One-dimensional geometric locality for total fermionic cohomology

Establish the $QMA_1$-completeness of the explicit-list total fermionic cohomology problem on the full Fock space under one-dimensional geometric locality, extending the constant-mode-arity classification proved for monomials acting on at most 41 modes.

Background

The paper proves that deciding whether an explicitly listed charge-one fermionic differential has nonzero total cohomology on the full Fock space is $QMA_1$-complete when each monomial acts on at most 41 modes. The hardness construction begins with a chain-local source Hamiltonian, but the number-penalized lift used to convert specified-degree cohomology into total cohomology introduces all-to-all and star interactions.

The unresolved issue is whether the same complexity classification can be obtained while preserving one-dimensional geometric locality, rather than merely bounded mode arity. Such a result would strengthen the paper's full-Fock completeness theorem by imposing a spatial locality structure on the final fermionic differential.