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.