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We show that the semidefinite programs involved in the computer proofs forKazhdan's property (T) satisfy strong duality and that the dual programs have ageometric interpretation in terms of harmonic cocycles. By dualizing geometricarguments about cocycles, we are able to simplify the property (T) SDP in thecase where it carries a symmetry by finite-order inner automorphisms. As anapplication, we simplify the SDP proof for $SL(n,\mathbb{Z})$ and we prove that$Aut(F_4)$ has property (T).

Computer proofs for Property (T), and SDP duality