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Corollary 1: the sharp global rho-minus-xi maximum

Exact coefficient evaluation and uniqueness over all bivariate copulas.

theorem Papers.AnsariRockel2026XiRho.unitDiagonalBand_xi : unitDiagonalBand.chatterjeeXi = 3 / 10

Exact xi value of the source's unit-slope diagonal band.

theorem Papers.AnsariRockel2026XiRho.unitDiagonalBand_rho : unitDiagonalBand.spearmanRho = 7 / 10

Exact rho value, evaluated from the actual conditional distribution.

theorem Papers.AnsariRockel2026XiRho.rho_sub_xi_le (C : ProbabilityTheory.Copula 2) : C.spearmanRho - C.chatterjeeXi ≤ 2 / 5

Corollary 1: universal sharp bound, including singular copulas.

theorem Papers.AnsariRockel2026XiRho.rho_sub_xi_eq_iff (C : ProbabilityTheory.Copula 2) : C.spearmanRho - C.chatterjeeXi = 2 / 5 ↔ C = unitDiagonalBand

Corollary 1: equality identifies the unique actual maximizing copula.

theorem Papers.AnsariRockel2026XiRho.sharp_gap_attained : ∃ (D : ProbabilityTheory.Copula 2), D.chatterjeeXi = 3 / 10 ∧ D.spearmanRho = 7 / 10 ∧ (∀ (C : ProbabilityTheory.Copula 2), C.spearmanRho - C.chatterjeeXi ≤ D.spearmanRho - D.chatterjeeXi) ∧ ∀ (C : ProbabilityTheory.Copula 2), C.spearmanRho - C.chatterjeeXi = D.spearmanRho - D.chatterjeeXi ↔ C = D

The maximum is attained, with the source's exact coefficient pair.