copy (called A) and a duplicated mirrored copy (called B); the two worlds A and B together are non-chiral. And adding some interactions to the mirrored copy(B) to cause the mass-gap to decouple the mirrored sector (B). Thus the left-over sector can be the desired chiral fermion / gauge theory.
This is called a mirrored fermion decoupling appraoch. One framework is this model. There are several results against the thinkings - mostly are based on perturbative arguments: 1. Absence of Chiral Fermions in the Eichten--Preskill Model-Maarten Golterman, Don Petcher, Elena Rivas; 2. Nondecoupling of Heavy Mirror-Fermion - Lee Lin'
Here is one for I found for non-perturbative arguments: 3. Decoupling a Fermion Whose Mass Comes from a Yukawa Coupling: Nonperturbative Considerations, T. Banks, A. Dabholkar But those are just arguments, not theorems or not any proofs. I read those cited papers beforehand. Some numerical attempts had fallen to see the decoupling: 4. On the decoupling of mirror fermions.
Some of the papers/arguments above are about SO(10) or SU(5) non-Abelian gauge theory.
Here instead let us take a simplest case: U(1) chiral symmetry and a 1+1D system.
$\star$ My inquiry is that: What are the SOLID objections on "there does not exist any lattice realization of 1+1D chiral fermion model with U(1) chiral symmetry, using mirrored fermion decoupling appraoch?"
The reader (you, and of course I) should know that: If we just focus on 1+1D, there is no spontaneous broken global symmetry due to Coleman-Mermin-Wagner theorem. Then one needs not to worry about spontaneous break down the chiral symmetry (such as U(1)).
ps. please feel free correct me if my understanind above has any ambiguity or error.