The Deterministic Hare Core Is Nonempty for Nine-Seat Approval Elections

Abstract

Core stability gives approval-based committee elections a strong form of coalitional proportionality, but deterministic existence is not known for an arbitrary number of seats. We prove that every finite election with nine seats has a deterministic Hare-quota core committee. Starting from an eight-seat committee that is both a global Proportional Approval Voting (PAV) maximizer and core stable, we suppose that all one-candidate extensions are blocked and extract an inclusion-minimal closed response system. Singleton responses are eliminated by exact equality rigidity. For the remaining responses, rational Farkas certificates bound every nonpositive PAV add-marginal drift below by $-n/189$ and force every successor drift above $n/126$. The uniform response chain is irreducible, so stationarity makes these bounds incompatible. The computer-assisted component comprises an exact pointwise check over 7,356 voter types, 36 one-blocker cells, and a symmetry-complete family of 19 two-blocker motifs. Exact certificate archives and standard-library verifiers support these checks. The theorem settles the nine-seat case but does not decide deterministic core existence for arbitrary committee size.