Sharp Self-Chollet Inequalities for Rank-Two Correlation Matrices of Order Five

For a real correlation matrix \(C\), the self-conjugate specialization of Chollet's conjecture asserts \(\operatorname{per}(C\circ C)\leq \operatorname{per}(C)^2\). We determine the sharp constant in order five under the rank-two constraint: \begin{array}{c} \operatorname{per}(C\circ C)\leq \dfrac{137}{1440}\,\operatorname{per}(C)^2,\\ C\in\mathbb{R}^{5\times5},\qquad C\succeq0,\qquad \operatorname{diag} C=\mathbf1,\qquad \operatorname{rank} C\leq2. \end{array} Equality holds exactly when the five Gram lines form a regular pentagon in \(\mathbb{RP}^1\), up to permutation and sign switching. The proof reduces the two permanents to elementary symmetric functions of five unimodular numbers, solves the remaining two-phase minimization, and ends with explicit sum-of-squares and positive-semidefinite Gram certificates. We also prove a uniform order-five result without a rank assumption: Chollet's inequality is strict away from the identity whenever \(\lambda_{\min}(C)\geq3/4\). All finite algebraic certificates are supplied in exact arithmetic.