Sharp Self-Chollet Inequalities in Order Five: Rank Two and Structured Rank Three

For a real correlation matrix C, the self-conjugate specializa- tion of Chollet’s conjecture asserts per(C ◦ C) ≤ per(C)2. We determine the sharp constant in order five under the rank-two constraint: per(C ◦ C) ≤ 137 1440 per(C)², C ∈ R^5×5, C ⪰ 0, diag C = 1, rank C ≤ 2. Equality holds exactly when the five Gram lines form a regular pentagon in RP¹, 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 then study the rank-three stratum. The matrix J2 ⊕ ( 1 −1/2 −1/2 −1/2 1 −1/2 −1/2 −1/2 1 ) has self-Chollet ratio 13/48. We prove that this value is sharp in two infinite structured rank-three families and that the displayed configura- tion is a strict local maximum in the full rank-three configuration space. We also prove Chollet’s inequality whenever |cij | ≤ 27/50 for i̸ = j, with the sharper estimate R(C) ≤ 621/800 at coherence 1/2. Finally, without a rank assumption, Chollet’s inequality is strict away from the identity whenever λmin(C) ≥ 3/4. The proposed global rank-three value 13/48 remains open. All finite algebraic certificates are supplied in exact arithmetic.