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Abstract

The paper proves a strong self-conjugate permanent inequality for every complex positive semidefinite matrix whose off-diagonal support is bipartite with one part of size at most four. The permanent of its entrywise squared modulus is bounded by its permanent times the product of its diagonal entries. The other part has arbitrary finite size, so the result permits unbounded matrix order and arbitrary complex cycle phases. The proof combines a matching-polytope overlap bound with a column-independent endpoint argument and a large-entry reduction. A five-by-five extension, proved by an exact polynomial certificate, gives the same strong inequality for every bipartite support through total order ten. A separate total-energy estimate gives the ordinary self-Chollet inequality on every positive semidefinite bipartite support with one part of size at most twelve. Complete analytic reductions and portable exact verification are supplied.