The Fermionic Cohomology Problem on the Full Fock Space Is $\mathrm{QMA}_1$-Complete

Abstract: 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 $\mathrm{QMA}_1$-hard and belongs to $\mathrm{QMA}$. 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 $\mathrm{QMA}_1$-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 $\mathrm{QMA}_1$-completeness for the specified-degree problem with $30$-mode terms whose hard instances admit a one-dimensional block-chain realization.