The Fermionic Cohomology Problem on the Full Fock Space Is QMA₁-Complete

Fermionic cohomology characterizes the zero-energy states of the supersymmetric Hamiltonian associated with a fermionic differential. Previous work showed that the problem restricted to a particle-number sector specified with the input is QMA₁-hard and belongs to QMA. We study the global problem, in which no sector is specified and cohomology may occur anywhere in the full Fock space. This formulation directly matches the whole-space ground-state question: a specified-degree NO instance may still have zero-energy states in another sector, whereas the global NO promise excludes them across all sectors and their superpositions. We prove that this full-Fock problem is QMA₁-complete for differentials given as exact lists of local monomials, even when each monomial acts on at most 41 modes. As a companion result, we prove QMA₁-completeness for the specified-degree problem with 30-mode terms whose hard instances admit a one-dimensional block-chain realization.