Local Ocneanu rigidity for separable algebra objects
We establish local Ocneanu rigidity for separable algebra objects in Hom-finite monoidal categories C over an algebraically closed field k. A separability morphism contracts the first two Hochschild cohomology groups without requiring an abelian ambient category. Separable algebra structures and homomorphisms from separable sources have open algebraic-group orbits, and affine Bezout estimates give effective finiteness results. For Frobenius subalgebras, exchange relations replace the source and embedding data by a single self-dual idempotent. We prove that their separable inner-conjugacy classes are open in the exchange locus, yielding a bound of 2^(dim_k(End_C(X))) for any ambient Frobenius algebra X; connectedness gives the same bound on the actual number of subalgebras. As an application, we obtain an effective form of the Etingof-Walton finiteness theorem: every finite-dimensional semisimple Hopf algebra H over C has fewer than 2^(dim_C(H)) left coideal subalgebras. In the unitary setting, we bound E-compatible intermediates of C*-algebra and von Neumann algebra inclusions up to unitary conjugacy, under finite-index and finite-center hypotheses. For irreducible finite-index subfactors N < M, we improve the 9^[M:N] bound of Bakshi-Das-Liu-Ren to fewer than 2^[M:N] intermediate subfactors.