We prove that there exist only finitely many commuting squares of finite dimensional *-algebras of fixed dimension, satisfying a "large second relative commutant" condition. We show this by studying the local minima of w dim(A ∩ wBw~*), where A, B are fixed subalgebras of some *-algebra C and w ∈ C is a unitary. When applied to lattices arising from subfactors satisfying a certain extremality-like condition, our result yields Ocneanu's finiteness theorem for the standard invariants of such finite depth subfactors.
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