Strong permanent inequalities for complex books and orthogonal three-center extensions

We prove that the permanent of the entrywise squared-modulus matrix is at most the permanent of a complex positive semidefinite correlation matrix whose support is contained in a book graph, that is, per(A ∘ Ā) ≤ per(A) for such A. This strong inequality is the self-conjugate case of a question of Bapat and Sunder; it fails for general positive semidefinite matrices by a counterexample of Drury, so structural hypotheses are essential. The number of leaves is arbitrary, and no rank or phase restriction is imposed. We determine all equality cases and obtain the corresponding inequality for one-point assemblies of book blocks. We also treat matrices with three centers and pairwise orthogonal leaf rows. Every feasible path on the centers satisfies the same strong inequality; a full center triangle is allowed under explicit row-energy and cyclic-product conditions. The proofs retain exact occupation contributions, use squared Gram norms to quantify slack, and reduce the two-center phase dependence to homogeneous quartic certificates. For three centers, a weighted deletion inequality and Schur-complement estimates control the phase terms without allocating the same slack more than once. The conclusions concern the stated structured classes and do not resolve the unrestricted Chollet conjecture. Verification programs are attached: Verification_Supplement.zip contains exact-arithmetic checkers for every displayed certificate and identity (Python standard library) and a second, independently written set of programs (exact and 50-digit permanents, symbolic re-derivation of the certificates, adversarial searches), with logs and a README.